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Research Papers

Bending Instability of a General Cross Section Thin-Wall Tube for Minimal Radius of Curvature Passage

[+] Author and Article Information
Hadas Ziso

Robotics Laboratory,
Department of Mechanical Engineering,
Technion,
Haifa 32000, Israel
e-mail: hadasz@tx.technion.ac.il

Moshe Shoham

Fellow ASME
Tamara and Harry Handelsman Academic Chair
US National Academy of Engineering
Foreign Member, Robotics Laboratory Director,
Department of Mechanical Engineering,
Technion,
Haifa 32000, Israel
e-mail: shoham@technion.ac.il

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 28, 2014; final manuscript received August 6, 2014; accepted manuscript posted August 8, 2014; published online August 27, 2014. Assoc. Editor: George Kardomateas.

J. Appl. Mech 81(10), 101008 (Aug 27, 2014) (7 pages) Paper No: JAM-14-1232; doi: 10.1115/1.4028220 History: Received May 28, 2014; Revised August 06, 2014; Accepted August 08, 2014

This paper describes an analytical tool for the design of thin-wall tubes for passage through minimal radius of curvature trajectory. The design is based on a model of thin-wall tube buckling under pure bending. An extended analytical solution for general initial cross section is found based on Brazier method by energy theory of elastic stability. The model predicts the critical moment, curvature, flattening, and stress and allows choosing the most suitable cross section shape for a specific purpose. For example, tubes with ocular and rounded-ocular cross sections were found suitable for semiflexible applications such as endoscopy, where they elastically cross a sharp corner.

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Figures

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Fig. 1

(a) A common endoscope with a fixed, low curvature joint (arrow). (b) Our mechanism as a part of an endoscopic device with high curvature, “floating” joint.

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Fig. 2

The process of thin-wall tube buckling as a result of pure bending moment. The solid line represents the primary equilibrium path; the dashed line—secondary equilibrium path. Point 1: initial state with no external moment; point 2: beginning of ovalization as a result of pure moment; point 3: bifurcation point, in which the secondary equilibrium path will be followed if buckling as a result of compression occurs; point 4: ovalization critical point, where the tube becomes unstable and snaps to point 5 (dashed arrow).

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Fig. 6

Different initial cross section shapes (solid line) and the flattened shapes in the critical state (dashed line). (a) Oval shape, (b) elliptic shape, (c) ocular shape, and (d) rounded-ocular shape.

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Fig. 7

The moment as a function of curvature (normalized) for different initial cross section shapes solved in Extended Brazier method

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Fig. 5

The moment as a function of curvature (normalized) of an initially circular cross section shape solved in four different methods

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Fig. 4

The general cross section shape, represented by a(θ)

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Fig. 8

The moment as a function of curvature (normalized) for oval initial cross section shape solved in Extended Brazier method and numerically by Houliara

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