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Abstract This article presents a novel derivation for the governing equations of geometrically curved and twisted three-dimensional Timoshenko beams. The kinematic model of the beam was derived rigorously by adopting a parametric description of the axis of the beam, using the local Frenet–Serret reference system, and introducing the constraint of the beam cross ection planarity into the classical, first-order strain versus displacement relations for Cauchy’s continua. The resulting beam kinematic model includes a multiplicative term consisting of the inverse of the Jacobian of the beam axis curve. This term is not included in classical beam formulations available in the literature; its contribution vanishes exactly for straight beams and is negligible only for curved and twisted beams with slender geometry. Furthermore, to simplify the description of complex beam geometries, the governing equations were derived with reference to a generic position of the beam axis within the beam cross section. Finally, this study pursued the numerical implementation of the curved beam formulation within the conceptual framework of isogeometric analysis, which allows the exact description of the beam geometry. This avoids stress locking issues and the corresponding convergence problems encountered when classical straight beam finite elements are used to discretize the geometry of curved and twisted beams. Finally, this article presents the solution of several numerical examples to demonstrate the accuracy and effectiveness of the proposed theoretical formulation and numerical implementation.
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A viscoelastic Timoshenko beam: Model development, analysis, and investigation
Vibrations are ubiquitous in mechanical or biological systems, and they are ruinous in numerous circumstances. We develop a viscoelastic Timoshenko beam model, which naturally captures distinctive power-law responses arising from a broad distribution of time-scales presented in the complex internal structures of viscoelastic materials and so provides a very competitive description of the mechanical responses of viscoelastic beams, thick beams, and beams subject to high-frequency excitations. We, then, prove the well-posedness and regularity of the viscoelastic Timoshenko beam model. We finally investigate the performance of the model, in comparison with the widely used Euler–Bernoulli and Timoshenko beam models, which shows the utility of the new model.
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- Award ID(s):
- 2012291
- PAR ID:
- 10354323
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 63
- Issue:
- 6
- ISSN:
- 0022-2488
- Page Range / eLocation ID:
- 061509
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0091043
