Locking-free finite element method for a bending moment formulation of Timoshenko beams

Lepe, Felipe; Mora, David; Rodrı́guez, Rodolfo

doi:10.1016/j.camwa.2014.05.011

Cited by 16 publications

(17 citation statements)

References 21 publications

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“…Since all the results from [23] used for the theoretical analysis remain valid for t = 0, the same happens with the results of the present paper. In particular the O(h) estimates for the eigenspaces from Theorem 1 and the O(h 2 ) estimate for the eigenvalues from Theorem 2 hold true for the finite element approximation (41)-(42) of the Euler-Bernoulli beam vibration problem.…”

Section: The Euler-bernoulli Beamsupporting

confidence: 60%

“…holds true with a positive constant C independent of t and h (see [23,Lemma 3.2]). Consequently, the discrete mixed problem (26)- (27) has a unique solution and there holds…”

Section: Spectral Approximationmentioning

confidence: 93%

“…On the other hand, since problem (26)- (27) is just the finite element discretization of problem (13)- (14), using again the results from [23] (in particular, Theorem 3.3), we have that…”

Section: Remarkmentioning

confidence: 97%

“…We follow the approach proposed in [23] for the load problem. We introduce the bending moment together with the shear stress as new unknowns in the model (we note that the former usually represents a quantity of major interest in engineering applications), which together with the rotation and the transverse displacement lead us to a mixed variational formulation.…”

Section: Introductionmentioning

confidence: 99%

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Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

2015

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In this paper we analyze a low-order finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well as a double order of convergence for the vibration frequencies.These estimates are independent of the beam thickness, which leads to the conclusion that the method is locking free. For its implementation, displacements and rotations can be eliminated leading to a well posed generalized matrix eigenvalue problem for which the computer cost of its solution is similar to that of other classical formulations. We report numerical experiments which allow us to assess the performance of the method.

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Section: The Euler-bernoulli Beamsupporting

confidence: 60%

“…holds true with a positive constant C independent of t and h (see [23,Lemma 3.2]). Consequently, the discrete mixed problem (26)- (27) has a unique solution and there holds…”

Section: Spectral Approximationmentioning

confidence: 93%

“…On the other hand, since problem (26)- (27) is just the finite element discretization of problem (13)- (14), using again the results from [23] (in particular, Theorem 3.3), we have that…”

Section: Remarkmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

2015

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“…developed a plane curved Timoshenko beam element based on NURBS interpolation for both geometry and displacements. Based on mixed finite element formulation, Lepe et al [12] proposed a locking-free element for Timoshenko beams analysis. For analyzing such geomechanics problems as beam-type structures and deep pile foundations, Caillerie et al [13] proposed a Timoshenko straight beam element with internal degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

An efficient finite element formulation for bending, free vibration and stability analysis of Timoshenko beams

Karkon

2018

J Braz. Soc. Mech. Sci. Eng.

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In this paper, a new three-node element is proposed for analysis of beams with shear deformation effect. In each node of this element, there exist translation and rotation degrees of freedom. The element's formulation is based on the first-order shear deformation theory. For this aim, the displacement field of the element is approximated by a fifth-order polynomial. The shear strain is varied as a quadratic function within the element. It is worth noting that the quadratic function can be used for axial displacement field as well. By employing curvature and shear strain relations of Timoshenko beam theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing these shape functions, the stiffness matrix and the geometric stiffness matrix of the element are calculated. The mass matrices of the proposed element are derived from kinetic energy relation of the beam. Finally, several numerical tests are performed to assess the robustness of the developed element. The results of the numerical tests prove the absence of the shear locking and demonstrate high accuracy and efficiency of the proposed element for bending, free vibration and stability analysis of Timoshenko beams.

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A comprehensive mechanical response and dynamic stability analysis of elastically restrained bi-directional functionally graded porous microbeams in the thermal environment via mixed finite elements

Mollamahmutoglu

Mercan

Levent

2022

J Braz. Soc. Mech. Sci. Eng.

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Locking-free finite element method for a bending moment formulation of Timoshenko beams

Cited by 16 publications

References 21 publications

Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

An efficient finite element formulation for bending, free vibration and stability analysis of Timoshenko beams

A comprehensive mechanical response and dynamic stability analysis of elastically restrained bi-directional functionally graded porous microbeams in the thermal environment via mixed finite elements

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