On one-dimensional finite-strain beam theory: The plane problem

Reissner, E.

doi:10.1007/bf01602645

Cited by 572 publications

(392 citation statements)

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“…In that paper the author generalized to the three-dimensional dynamic case the rod models earlier proposed in [35,36]. In these works, the Euler-Bernoulli hypothesis, for which there can be no shears of the beam cross section with respect to the axis, is removed and it is assumed that the cross section remains planar during the deformation but not necessarily orthogonal to the rod axis (Timoshenko-Reissner model).…”

Section: Introductionmentioning

confidence: 99%

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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Highlights• Isogeometric collocation is extended to geometrically exact beams.• Consistent linearization of the strong form of the governing equations is derived.• Incremental rotations are parametrized through Eulerian rotation vectors.• Numerical tests show efficiency and high accuracy. AbstractWe extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.

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Section: Introductionmentioning

confidence: 99%

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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“…(6)- (8) and retaining terms of the same order, approximated equation are deduced. Finally, we stress that the advantage of having a hierarchy of approximated models relies on the possibility of choosing the more suitable set of equations for a given problem, at the less computational cost for the desired level of accuracy.…”

Section: Discussionmentioning

confidence: 99%

“…Thus, constitutive equations must be either introduced on an axiomatic way or experimentally established. For considerations on how to obtain constitutive equations, see [8].…”

Section: A Remark On the Moment-curvature Relationmentioning

confidence: 99%

The role of parameters of smallness in deduction of approximated theories for deterministic dynamics of beams

Babilio

Lenci

2016

MATEC Web Conf.

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“…This theory mainly targets the set of structures that are subjected to small strains (elastic response) but arbitrarily large deformations, such a scenario occurs in such essentially one-dimensional structures, implying that the cross-sectional dimension is much smaller than its length [2]. The work on the theory of geometrically exact beams was mainly contributed by Timoshenko, Iura and Atluri [3], Simo and Vu-Quoc [4], and Reissner [5][6][7], which are all based on the work of Euler beam theory, with more modern work by Kirchhoff, Love [8], and most importantly by Cosserat and Cosserat [9]. There is substantial literature on Cosserat theories including that by Rubin [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

A Generalized Approach for Reconstructing the Three-Dimensional Shape of Slender Structures Including the Effects of Curvature, Shear, Torsion, and Elongation

Chadha

Todd

2017

Journal of Applied Mechanics

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This paper extends the approach for determining the three-dimensional global displaced shape of slender structures from a limited set of scalar surface strain measurements. It is an exhaustive approach that captures the effect of curvature, shear, torsion, and elongation. The theory developed provides both a determination of the uniaxial strain (in a given direction) anywhere in the structure and the deformed shape, given a set of strain values. The approach utilizes Cosserat rod theory and exploits a localized linearization approach that helps to obtain a local basis function set for the displacement solution in the Cosserat frame. For the assumed deformed shape (both the midcurve and the crosssectional orientation), the uniaxial value of strain in any given direction is obtained analytically, and this strain model is the basis used to predict the shape via an approximate local linearized solution strategy. Error analysis due to noise in measured strain values and in uncertainty in the proximal boundary condition is performed showing uniform convergence with increased sensor count.

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On one-dimensional finite-strain beam theory: The plane problem

Cited by 572 publications

References 0 publications

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

The role of parameters of smallness in deduction of approximated theories for deterministic dynamics of beams

A Generalized Approach for Reconstructing the Three-Dimensional Shape of Slender Structures Including the Effects of Curvature, Shear, Torsion, and Elongation

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