Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

Santos, H.A.F.A.; Pimenta, Paulo de Mattos; Almeida, J. P. Moitinho de

doi:10.1016/j.ijnonlinmec.2010.06.003

Cited by 30 publications

(26 citation statements)

References 44 publications

(35 reference statements)

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“…Application of variational principle to governing equation [87,88] is the basis of semi analytical method. This method expresses governing equation in variational form and solution of such formulation is obtained by minimization of residual error [89][90][91] encountered in the analysis of system response.…”

Section: Semi Analytical Methodmentioning

confidence: 99%

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A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

Ghuku

Saha

2017

IJET

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Abstract. The paper presents a review on large deflection behavior of curved beams, as manifested through the responses under static loading. The term large deflection behavior refers to the inherent nonlinearity present in the analysis of such beam system response. The analysis leads to the field of geometric nonlinearity, in which equation of equilibrium is generally written in deformed configuration. Hence, the domain of large deflection analysis treats beam of any initial configuration as curved beam. The term curved designates the geometry of center line of beam, distinguishing it from the usual straight or circular arc configuration. Different methods adopted by researchers, to analyze large deflection behavior of beam bending, have been taken into consideration. The methods have been categorized based on their application in various formats of problems. The nonlinear response of a beam under static loading is also a function of different parameters of the particular problem. These include boundary condition, loading pattern, initial geometry of the beam, etc. In addition, another class of nonlinearity is commonly encountered in structural analysis, which is associated with nonlinear stress-strain relations and known as material nonlinearity. However the present paper mainly focuses on geometric nonlinear analysis of beam, and analysis associated with nonlinear material behavior is covered briefly as it belongs to another class of study. Research works on bifurcation instability and vibration responses of curved beams under large deflection is also excluded from the scope of the present review paper.

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Section: Semi Analytical Methodmentioning

confidence: 99%

“…Three major categories of loading are concentrated, uniformly or non-uniformly distributed and combined distributed and concentrated loads. In addition, a beam can also be subjected to couple or moment [58,63,73,87,92,111].…”

Section: Loading Conditionmentioning

confidence: 99%

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

Ghuku

Saha

2017

IJET

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“…For details on the derivation of these equations and also the form of their corresponding operators, the reader is referred to [39,38,37]. It is only worth noting that d represents a generalized displacement vector, including both the displacement vector of a point lying in the centroidal axis of the beam, u, and also the rotation vector of the beam cross-section attached to that point, θ.…”

Section: The Geometrically Exact Finite Strain Beam Theory: Boundary-mentioning

confidence: 99%

“…Q and Γ represent the cross section rotation tensor and a transformation tensor, respectively, both defined as functions of the rotation vector θ. For details on their expressions the reader is referred to [39,38]. I denotes the standard second-order identity tensor.…”

Section: The Geometrically Exact Finite Strain Beam Theory: Boundary-mentioning

confidence: 99%

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Dual extremum principles for geometrically exact finite strain beams

Santos¹,

Almeida²

2011

International Journal of Non-Linear Mechanics

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Both necessary and sufficient conditions for the existence of two complementary-dual extremum principles for geometrically exact finite strain (one-dimensional) beam models are investigated by means of two different approaches. One is based on the results published by Gao and Strang, and the other relies on the approach proposed by Noble and Sewell. While the former is limited to beam models restricted to moderate large deformations, the latter is valid for arbitrarily large deformations (and strains). The numerical implementation of the complementary-dual extremum principles can lead to simple true global upper bounds of the error of the approximate solutions.

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A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

Eugster

Harsch

2023

GAMM-Mitteilungen

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The standard in rod finite element formulations is the Bubnov-Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov-Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

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Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

Cited by 30 publications

References 44 publications

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

Dual extremum principles for geometrically exact finite strain beams

A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

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