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Research Papers

On Saint-Venant's Problem for Helicoidal Beams

[+] Author and Article Information
Shilei Han

Department of Mechanical Engineering,
University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai 200240, China
e-mail: shilei.han@outlook.com

Olivier A. Bauchau

Fellow ASME
Professor
Department of Aerospace Engineering,
University of Maryland,
College Park, MD 20742

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 11, 2015; final manuscript received October 28, 2015; published online December 4, 2015. Assoc. Editor: Daining Fang.

J. Appl. Mech 83(2), 021009 (Dec 04, 2015) (14 pages) Paper No: JAM-15-1498; doi: 10.1115/1.4031935 History: Received September 11, 2015; Revised October 28, 2015

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

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Figures

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Fig. 1

Configuration of a helicoidal beam

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Fig. 2

Semidiscretization of the beam

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Fig. 3

Incomplete ring subjected to shearing and torsion

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Fig. 4

Nondimensional shear stress σ¯13 along diameter PQ

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Fig. 5

Nondimensional circumferential shear stress σ¯1ϕ along diameter PQ

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Fig. 6

Cross section of initially curved or twisted beam

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Fig. 7

Compliance components S¯ij versus radius of curvature, case (a), S¯11 (○), S¯22 (□), S¯33 (⋄), S¯44 (▷), S¯55 (◁), S¯66 (△)

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Fig. 8

Compliance components versus radius of curvature, case (a), S¯16 (○), S¯34 (⋄)

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Fig. 9

Compliance components versus radius of curvature, case (b), S¯11 (○), S¯22 (□), S¯33 (⋄), S¯44(▷), S¯55 (◁), S¯66 (△)

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Fig. 10

Compliance components versus radius of curvature, case (b), S¯14 (○), S¯25 (□), S¯36 (⋄)

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Fig. 11

Helicoidal beam subjected to a tip force

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Fig. 12

Sectional rigid-body motion along axial coordinate: u1* (○), u2* (⋄), u3* (▷), ϕ1*(◁), ϕ2* (△), ϕ3* (▽)

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Fig. 13

Stress resultants along axial coordinate, F1*: (○), F2*: (⋄), F3*: (▷), M1*:(◁), M2*: (△), M3*: (▽)

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Fig. 14

Distribution of stress component σ11* along axial coordinate

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Fig. 15

Distribution of stress component σ12* along axial coordinate

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Fig. 16

Distribution of stress component σ13* along axial coordinate

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Fig. 17

Distribution of stress component σ22* along axial coordinate

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Fig. 18

Distribution of stress component σ33* along axial coordinate

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Fig. 19

Distribution of stress component σ23* along axial coordinate

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Fig. 20

Distribution of displacement component μ1 along axial coordinate

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Fig. 21

Distribution of displacement component μ2 along axial coordinate

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Fig. 22

Distribution of displacement component μ3 along axial coordinate

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Fig. 23

Distribution of stress component σ11* (Pa), present solution

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Fig. 24

Distribution of stress component σ11* (Pa), abaqus

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Fig. 25

Distribution of stress component σ22* (Pa), present solution

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Fig. 26

Distribution of stress component σ22* (Pa), abaqus

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