Research Papers

Post-Buckling Analysis of a Rod Confined in a Cylindrical Tube

[+] Author and Article Information
Jia-Peng Liu, Xiao-Yu Zhong

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Zai-Bin Cheng

Drilling & Production Research Institute,
CNOOC Research Institute Ltd.,
Beijing 100028, China

Xi-Qiao Feng

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: fengxq@tsinghua.edu.cn

Ge-Xue Ren

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: rengx@tsinghua.edu.cn

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 11, 2018; final manuscript received March 13, 2018; published online April 12, 2018. Assoc. Editor: Pedro Reis.

J. Appl. Mech 85(7), 071001 (Apr 12, 2018) (11 pages) Paper No: JAM-18-1088; doi: 10.1115/1.4039622 History: Received February 11, 2018; Revised March 13, 2018

Understanding the buckling and post-buckling behavior of rods confined in a finite space is of both scientific and engineering significance. Under uniaxial compression, an initially straight and slender rod confined in a tube may buckle into a sinusoidal shape and subsequently evolve into a complicated, three-dimensional (3D) helical shape. In this paper, we combine theoretical and numerical methods to investigate the post-buckling behavior of confined rods. Two theoretical models, which are based on the inextensible and extensible rod theories, respectively, are proposed to derive the analytical expressions for the axial compressive stiffness in the sinusoidal post-buckling stage. The former is concise in formulation and can be easily applied in engineering, while the latter works well in a broader scope of post-buckling analysis. Both methods can give a good approximation to the sinusoidal post-buckling stiffness and the former model is proved to be a zeroth-order approximation of the latter. The flexible multibody dynamics method based on the Timoshenko's geometrically exact beam theory is used to examine the accuracy of the two models. The methods presented in this work can be used in, for example, drilling engineering in oil and gas industries.

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Fig. 1

Schematic representation of (a) coiled tubing, (b) drilling, and (c) a typical load–displacement curve [18] and the corresponding morphological transition of the rod from the straight shape to the sinusoidal, and the helical buckling shape

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Fig. 2

The compressive load F versus axial displacement Δ relation assumed in different buckling models (adapted from Ref. [13])

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Fig. 3

A typical sinusoidal buckling morphology of a rod: (a) 3D view and (b) side view. For clear illustration, only the center-line of the rod is given in the 3D view, and its deflections are magnified in the figure, with the data aspect ratio x: y: z = 1: 1: 150.

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Fig. 4

(a) Displacement decomposition Δ = Δb + Δc and (b) the load–displacement curve

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Fig. 5

Energy analysis in Eqs. (12) and (13). (a) The work done by the axial load F is calculated by the area under the load–displacement curve, S1 + S2. (b) The elastic strain energy of compression, Uc = S3, and (c) the elastic strain energy of bending and the gravitational energy, Ub + Ug = S4.

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Fig. 6

Schematic of the numerical, flexible multibody dynamics model of a rod confined in a cylindrical tube

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Fig. 7

(a) Schematic of a rod constrained in a circular cylinder and (b) the numerical results for the reaction forces F1 and F2 with respect to the externally applied displacement

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Fig. 8

Comparison between the theoretical and numerical results: (a) 3D view (data aspect ratio x: y: z =1: 1: 1), (b) 3D view (data aspect ratio x: y: z = 1: 1: 150), (c) front view, (d) top view, (e) side view, and (f) angular displacement

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Fig. 9

(a) Variations in the different energy components in the rod, including compressive energy Uc, bending energy Ub, torsional energy Ut, shearing energy Us, gravitational energy Ug and (b) their percentages in the total potential energy U = Uc + Ub + Us + Ut + Ug

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Fig. 10

Distributions of (a) internal forces and (b) moments in the rod

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Fig. 11

Comparison between the numerical method and the two theoretical results: (a) load–displacement curve and (b) axial compressive stiffness

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Fig. 12

Comparison between the numerical method and the two theoretical results: (a) bending energy Ub, (b) compression energy Uc, (c) gravitational energy Ug, and (d) total potential energy U




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