Research Papers

Asymptotic Approach to Oblique Cross-Sectional Analysis of Beams

[+] Author and Article Information
Anurag Rajagopal

e-mail: r_anurag87@gatech.edu

Dewey H. Hodges

e-mail: dhodges@gatech.edu
Daniel Guggenheim School
of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150

1Corresponding author.

Manuscript received June 24, 2013; final manuscript received September 4, 2013; accepted manuscript posted September 12, 2013; published online October 16, 2013. Assoc. Editor: Anthony Waas.

J. Appl. Mech 81(3), 031015 (Oct 16, 2013) (15 pages) Paper No: JAM-13-1257; doi: 10.1115/1.4025412 History: Received June 24, 2013; Revised September 04, 2013; Accepted September 12, 2013

Structural and aeroelastic analyses using beam theories by default choose a cross section that is perpendicular to the reference line. In several cases, such as swept wings with high AR, a beam theory that allows for the choice of a cross section that is oblique to the reference line may be more convenient. This work uses the variational asymptotic method (VAM) to develop such a beam theory. The problems addressed are the planar deformation of a strip and the full 3D deformation of a solid, prismatic, right-circular cylinder, both made of homogeneous, isotropic material. The motivation for choosing these problems is primarily the existence of 3D elasticity solutions, which comprise a complete validation set for all possible deformations and which are shown to be accurately captured by the current analysis. A secondary motivation was that the development and final results of the beam theory, i.e., the cross-sectional stiffness matrix and stress-strain-displacement recovery relations, are obtainable as closed-form analytical expressions. These results, coupled with the VAM-based beam analysis being devoid of ad hoc assumptions, culminate in what is expected to be of significance when formulating a general oblique cross-sectional analysis for beams with anisotropic material and initial curvature/twist, the detailed treatment of which will be alluded to in a later paper.

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Fig. 1

Schematic of swept blade or wing with normal and oblique cross sections

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Fig. 2

Stress recovery over an oblique plane using an orthogonal cross-sectional analysis

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Fig. 3

Reference frames used in the cross-sectional analysis

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Fig. 4

Schematic of the isotropic strip

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Fig. 5

In-plane shear stiffness of the strip versus obliqueness angle (in degrees) for ν = 0.3. Shear stiffnesses have been normalized by the corresponding elasticity values.

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Fig. 6

Various loading cases for the strip which possess elasticity solutions

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Fig. 7

Variation of cross-sectional stresses for flexure; b/l = 0.1 and Λ = 0 deg

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Fig. 8

Variation of cross-sectional stresses for flexure; b/l = 0.1 and Λ = 15 deg

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Fig. 9

Variation of cross-sectional stresses for flexure; b/l = 0.1 and Λ = 30 deg

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Fig. 10

Schematic prismatic, isotropic beam with a circular cross-section

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Fig. 11

Shear stiffnesses associated with SGT(B) versus obliqueness angle, Λ (deg) for ν = 0.3. All values are normalized by the corresponding elasticity solutions.

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Fig. 12

Loading cases for which the beam has elasticity solutions. Some of the figures have multiple loading cases depicted on them.

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Fig. 13

Variation of cross-sectional stresses (σ11, σ12, and σ13) for 2-flexure: VAM versus elasticity for r/l = 0.1 and Λ = 30 deg

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Fig. 14

Variation of cross-sectional stresses (σ22, σ23 and σ33) for 2-flexure: VAM versus elasticity for r/l = 0.1 and Λ = 30 deg




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