Convergence properties and derivative extraction of the superconvergent Timoshenko beam finite element

Mukherjee, Somenath; Reddy, J. N.; Krishnamoorthy, C. S.

doi:10.1016/s0045-7825(00)00280-2

Cited by 23 publications

(26 citation statements)

References 12 publications

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“…These equations may thus be expressed in terms of the known boundary values for θ and r leading to a finite element capable of providing the analytical solution. Specific to the interpolation obtained in this way is that it becomes dependent on the material and cross-sectional properties [2,3,5,6] as well as the loading functions μ(ξ ) and ν(ξ ) [8].…”

Section: Exact Representation Of the Solution In Terms Of Nodal Positmentioning

confidence: 99%

“…Such exact interpolation has been also widely reported [2][3][4][5][6] and often praised for automatically eliminating the notorious shear-locking anomaly [4,7]. It turns out that there exist a number of forms which this interpolation can take, from a highly coupled one in which the displacement and the rotation field both depend on the nodal displacements and rotations as well as the problem geometric, material and loading data [2,3,5,8] to a completely uncoupled interpolation of the displacement and the rotation fields.…”

Section: Introductionmentioning

confidence: 96%

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Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order

Jelenić

Papa

2009

Arch Appl Mech

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For arbitrary polynomial loading and a sufficient finite number of nodal points N , the solution for the 3D Timoshenko beam differential equations is polynomial and given as θ = N i=1 I i θ i for the rotation field and u = N +1 i=1 J i u i for the displacement field, where I i and J i are the Lagrangian polynomials of order N − 1 and N , respectively. It has been demonstrated in this work that the exact solution for the displacement field may be also written in a number of alternative ways involving contributions of the nodal rotations includingwhere R i are the beam nodal positions.

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Section: Exact Representation Of the Solution In Terms Of Nodal Positmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order

Jelenić

Papa

2009

Arch Appl Mech

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“…Therefore, the suggested FE can be used for both thick and thin beam. This FE was also proposed in [26][27][28][29]. Furthermore, Friedman and Kosmatka [30], Kosmatka [31] use this FE to solve dynamic and buckling problems, whereas in [32,33] thin-walled beam problems are studied.…”

mentioning

confidence: 99%

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral

Tullini

Tralli

2009

Comput Mech

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Making use of a mixed variational formulation including the Green function of the soil and assuming as independent fields both the structure displacements and the contact pressure, a finite element model is derived for the static analysis of a foundation beam resting on elastic half-plane. Timoshenko beam model is adopted to describe structural foundations with low slenderness and to impose displacement compatibility between beam and half-plane without requiring the continuity of the first order derivative of the surface displacements enforced by Euler-Bernoulli beam. Numerical results are obtained by using locking-free Hermite polynomials for the Timoshenko beam and constant reaction over the soil. Foundation beams loaded by many load configurations illustrate accuracy and convergence properties of the proposed formulation. Moreover, the different behaviour of the Euler-Bernoulli and Timoshenko beam models is thoroughly discussed. Rectangular pipe loaded by a force in the upper beam exemplifies the straightforward coupling of the foundation finite element with a structure described by usual finite elements

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“…These polynomials, belonging to the family of Hermitian shape functions, produce a superconvergent locking-free finite element [17,19,21]. In fact, when the beam slenderness goes to infinity, a proper parameter vanishes and the stiffness matrix of the element reduces to the classical Euler-Bernoulli stiffness matrix.…”

Section: 'Modified' Hermitian Shape Functionsmentioning

confidence: 99%

“…If inner nodes are condensed out, it is easy to demonstrate [21] that the finite element n. 2 (L32) is characterized by both the same bending stiffness matrix [15] (see Appendix B) and nodal load vector as elements adopting Hermitian polynomials. On the other hand, element n. 1 (L21) coincides, for the flexural part, with the 'consistent interpolation Timoshenko beam element (CIE)' introduced by Reddy [19].…”

Section: Lagrangian Elements No 1 and 2 (L21 And L32)mentioning

confidence: 99%

Locking‐free finite elements for shear deformable orthotropic thin‐walled beams

Minghini

Tullini

Laudiero

2007

Numerical Meth Engineering

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Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are\ud presented. The assumed kinematical model is based on Timoshenko-Reissner theory so as to take shear\ud strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms\ud arise between bending in the two principal planes and between bending and torsion. The adopted model\ud holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared\ud with the available numerical examples. In particular, some shape functions are obtained from ‘modified’\ud Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical\ud interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and\ud six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections\ud are performed to verify convergence rate and accuracy of the proposed formulations, especially in the\ud presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles\ud made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models\ud are compared with results given by plate-shell models

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Convergence properties and derivative extraction of the superconvergent Timoshenko beam finite element

Cited by 23 publications

References 12 publications

Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order

Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral

Locking‐free finite elements for shear deformable orthotropic thin‐walled beams

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