Research Papers

Finite Element Solutions to the Bending Stiffness of a Single-Layered Helically Wound Cable With Internal Friction

[+] Author and Article Information
Dansong Zhang

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: dzhang31@illinois.edu

Martin Ostoja-Starzewski

Department of Mechanical Science
and Engineering,
Institute for Condensed Matter Theory
and Beckman Institute,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: martinos@illinois.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 3, 2015; final manuscript received November 10, 2015; published online December 10, 2015. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 83(3), 031003 (Dec 10, 2015) (8 pages) Paper No: JAM-15-1469; doi: 10.1115/1.4032023 History: Received September 03, 2015; Revised November 10, 2015

This paper is focused on the bending stiffness of a cable consisting of a straight core wound around by a layer of helical wires, taking friction into account. Depending on the interaction between the cable components, the effective bending stiffness of the cable lies between an upper bound Bmax and a lower bound Bmin according to analytic models in literature. Two finite element models are created. The first aims to determine the maximum obtainable bending stiffness, whereby two contact types are tested: one bonding together all the touching surfaces and the other one only bonding together the wire–core contact surfaces. The numerical results show that Bmax is achieved for the first contact type, while neglecting the wire–wire contact lowers the bending stiffness due to the rotation of wire cross sections. In the second model, the wires are allowed to slip, while the cable is subjected to tension and bending. The effects of the tension level, the friction coefficient, and the contact types are investigated. The numerical results are able to capture the increase of bending stiffness with increasing tension and decreasing curvature, consistent with experimental observations and analytic models. The initial bending stiffness is sensitive to the imperfect contact between the components and is lower than Bmax. The final bending stiffness is higher than Bmin because of the contribution of friction and it increases with the friction coefficient.

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Fig. 1

Geometry of a 1 + 6 cable: (a) cross section and (b) lateral view. β is the lay angle. ϕ is the angle from x1-direction to the radial direction that passes through the center of the cross section of a wire.

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Fig. 2

Bending stiffness versus curvature for a 1 + 6 cable according to the theory in Ref. [21] at different tension levels and friction coefficients. The two horizontal dashed lines represent Bmax and Bmin, respectively.

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Fig. 3

Bending moment versus angle of rotation at the ends obtained by solving Eq. (10) subjected to Eq. (11)

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Fig. 4

The mesh of the cable with a lay angle of 17.03 deg used for the fully bonded model

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Fig. 5

The coarse mesh of the cable used for the stick-to-slip model

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Fig. 6

Contours of u3, with the directions of isolines indicated by arrows: (a) x2=0.00376m, in D1; (b) x2=0.00376m, in D2; (c) x2=−0.00376m, in D1; and (d) x2=−0.00376m, in D2

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Fig. 7

Finite element solutions corresponding to two different ways of applying friction (μ = 0.5). The coarse mesh is used. Contact between all surfaces is considered.

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Fig. 8

Finite element and theoretical solutions: (a) T varies, μ = 0.5 and all contact considered, (b) μ varies, T = 20 kN and all contact considered, and (c) two contact types, T = 20 kN and μ = 0.5

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Fig. 9

Contact pressure on a wire at the beginning of bending: (a) T = 20 kN, coarse mesh, (b) T = 40 kN, coarse mesh, and (c) T = 20 kN, fine mesh. Both wire–core contact and wire–wire contact are taken into account.




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