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IssuesArchive of Issues2011-6pp.888-897

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P.E. Tovstik and T.P. Tovstik, "One-Dimensional Models of a Beam Made of an Anisotropic Material with Oblique Anisotropy," Mech. Solids. 46 (6), 888-897 (2011)
Year 2011 Volume 46 Number 6 Pages 888-897
DOI 10.3103/S0025654411060082
Title One-Dimensional Models of a Beam Made of an Anisotropic Material with Oblique Anisotropy
Author(s) P.E. Tovstik (St. Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia, peter.tovstik@mail.ru)
T.P. Tovstik (St. Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia)
Abstract One-dimensional equations of the statics and free vibration of a strip beam made of an anisotropic material of general form (oblique anisotropy) are derived. It is shown that neither the Kirchhoff-Love hypotheses nor the classical Timoshenko-Reissner hypotheses lead to well-posed one-dimensional equations. A variant of the generalized Timoshenko-Reissner model is proposed that permits one to satisfy the boundary conditions on the beam surface exactly and leads to an asymptotically correct one-dimensional model. The solutions of the one- and two-dimensional equations of the statics and free vibration problems are compared and the dispersion equation is analyzed.
Keywords beams, oblique anisotropy, approximate models, vibration, dispersion equation
References
1.  L. H. Donnell, Beams, Plates, and Shells (McGraw-Hill, New York, 1976; Nauka, Moscow, 1982).
2.  L. A. Agalovyan, Asymptotic Theory of Anisotropic Plates and Shells (Nauka, Moscow, 1997) [in Russian].
3.  V. A. Rodionova, B. F. Titaev, and K. F. Chernykh, Applied Theory of Anisotropic Plates and Shells (Izd. SPbGU, St. Petersburg, 1996) [in Russian].
4.  P. E. Tovstik, "On the Asymptotic Nature of Approximate Models of Beams, Plates, and Shells," Vestnik Sankt-Peterburgskogo Univ. Ser. 1. Mat. Mekh. Astr., No. 3, 49-54 (2007) [Vestnik St. Petersburg Univ. Math. (Engl. Transl.) 40 (3), 188-192 (2007)]
5.  P. E. Tovstik and T. P. Tovstik, "On the 2D Models of Plates and Shells Including the Transversal Shear," ZAMM 87 (2), 160-171 (2007).
6.  P. E. Tovstik and T. P. Tovstik, "Two-Dimensional Models of Plates Made of an Anisotropic Material," in Proc. Seminar "Computer Methods in Continuum Mechanics, No. 3 (Izd. SPbGU, St. Petersburg, 2008), pp. 4-16 [in Russian].
7.  R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko, "To the Description of Multi-Layered Nanotubes in Models of Cylindrically Anisotropic Elasticity," Fizich. Mezomekh. 12 (5), 5-14 (2009) [Phys. Mesomech. (Engl. Transl.) 13 (1-2), 12-20 (2010)].
8.  P. E. Tovstik, "Models of Plates Made of an Anisotropic Material," Dokl. Ross. Akad. Nauk 425 (4), 487-491 (2009). [Dokl. Phys. (Engl. Transl.) 54 (4), 205-209 (2009)].
Received 04 March 2010
Link to Fulltext http://www.springerlink.com/content/9xu67576301r378g/
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