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Research Papers

Proper Inclusion of Interiorly Applied Loads With Beam Theory

[+] Author and Article Information
Jimmy C. Ho

Research Scientist
Science and Technology Corporation,
Ames Research Center,
Moffett Field, CA 95054
e-mail: jimmy.c.ho@us.army.mil

Wenbin Yu

Associate Professor
Mem. ASME
Department of Mechanical and
Aerospace Engineering,
Utah State University,
Logan, UT 84322
e-mail: wenbin@engineering.usu.edu

Dewey H. Hodges

Professor
Mem. ASME
Guggenheim School of
Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30338
e-mail: dhodges@gatech.edu

1Corresponding author.

Manuscript received June 6, 2011; final manuscript received May 8, 2012; accepted manuscript posted June 6, 2012; published online October 29, 2012. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 80(1), 011007 (Oct 29, 2012) (6 pages) Paper No: JAM-11-1176; doi: 10.1115/1.4006940 History: Received June 06, 2011; Revised May 08, 2012; Accepted June 06, 2012

An error is introduced by the conventional approach of applying beam theory in the presence of interiorly applied loads. This error arises from neglecting the influence of the precise distribution of surface tractions and body forces on the warping displacements. This paper intends to show that beam theory is capable of accounting for this influence on warping and accomplishes this by the variational asymptotic method. Correlations between elasticity solutions and beam solutions provide not only validations of beam solutions, but also illustrate the resulting errors from the conventional approach. Correlations are provided here for an isotropic parallelepiped undergoing pure extensional deformations and for an isotropic elliptic cylinder undergoing pure torsional deformations.

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References

Giavotto, V., Borri, M., Mantegazza, P., Ghiringhelli, G., Carmaschi, V., Maffioli, G. C., and Mussi, F., 1983, “Anisotropic Beam Theory and Applications,” Comput. Struct., 16(1-4), pp.403–413. [CrossRef]
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Yu, W., and Hodges, D. H., 2005, “Generalized Timoshenko Theory of the Variational Asymptotic Beam Sectional Analysis,” J. Am. Helicopter Soc., 50(1), pp.46–55. [CrossRef]
Vlasov, V. Z., 1961, Thin-Walled Elastic Beams, National Science Foundation and Department of Commerce, Washington, DC.
Hodges, D. H., 2003, “Contact Stress From Asymptotic Reissner-Mindlin Plate Theory,” AIAA Journal, 41(2), pp. 329–331 [CrossRef].
Yu, W., and Hodges, D. H., 2004, “Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams,” ASME J. Appl. Mech., 71(1), pp.15–23. [CrossRef]
Berdichevsky, V. L., 1979, “Variational-Asymptotic Method of Constructing a Theory of Shells,” Prikl. Mat. Mekh., 43(4), pp.664–687.
Hodges, D. H., 2006, Nonlinear Composite Beam Theory, Vol.213(Progress in Astronautics and Aeronautics), AIAA, Reston, VA.
Yu, W., Hodges, D. H., and Ho, J. C., “Variational Asymptotic Beam Sectional Analysis—An Updated Version,” Int. J. Eng. Sci.(in press). [CrossRef]
Love, A. E. H., 1934, A Treatise on the Mathematical Theory of Elasticity, 4th ed., Cambridge University Press, London, UK.

Figures

Grahic Jump Location
Fig. 1

Beam theory predictions of uB, normalized by elasticity solution, for varying ν

Grahic Jump Location
Fig. 2

Beam theory predictions of θ, normalized by elasticity solution, for b = 1.0 m and C1→0

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