NON-LINEAR ANALYSIS OF ALL-DIELECTRIC SELF-SUPPORTING LONG SPAN OPTICAL CABLES

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1 GIMC_2002 Third Joint Conference of Italian Group of Computational Mechanics and Ibero-Latin American Association of Computational Methods in Engineering NON-LINEAR ANALYSIS OF ALL-DIELECTRIC SELF-SUPPORTING LONG SPAN OPTICAL CABLES P.A.OLIVEIRA 1, M.B.HECKE 1, R.D.MACHADO 1, M.SCHNEIDER 2 1 Numerical Methods in Engineering Graduate Program, Federal Universit of Paraná, Curitiba, Brazil 2 Furukawa Industrial S.A., Curitiba, Brazil. ABSTRACT Nowadas, most of communication is strongl based on optical fiber nets. It is ver important to epand the optical nets, even in long distances. It has become common for utilities to locate optical fiber communication sstems on transmission line towers, as an alternative to conventional earthwire. There are several cable options that use eisting power lines. One of them, the All-Dielectric Self-Supporting (ADSS) cable, which is mounted below the phase conductors, is the preferable option due to his cost effective relation. The installed ADSS cable is independent of power lines and provides an attractive solution related to maintenance of both power lines and fiber optic cable. The ADSS cables are much easier to install and to repair on energized circuits. The cable design must ensure that there are no long-term problems with fibers or cable construction, which might otherwise lead to the collapse of fiber-optic communication sstem. Man alread-in-use power transmissions lines in Brazil have long span between towers. The mechanical behavior of this kind of cables is non-linear due to geometric and constitutive response and submitted to changes in temperature and dnamic loads. The objective of this work is to analze the static non-linear behavior of ADSS optical cables. Some cable elements were reviewed from the literature. A finite element code was developed and some non-linear cable elements were implemented. Those formulations are compared and their efficienc and accurac are analzed, even in their usual design, i.e., the linear, static and non-strained cable element. 1. INTRODUCTION The increasing demand for data transmission has recentl given a great impulse to the stud and manufacturing of optical cables. In man cases, the use of directl buried cables or cables installed in ducts is neither technicall nor economicall feasible. This holds particularl true in rural areas with low population densit or in regions with rock grounds or in mountains. In these cases, the use of aerial cables is indicated. It has become common for utilities to locate optical fiber communication sstems on their transmission line towers, as an alternative to conventional earthwire. 1

2 There are several cable options that use eisting power lines. Two of them are ver popular: the Optical Fiber Ground Wires (OPGW) and the All-Dielectric Self-Supporting (ADSS) cables [1]. The first one has two functions: the protect the line conductors against lightning and serve as telecommunication lines. In most constructions, the optical fibers are enclosed in the core of the cable and are designed to be stress free, or to resist onl low stresses under normal operation loads. The OPGW has been found as the best solution when installed as replacement to conventional earthwire on new transmission lines or when there is a need to change the old groundwire of an eisting line. The second, ADSS cable, which is mounted below the phase conductors (Figure 1b), is the preferable option due to its cost effective relation. The installed ADSS is independent from power lines and provides an attractive solution concerning maintenance of both power lines and fiber optic cable. The ADSS cables are much easier to install and to repair on energized circuits. The ADSS cables are made b several materials, like the eternal and internal polmeric tubes, the aramid fiber reinforcement (traction element), the FRP central element and the loose tube with optical fibers unit inside (Figure 2b [2]). Fig 1a - Transmission line sstem of 230kV with span of 300meters in São Paulo site, Brazil. Fig 1b - Positions of OPGW and ADSS cables The cable design must ensure that there are no long-term problems with fibers or cable construction, which might otherwise lead to the collapse of fiber-optic communication sstem. The determination of the specific location of an optical cable in a tower is determined b the electrical-field intensit around the phase conductor [3] and mechanical behavior of the cable. These structures are highl nonlinear in their geometric sense, especiall for long-span cables (Figure 2a). The stress-strain relationship is non-linear. Changes in air temperature must be considered in design. Wind and ice loads are also ver important to be considered [4]. Galloping of cables is an aerodnamic instabilit phenomenon. Galloping is tpicall associated with ice formation on a circular cable. The resulting cross-sectional shape can become a quasi-elliptical profile. Under high velocit wind, these profile causes forces promoting galloping that can be sustained for hours [5]. The self weight is a long time load and it ma produce creep deformations [3]. Modeling cables has attracted the attention of several researches. The first half of the centur has seen the development of analtical solutions for the fleible cable under distributed and concentrated loads. More recentl, with the advent of the digital computer, numerical solutions have been presented [6-10]. Although efforts in analtical treatment of cable problems still continue [10], a number of works have alread been devoted to cable finite element in static and dnamics analsis [11-17] and the stud of mechanics and geometr of suspension sstems [18]. 2

3 Fig 2a - ADSS cable installed in Cruz Machado site (power line up to 230kV and 1137 meters of span) Fig 2.b) Cross-section of ADSS cable ADSS optical cables began to be installed on overhead power lines in around These earl cables were mostl used on 110kV lines in Europe and have performed satisfactoril ever since. A few ears later, reports about man problems on ADSS cables began to be received and man researches have attracted the attention for this problem [1-5]. Various laborator eperiments and field trials were done to solve these problems. Mechanical behavior of optical cables has been studied b some researches in Brazil [19, 20] and currentl long span cables are being studied [21]. The objective of this work is to construct a computational code to analze the behavior of ADSS cable under all non-linearities mentioned before. The first step is to consider the static, elastic, geometric non-linear behavior of ADSS optical cables. This paper describes a review of some cable elements from the literature. A finite element code was developed and some non-linear cable elements were implemented. The are compared and the efficienc and accurac of these formulations will be analzed, even with the usual design, i.e., the linear, static and non-strained cable element. 2. PROBLEM AND CABLE CONFIGURATIONS Long span cables are structural elements that have such a span-diameter ratio that the ma be assumed to be perfectl fleible (the are devoid of fleural rigidit) [8]. These kind of structural elements use a mechanism of changes in geometr to resist ais-normal forces. This characteristic gives a strong geometric non-linearit. Consider a long span elastic cable like the one described before. The cable is assumed to be of uniform cross-section, with area A, and is made of an "equivalent" isotropic material with uniform densit (with same mechanical behavior as the ADSS cable). The stress-strain relation must be determined either eperimentall or calculated from finite element analsis that has a contribution of individual relevant cable component. It is assumed that the displacement is large, but the strain is small in the problem Catenar s element The cable ma be represented b a single two-node catenar element shown in figure 3, which eactl considers the curved geometr of the cable. It was proposed b Perot and Goulois [9] 3

4 and used b Karoumi [16] and the basic considerations are that the cable is perfectl fleible. Hooke's law is applicable to the cable material. Consider an elastic cable element, stretched in the vertical plane, with an unstressed length 0 L, modulus of elasticit E, cross section area A, and weight per unit length w, as shown in figure 3. The eact relations between the element projections and cable force components at the end of the element are [9] a 0 j + i L 1 T T T + ln i EA w T T = i j b = 2EAw j j 2 i 2 T ( T T ) + w i 1 (1) T where T i and T j are the cable tension at the two nodes of the element, and T i and T i, are the components T i wa i w bcoshϕ 0 L b = T = + where = 3 1 L ϕ 2 2ϕ 2 senhϕ a (2) The components of tangent stiffness matri would be evaluated from equilibrium relation of cable. j u 2, T j j j u 1,T node j i i u 2,T i i u 1,T node i w Figure 3 Catenar s two-node element Lagrangian finite elements Consider a bod (cable) that moves through space and occupies a region of Euclidean space that is called configuration. As the cable moves and deforms, it is important to correctl observe the different configuration that the cable assumes. For this, a Lagrangian or material formulation of the problem can be use. In the total Lagrangian formulation (TL), the initial configuration is used as the reference placement throughout. In the updated Lagrangian formulations (UL), the choice of reference configuration is changed. It is "updated" to become the configuration currentl occupied b the bod after each increment of motion. The cable ma be represented b a two-node finite element shown in figure 4, that was proposed b 4

5 Ozdemir [14] and presented b Bathe [23] either in total Lagrangian formulation (TL) or updated Lagrangian formulations (UL). t + t 1 t 1 t 2 Configuration at timet t + t 2 Configuration at time t + t Configuration at time 0. z Figure 4 - Elements TL and UL formulations 3. EXAMPLES 3.1. Eample 1 A cable hanging submitted to uniform load. A cable, presented b Judd and Wheen [13], is suspended between two points at same level, as shown in figure 5. Figure 5 Cable hanging submitted to uniform load. A cable has a cross-sectional area of m², and a self-weight of 5.28 kn/m. It is suspended between two points at same level of 488m apart. It has an elastic modulus E=20610³ MN/m². The cable will be considered in each of the conditions where it is initiall suspended with midspan sag of 6.1 m, 12.2m, 24.4m and 48.8 m, respectivel. An iterative Newton method has been used to obtain the span of the cable under selfweight. The cable will be submitted to a uniform load of kn/m. The figure 6 has shown the behavior of cable suspended under uniform load when a different sag has been adopted. 5

6 30000 Initial Sag = 48.8m Uniform Load [N/m] i Initial Sag = 24.4m Catenar Cable Element Total Lagrangian Formulation Updated Lagrangian Formulation ,0 1,0 2,0 3,0 Midspan Deflection [m] Initial Sag = 12.2m Initial Sag = 6.1m Figure 6 Behavior under uniform load of suspended cable Eample 2 A cable under its self weight subject to tensile force at both ends. A suspended cable under its self weight is subject to a tensile force at both ends along its chord, as shown in figure 7. This problem was previousl studied b Ali and Ghaffar[15] and Karoumi[16], using isoparametric cable elements with four nodes and catenar s element. The published results can now be compared to results obtained here with the two-node element using total Lagrangian (T.L) and updated Lagrangian (U.L.) formulations and also the twonode catenar s cable element [16]. A cable with an unstressed length L u =312.7 m, modulus of elasticit E = N/m 2, cross section area A = m 2, and weight per unit length w = N/m, is suspended between two points at the same level. To span the distance of m, using a cable with the above given properties, a horizontal force of T 0 = N was needed at both ends. T T ,8 m Figure 7 - Cable under its self weight subjected to tensile force at both ends. The horizontal force of T 0 gave a mid point cable sag of m and was adopted as the initial force when calculating the curves in Figure 7. Using the different sag and the longitudinal displacement along the cord of the cable were determined for different value of the tensile force (T 0 = N, 2T 0, 3T 0, 10T 0, 12.5T 0, 15T 0, 17.5T 0, 20T 0 ). Ten elements (T.L), 10 elements (U.L) and one catenar element have been used. The sag and the horizontal 6

7 displacement along the cord of the cable were determined for different values of tensile force T and the results are plotted in figure 8 and 9, respectivel. (Sag / Horizontal length) ,0 8,0 6,0 4,0 2,0 Catenar Cable Element Updated Lagrangian Formulation Total Lagrangian Formulation 0,0 0,0 2,5 5,0 7,5 10, 12, 15, 17, Tension / Initial Tension 20, 0 Figure 8 Response of cable under its self weight subject to tensile force at both ends. 2,0 (Displacement / Horizontal length) ,5 1,0 0,5 Updated Lagrangian Formulation Total Lagrangian Formulation Catenar Cable Element 0, Tension / Initial Tension Figure 9 Response of the cable due to horizontal displacement. The results obtained b the behavior of cable elements used in this work are compared to the results shown b H. M. Ali and A. M. Abdel-Ghaffar [15] and R. Karoumi [16]. The third result was determined b using an equivalent modulus E eq obtained from the use of practical 7

8 formulae often adopted when modeling cables in cable-staed bridges. The fourth and fifth results were determined b the use of a beam element. Figure 10 Comparing results of displacements/horizontal length tension/initial tension 4. DISCUSSIONS AND CONCLUSION Optical aerial cables have to be designed so that while operating, the fibers within the cable are kept free from microbending, macrobending and elongation which would increase attenuation and reduce lifetime of the fibers. Long span cables use a mechanism of changes in geometr to resist ais-normal forces. This characteristic gives a strong geometric nonlinearit. It s important to take into account this non-linearit to optimize the sag performance b avoiding damaged optical fibers and attenuation. B studing the effect of this nonlinearit, the cable presented in the first eample has been supended in tight, medium and large sag conditions. So, the ADSS cable must have large initial sag to present low etensibilit for load variation (figure 7). For small initial sag, the results obtained with Catenar s and Lagrangian elements are close. A good coherence is observed when comparing the curves in figure 9, for the used elements. In figure 10, a significant difference can be observed when comparing the one bar equivalent modulus curve (frequentl adopted in practical cases when modeling cables in cable-staed bridges) with the elements used. This difference is due to the fact that the equivalent modulus approach accounts for the sag effect but does not account for the stiffening effects due to large displacements. This leads to a softer cable model. For long span (~1000m) these differences increase. This stud has shown that the Lagrangian elements are accurate for non-linear analsis of long span cables. The eperimental check would be convenient. 8

9 ACKNOWLEDGMENT The support of Furukawa Industrial S.A., industr of optical cables, for the development of this research and the permission to publish this work are gratefull acknowledged. REFERENCES [1] Sharma, S.C.: Solutions for fibre-optic cables installed on overhead power transmission lines - A review, IETE Technical Review, Vol 11, pp (1994). [2] Simião, A.M., Medeiros Neto, J..A., Kunioshi, C., Furtado, J.M.I., Silvério, L.: All-dielectric, selfsupporting, tracking resistant cable for use on long span, high voltage overhead power lines, 48 th IWCS - International Wire and Cable Smposium, pp (1999). [3] Olsen, R.G.: An improved model for the electromagnetic compatibilit off all-dielectric self-supporting fiber-optic cable and high-voltage power lines, IEEE Transactions on Electromagnetic Compatibilit, Vol 41 pp (1999). [4] Cigada, A., Consonni, E., Falco, M., Marelli, P., Sutehall, R., Vanali, M.: All-dielectric, self-supporting, cables: Mechanical Features and Aeolian Vibration, 47 th IWCS - International Wire and Cable Smposium, pp (1998). [5] Zhang, Q., Popplewell, N., Shah, A.H.: Galloping of Bundle Conductor, Journal of Sound and Vibration, Vol 234 pp (2000). [6] O'Brien, W.T.: General solution of suspended cable sstem. Journal of Structural Division-ASCE, Vol 94, pp.1-26 (1967). [7] Irvine, M.: Statics of suspended cable, Journal of Engineering Mechanics Division - ASCE, Vol 101, pp (1975). [8] Irvine, M.: Cable Structures; Dover Publications, Inc., New York (1981). [9] Peirot, A. H., Goulois, A. M., Journal of Structural Division-ASCE, Vol 104, pp (1978). [10] Lu, L. Y., Chan, S.L., Lu, Z. H.: An analtical approach for non-linear response of elastic cable under comple loads, Structural Engineering and Mechanics, Vol 5, pp (1997). [11] Henghold, W. M., Russel, J. J., Equilibrium and natural frequencies of cable structure ( a non-linear finite element approach), Computers and Structures, Vol 6, pp (1975). [12] Maier, G., Contro, R., Energ approach to inelastic cable-structure analsis, Journal of Engineering Mechanics Division - ASCE, Vol 101, pp (1975). [13] Judd, B. J., Wheen, R.J., Nonlinear cable behavior, Journal of Structural Division-ASCE, Vol 104, pp (1978). [14] Ozdemir, H.: A finite element approach for cable problems, International Journal of Solids and Structures, Vol 15, pp (1979). [15] Ali, H. M., Abdel-Ghaffar, A. M.: Modeling the non-linear seismic behavior of cable-staed bridges with passive control bearings, Computers and Structures, Vol 54, pp (1995). [16] Karoumi, R.: Some modeling aspects in the nonlinear finite element analsis of cable supported bridges, Computers and Structures, Vol 71, pp (1999). [17] Gosling, P.D., Korban, E.A.: A bendable finite element for the analsis of fleible cable structures, Finite Elements in Analsis and Design, Vol 38 pp (2001). [18] Aufaure, M. A finite element of cable passing through a pulle, Computers and Structures, Vol 46, pp (1993). [19] Tormena, F. V.: Modelagem Computacional Bidimensional de Cabos Ópticos utilizando o Método dos Elementos Finitos, Master's Dissertation, Numerical Methods in Engineering Graduate Program, Federal Universit of Paraná, Brazil, p91 (1999). [20] Vasconcellos, C. A. M.: Modelagem Computacional Tridimensional da Estrutura de um Cabo de Fibras Ópticas Utilizando o Método dos Elementos Finitos, Master's Dissertation, Numerical Methods in Engineering Graduate Program, Federal Universit of Paraná, Brazil, p59 (1999). [21] Oliveira, P. A.: Análise estática não-linear de cabos suspensos utilizando o Método dos Elementos Finitos Master's Dissertation, Numerical Methods in Engineering Graduate Program, Federal Universit of Paraná, Brazil, p91 (2002). 9

10 [22] Hecke, M. B., Machado, R. D., Arndt, M., Oliveira, P.A. - Technical Report (2001). [23] Bathe, K.J.: Finite Element Procedures, Prentice-Hall (1986). 10