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1、A finite element model for independent wire rope core with double helical geometry subjected to axial loads CENGIZ ERDONMEZ1,and C ERDEM IMRAK21Istanbul T echnical University, Institute of Informatics, Computational Science and Engineering Program, 34469 Maslak, Istanbul, Turkey2 Istanbul T echnical

2、 University, Faculty of Mechanical Engineering, 34394 Gumussuyu, Istanbul, Turkeye-mail: cerdonmez; .trMS received 27 March 2009; revised 26 March 2010; accepted 1 June 2010Abstract. Due to the complex geometry of wires within a wire rope, it is difficultto model and analyse independent

3、wire rope core accurately (IWRC. In this paper,a more realistic three-dimensional modelling approach and finite element analysis ofwire ropes are explained. Single helical geometry is enough to model simple straight strand while IWRC has a more complex geometry by inclusion of double helical wiresin

4、 outer strands. Taking the advantage of the double helical wires, three-dimensional IWRCs modelling is applied for both right regular lay and lang lay IWRCs. Wire-bywire based results are gathered by using the proposed modelling and analysis methodunder various loading conditions. Illustrative examp

5、les are given for those show the accuracy and the robustness of the present FE analysis scheme with considering frictional properties and contact interactions between wires. FE analysis results are compared with the analytical and available test results and show reasonable agreement with a simpler a

6、nd more practical approach.Keywords. Wire strand; independent wire rope core; double helices; wire-by-wire analysis.1. IntroductionOn the design of wire ropes, a simple straight strand is used as a core strand and it is wound by outer strands to complete the whole geometry. Independent wire rope cor

7、e (IWRC is a special type of wire rope, which can be used in applications and becomes the core for some other types of ropes known as Seale IWRC or Warrington IWRC. Large tensile force strength of the wire ropes is very important in application areas wherein the small bending and torsional stiffness

8、.In general, two types of lay construction are used in wire rope strands: regular lay and lang lay. In regular lay wire rope constructions, lay directions of the wires within the outer strand are opposite in direction to the core strand. In lang lay wire rope constructions, on the other hand, the di

9、rections of the wires are the same as the direction of the core strand.Under axial loading conditions, regular lay wire ropes tend to tighten up while lang lay ropes have the tendency to unwind; for this reason lang lay ropes are used only whenrope ends are restrained from rotating. When the wire ro

10、pe is run under large lateral compressive loads and when additional axial loading capacity is required IWRCs are preferred (Velinsky 1989.Wire rope theory is based on a well-known classical treatise on elasticity by Love (1944 and general nonlinear equilibrium equations are derived and presented. It

11、 is found that the frictionless theory is widely used to solve equilibrium equations because of the nonlinear behaviour and the complex nature of the wire ropes.Although considerable research studies have been carried out about the analytical solutions of IWRCs by Costello & Sinha (1977, Costell

12、o (1990, Velinsky et al (1984, Velinsky (1985,Velinsky (1989, Jolicoeur & Cardou (1991, Elata et al (2004, Usabiaga & Pagalday (2008, a little work has been done using the double helical geometry and real modelling approaches via numerical analysis. All the above-mentioned analytical models

13、takes into account a simple straight strand as being the main component for modelling IWRC and more complex wire ropes by using IWRC.A frictionless theory is presented for the determination of the static response of a simple straight strand in Costello & Sinha (1977 which was the one of the firs

14、t analytical solutions using the theory of Love (1944. Costello (1990 has also presented the general behaviour of the wire ropes in different aspects such as static response of reduced rope rotation, simplified bending theory for wire rope.A wire rope with complex cross sections as a Seale IWRC is a

15、nalysed in a manner, which will predict the axial static response by Velinsky et al (1984. A general nonlinear theory to analyse complex wire rope is developed as an extension of the frictionless strand theory of Costello for a Seale IWRC by Velinsky (1985. Wire ropes with three types of cores: inde

16、pendent wire rope cores (IWRC, fibre-core (FC and wire strand core (WSC are investigated and Velinsky (1989 developed a design methodology for multi-lay wire strands.A number of analytical models of twisted wire ropes under axisymmetric loads are compared simultaneously with each other and with the

17、test results in Jolicoeur & Cardou (1991. Among them Phillips & Costellos (1985 model is remarkable which represents excellent correlation with the available experimental results in the literature. It is emphasized that the evaluation of the local effectssuch as interwire or interlayer press

18、ure and nonlinear behaviour should be undertaken while comparing to more advanced models.Complex structures such as IWRC are solved using rod theory and the nonlinear equilibrium equations of Love (1944 by taking the core strand as a straight wire and outer six strands as six single helical wires an

19、d assuming the whole system as a simple straight strand by using homogenization hypothesis.However,Elata et al (2004 proposed a new model simulating the mechanical response of a wire rope with an IWRC for open sieves that emulate a well-lubricated rope, and closed sieves that emulate infinite fricti

20、on between adjacent wires, which fully consider the double-helix configuration of individual wires within the IWRC. In the literature, the advantage of wire-by-wiremodelling approach is introduced on the basis of the general thin rod theory and compared with the fibre models in Usabiaga & Pagald

21、ay (2008. Most of the theoretical analyses over IWRCs are relied on some kind of homogenization hypothesis except thetheoretical studies in Elata et al (2004 and Usabiaga & Pagalday (2008. During the wire rope model subjected to axial loads,above-mentioned models are considered and a new finite

22、element model is proposed.Finite element approach has been embedded in wire rope analysis since 1999. An early approach was the termination of the end effects of a wire strand, which models 0.2% of a pitch length. It also showed the frictional effects of a simple straight strand by Jiang & Hensh

23、all (1999. Then a concise finite element model which took full advantage of the helical symmetry features of a simple straight strand is developed by Jiang et al (1999. Jiang et al (2000 has also extended their study to three-layered straight helical wire ropes under axial loads. Nawrocki & Labr

24、osse (2000 which takes into account every possible interwire motion have developed a finite element model of a simple straight strand based on a Cartesian isoparametric formulation.A three-dimensional finite element model of an optical ground wire is composed to predict the stress distribution in ea

25、ch component when the cable is subjected to a given elongation. Modelling considerations are discussed using ADINA software for one and two layers of metallic single helical wires wound around a core tube containing optical fibres. Results of stress analysis are presented in some detail for two exam

26、ples; six wires and ten wires wound around a central tube. Analyses with different cable lengths without friction for six-wire strand model are carried out for various pitches, and the one-pitch cable length was selected. For a length of one pitch, it has been shown by the authors that the stress st

27、ate is almost uniform in a cross-section at half pitch length of the cable. The problem with the shorter lengths was that they could not develop adequate contact between the wires and the tube. It has been presented by Fekr et al (1999 that the end effects are limited to few elements and end effects

28、 in helical wire are shown.The application of the homogenization method to practical situations appears to be limited, especially by means of a numerical approach. The numerical implementation of the homogenization method is performed using finite element analysis of the basic cell. Using the homoge

29、nization procedure principle, only one three-dimensional period of the cable (one pitch length has been modelled using Samcef FEM code by Cartraud &Messager (2006.The geometry of the strands are generated exactly by extruding circular surfaces along the centroidal helical curves of the wires as

30、in (Fekr et al 1999, geometrical approximation have a great influence on numerical results i.e., circular strand cross-sections instead of the real elliptical geometry (Cartraud & Messager 2006.The various applications of homogenization discussed in the literature have demonstrated its efficienc

31、y and usefulness for the overall modelling of beam-like structures exhibiting periodic geometrical or material heterogeneity. It is based on the asymptotic expansion method, and gives the first-order approximation of the three-dimensional heterogeneous solution. Starting from the homogenization theo

32、ry of periodic slender domains and taking benefit of the property of helical symmetry, the overall elastic behaviour is obtained from the solution of three-dimensional problems posed on a reduced basic cell that allows significant reduction of the size of the numerical model by Messager & Cartra

33、ud (2008.Validity domain of analytical models of steel wire strands is determined by Ghoreishiet al (2007. The elastic behaviour of a simple straight strand geometry having lay angles up to 35subjected to axial static load is analysed. Nine analytical model and FE model results compared by using gen

34、eral dimensionless stiffness coefficients. A complete 3D FE model is used to determine the validity domain of analytical models. The influence of contact conditions (wire/corecontact is examined for two limit cases: sliding without friction and merging. It has been shown that the overall static beha

35、viour is not sensitive to these modelling hypotheses and performed preliminary tests for models of lengths between two and ten pitches shows that end effects do not influence the overall axial response (Ghoreishi et al 2007.The behaviour of simple wire strand with multi-contact modes is analysed usi

36、ng finite element model by Jiang et al (2008. Radius of helical wires is such that the helical wires are in contact with both core and outer wires within a strand, which forms a statically indeterminate contact problem. Using the symmetric feature of the strand, a small basic sector of 1/6 of the co

37、mplete cross-section with a small sector length is used. Numerical results show that contacts can occur simultaneously at all possible contact points when the strand is subjected to extension with both ends restrained to rotation. The contacts with the conventional assumptions made in the analytical

38、 models of neglecting the local contact deformation. Loading and unloading curve is presented and it has been shown that for frictional analysis the loading and unloading curve indicates that a hysteretic phenomenon exists when the strand is subjected to cyclic loads. However,the influence on the gl

39、obal behaviour of the strand is shown to be very small by Jiang et al (2008.In previous studies, outer strands of IWRCs are modelled by means of some kind of homogenization hypothesis, as if, the wires were helically wrapped around the core. This phenomenonsuppresses the geometrical properties of do

40、uble helical trajectory of wires within an IWRC. In this paper, a new numerical method is used while geometry construction of double helical geometry of wires within an IWRC which is taken into account in the analytical models developed by Elata et al (2004 and Usabiaga & Pagalday (2008 for IWRC

41、. Construction issue of IWRC solid and meshed model, using definition of the double helical geometry, is presented first. Then the finite element model generation considering frictional effects and contacts between wires are explained over a more realistic 3-D solid numerical model. The benefit of t

42、his model is to obtain wire-by-wire numerical results of each wire within a simple straight wire strand and IWRC. Interactions between wires within a strand are established by definition of contacts between wires and both analytical and numerical models are compared. Analytical results of axial load

43、ing condition without rotation are investigated over an IWRC model.2. Modelling the independent wire rope coreDue to its complex geometry, it is not simple and straightforward to obtain wire-by-wire stressand strain analysis results rigorously considering theoretical models. Most of the finite eleme

44、nt based analyses carried on the modelling arc length and basic sector of a simple straight strand to see the mechanical behaviours. The present study considers a more realistic model of a wire strand and IWRC, instead of a model based on arc length or a basic sector of a simple straight strand. An

45、IWRC, consists of a simple straight strand as acore strand and six outer strands wrapped around it, is depicted using an accurate meshed form in figure 1.In this model, a straight wire strand composed by a straight core wire of radius R1 and six single helical wires of radius R2 around it as shown i

46、n figure 2(a. Helix angle 2 is determined by using the related equation of pitch length of p2 by solving the equation tan 2 = p2/2r2 using r2 = R1 + R2.To model the outer wires of an IWRC, outer wire geometry is treated with using special parametric equations. When the outer strand of the IWRC is co

47、nsidered, single helical wire is taken into account as the reference centerline for the outer double helical wires. to produce the double helical geometry of the IWRC it is necessary to formulate the centerline of the double helical geometry based on the single helical wire as a core. To do this cen

48、terline of a single helical wire is considered by using its parametric equation as,x s = r s cos(s ,y s = r s sin(s ,z s = r s tan(ss , (1Figure 1. (a A right lang lay (6 ×7 wire meshed IWRC. (b A real cross sectional view of section AA. where r s is the radius of the single helix, s is the sin

49、gle helix lay angle and s is the position of the wire within a strand. The outer double helical wires are wound around the given centerline of the single helical wire in Eqn. (1 by using the following parametric equations defined for double helical geometry,x d = x s(s + r d cos( d cos(s r d sin( d

50、sin(s sin(s , (2y d = y s(s + r d cos( d sin(s + r d sin( d cos(s sin(s , (3z d = z s r d sin( d cos(s , (4Figure 2. (a Simple straight strand. (b Wire lengths and helix angles of an IWRC.where d = ms + d0, d0 is the wire phase angle and r d is the distance along the double helix wire centerline and

51、 single helix strand centerline shown. The construction parameter m is a constant value that can be estimated by m = d/s . Parametric equations used to model an IWRC and solid model ling procedure is explained by Imrak & Erdönmez (2010. According to Eqns. (14 a right lang lay wire rope stru

52、cture can be constructed and to construct a left lang lay wire rope, it is enough to negate one of the coordinate values of x d , x d or z d given in Eq. (2(4.3. Numerical study of an IWRC under axial loadingOne of t he important issues while modelling a wire rope strand for numerical analysis is to

53、 select a correct wire strand or wire rope length, which is used to model and analyse. Jiang & Henshall (1999 showed that the region from 3 to 9 per cent of the pitch length is that in which both contact and sliding exist. In addition, the problem with the shorter lengths is mentioned by Fekr et

54、 al (1999 that they could not develop adequate contact between the wires and the end effects are limited to few elements. Due to these facts, wire lengths are selected within the range of effective length while the end effects are neglected during the numerical analysis.For the simple straight stran

55、d numerical model, strand length is selected enough to guarantee contacts between center and outer wires of the strand. Pitch length of the simple straight strand is defined as p = 115 mm. It is selected to be quarter of the pitch length of the simple straight strand as 28.75 mm in this study. Surfa

56、ce to surface contactinteractions between center and six outer single helical wires and between six helical wires are defined individually. During the numerical FEA, tangential and normal contact properties are defined. Contact property of tangential behaviour with penalty frictional formulation is

57、used with friction coefficient. The geometrical parameters of the simple straight strand are presented at the bottom of figure 2(b and it is selected to be used as the core strand for the IWRC in the following numerical application with different pitch length.An IWRC with18 mm length (6 ×7 is c

58、onsidered as the second numerical application in this study. Geometrical parameters are presented for the IWRC and presented in figure 2(b. Pitch lengths of the helical wires within the IWRC are selected as follows; inner helical wire p2 = 70 mm, outer center wire p2 = 193 mm, double helical wire p4

59、 = 70 mm.Materials elasticplastic properties used are listed as; You ngs modulus E = 188000 MPa, plastic modulus E p = 24600 MPa, yield stress R p0.2 = 1540 MPa, ultimate tensile stress R m = 1800 MPa, Poissons ratio v = 0.3 and friction coefficient = 0.115 (Jiang & Henshall 1999.Quadratic hexahedral brick elements with x, y and z displacements DOFs are preferred to analyse nonlinear effects of the complex geometry of wire strand and IWRC. The simple straight strand and IWRC are meshed using