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Classical Nonlinear Beam Theories

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Geometric Continuum Mechanics and Induced Beam Theories

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 75))

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Abstract

Classical nonlinear beams from the point of view of an induced theory are continuous bodies with a constrained position field which are described by the motion of a centerline and the motion of plane rigid cross sections attached to every point at the centerline. This restricted kinematics allows to determine resultant forces at each cross section and to reduce the equations of motion of a three-dimensional continuous body to a partial differential equation with only one spatial variable. The present chapter is partly based on the publication of Eugster et al. [1]. First, in Sect. 5.1, the kinematical assumptions are stated. Subsequently, in Sect. 5.2, the virtual work contributions of the internal forces, the inertia forces and the external forces are reformulated by the application of the restricted kinematics to the virtual work of the continuous body. In Sects. 5.3–5.5 we present the generalized constitutive laws of the geometrically nonlinear and elastic theories of Timoshenko, Euler–Bernoulli and Kirchhoff in the form of a semi-induced beam theory. Lastly, Sect. 5.6 closes the chapter with a concise literature survey of numerical implementations of nonlinear classical beam theories.

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  1. Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany

    Simon R. Eugster

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Eugster, S.R. (2015). Classical Nonlinear Beam Theories. In: Geometric Continuum Mechanics and Induced Beam Theories. Lecture Notes in Applied and Computational Mechanics, vol 75. Springer, Cham. https://doi.org/10.1007/978-3-319-16495-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-16495-3_5

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