A mixed stress model for linear elastodynamics of arbitrarily curved beams

Cannarozzi, Mario; Molari, Luisa

doi:10.1002/nme.2161

Cited by 14 publications

(11 citation statements)

References 21 publications

(43 reference statements)

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“…The arbitrariness of δθ(s, t) and δr(s, t) directly yields the strong form (32). Inverting the last step from (49) to (50), by partial integration of the weighted strong form (32), this time only along the beam length l, yields the weak form of the balance equations (33). The terms occurring in this weak form for an unloaded beam can already be identified in the third line of (49).…”

Section: Interlude: Variational Problem Formulation Of Simo-reissner mentioning

confidence: 99%

“…In order to simplify notation required for subsequent derivations, the weak form G (see e.g. (33) or (81)) is split into the contributions G int of internal forces, G kin of kinetic forces and G ext of external forces:…”

Section: Temporal Discretization Of Primary Fieldsmentioning

confidence: 99%

“…allows to investigate the mechanical power balance. Inserting (151) into the discrete versions of the weak forms (33) and (81)…”

Section: Conservation Propertiesmentioning

confidence: 99%

See 2 more Smart Citations

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

Meier

Popp

Wall

2017

Arch Computat Methods Eng

145

183

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The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and Simo-Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a C 1 -continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large-deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes, preservation of objectivity and path-independence, consistent convergence orders, avoidance of locking effects as well as conservation of energy and momentum by the employed spatial discretization schemes, but also a range of practically relevant secondary aspects will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff-Love beam elements proposed in this work are the first ones of this type that fulfill all these essential requirements. On the contrary, Simo-Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff-Love formulations can provide considerable numerical advantages such as lower spatial discretization error level, improved performance of time integration schemes as well as linear and nonlinear solvers or smooth geometry representation as compared to shear-deformable Simo-Reissner formulations when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff-Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo-Reissner element formulations from the literature.

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Section: Interlude: Variational Problem Formulation Of Simo-reissner mentioning

confidence: 99%

Section: Temporal Discretization Of Primary Fieldsmentioning

confidence: 99%

See 1 more Smart Citation

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

Meier

Popp

Wall

2017

Arch Computat Methods Eng

145

183

View full text Add to dashboard Cite

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“…Based on the extended Hamilton principle, Yang et al [2] developed a high-order Lagrangian-type element suitable for variable curvature curved beams. Cannarozzi and Molari [16] proposed a stress-mixed element based on the improved Hellinger-Reissner function and proved that the element can be used to calculate arbitrary geometrical curved beam structures through calculation examples. Saffari et al [17] regard the curved beam structure as a combination of arch elements, each basic arch element introduces shear and axial deformation energy, and the elements are connected by a transformation matrix constructed by curvature.…”

Section: Introductionmentioning

confidence: 99%

A General Fourier Formulation for In-Plane and Out-of-Plane Vibration Analysis of Curved Beams

Nie

Zhu

et al. 2021

Shock and Vibration

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Based on the principle of energy variation, an improved Fourier series is introduced as an allowable displacement function. This paper constructs a calculation model that can study the in-plane and out-of-plane free and forced vibrations of curved beam structures under different boundary conditions. Firstly, based on the generalized shell theory, considering the shear and inertial effects of curved beam structures, as well as the coupling effects of displacement components, the kinetic energy and strain potential energy of the curved beam are obtained. Subsequently, an artificial spring system is introduced to satisfy the constraint condition of the displacement at the boundary of the curved beam, obtain its elastic potential energy, and add it to the system energy functional. Any concentrated mass point or concentrated external load can also be added to the energy function of the entire system with a corresponding energy term. In various situations including classical boundary conditions, the accuracy and efficiency of the method in this paper are proved by comparing with the calculation results of FEM. Besides, by accurately calculating the vibration characteristics of common engineering structures like slow curvature (whirl line), the wide application prospects of this method are shown.

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“…Besides the many proposals limited to linear elastic material assumptions and usually employed for large displacements and/or vibration analyses [19][20][21][22][23][24], the models in [25,26] are worth to be mentioned for nonlinear material modeling, yet belonging to the DB formulations. For twodimensional (2D) arches, Molari and Ubertini [27] have proposed a robust and efficient FB beam model, where the curved axis geometry is approximated with a cubic interpolation scheme.…”

Section: Introductionmentioning

confidence: 99%

A multiscale force-based curved beam element for masonry arches

Addessi

Sacco

2018

Computers & Structures

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This paper presents a Timoshenko beam finite element for nonlinear analysis of planar masonry arches. Considering small displacement and strain assumption, the element governing equations are defined according to a force-based formulation that adopts three different parametrizations of the axis planar curve, permitting the exact description of the element geometry for arbitrarily curved arches. Specific quadrature techniques are illustrated to perform numerical integration over the curved axis. A two-scale arch-to-beam homogenization procedure reproduces the nonlinear response of periodic masonry materials, where an equivalent Timoshenko straight beam describes the behavior of the reference Unit Cell made of a single linear elastic brick and a nonlinear mortar layer. Formation of the hinges characterizing the collapse mechanism of the arch is detected taking advantage of the quadrature rule along the axis and a fracture energy based regularization technique is employed to avoid damage localization.

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A mixed stress model for linear elastodynamics of arbitrarily curved beams

Cited by 14 publications

References 21 publications

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

A General Fourier Formulation for In-Plane and Out-of-Plane Vibration Analysis of Curved Beams

A multiscale force-based curved beam element for masonry arches

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