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A Variational Formulation of Classical Nonlinear Beam Theories

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Developments and Novel Approaches in Nonlinear Solid Body Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 130))

Abstract

This article intends to present a concise theory of spatial nonlinear classical beams followed by a special treatment of the planar case. Hereby the considered classical beams are understood as generalized one-dimensional continua that model the mechanical behavior of three-dimensional beam-like objects. While a onedimensional continuum corresponds to a deformable curve in space, parametrized by a single material coordinate and time, a generalized continuum is augmented by further kinematical quantities depending on the very same parameters. We introduce the following three nonlinear spatial beams: The Timoshenko beam, the Euler–Bernoulli beam and the inextensible Euler–Bernoulli beam. In the spatial theory, the Euler–Bernoulli beam and its inextensible companion are presented as constrained theories. In the planar case, both constrained theories are additionally described using an alternative kinematics that intrinsically satisfies the defining constraints of these theories.

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Acknowledgements

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No. 405032572 as part of the priority program 2100 Soft Material Robotic Systems.

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Authors and Affiliations

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

    Simon R. Eugster & Jonas Harsch

Authors
  1. Simon R. Eugster
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  2. Jonas Harsch
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Correspondence to Simon R. Eugster .

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Editors and Affiliations

  1. Institute of Mechanics, Technische Universität Berlin, Berlin, Germany

    Bilen Emek Abali

  2. Department of Mechanical and Aerospace Engineering, University of Rome La Sapienza, LATINA, Latina, Italy

    Ivan Giorgio

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Eugster, S.R., Harsch, J. (2020). A Variational Formulation of Classical Nonlinear Beam Theories. In: Abali, B., Giorgio, I. (eds) Developments and Novel Approaches in Nonlinear Solid Body Mechanics. Advanced Structured Materials, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-030-50460-1_9

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