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**A new mixed finite element method for the Timoshenko beam problem.**
*(English)*
Zbl 0779.73059

The Timoshenko beam problem, with shear force and rotation as primal variables and displacement as Lagrange multiplier, is examined under the general hypotheses of Brezzi’s theorem concerning existence and uniqueness in saddle point problems. A new finite element approximation is proposed and shown to be convergent for various combinations of interpolations for the three variables.

The paper is organized as follows. In section 2 the differential equations governing the problem are presented, a mixed variational formulation in terms of nondimensional variables is derived, and its analysis is shown. In section 3 a finite element approximation is described and its convergence for a rather general family of finite element interpolations is proved. Numerical results in section 4 on equal-order linear and quadratic elements (shear discontinuous on element interfaces) are obtained by the Galerkin and the perturbed Galerkin methods. We believe the analysis presented here may prove itself useful in the analysis of mixed variational formulations with more than two variables.

The paper is organized as follows. In section 2 the differential equations governing the problem are presented, a mixed variational formulation in terms of nondimensional variables is derived, and its analysis is shown. In section 3 a finite element approximation is described and its convergence for a rather general family of finite element interpolations is proved. Numerical results in section 4 on equal-order linear and quadratic elements (shear discontinuous on element interfaces) are obtained by the Galerkin and the perturbed Galerkin methods. We believe the analysis presented here may prove itself useful in the analysis of mixed variational formulations with more than two variables.

Reviewer: L.Prášek (Plzeň)

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

### Keywords:

interpolation procedure; Galerkin method; perturbed Galerkin method; shear force; rotation; Lagrange multiplier; Brezzi’s theorem; mixed variational formulation; convergence; equal-order linear and quadratic elements
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\textit{L. P. Franca} and \textit{A. F. D. Loula}, RAIRO, Modélisation Math. Anal. Numér. 25, No. 5, 561--578 (1991; Zbl 0779.73059)

### References:

[1] | A. F. D. LOULA, T. J. R. HUGHES, L. P. FRANCA and L MIRANDA, Mixed Petrov-Galerkin Method for the Timoshenko Beam, Comput. Methods Appl. Mech. Engrg., 63, 133-154, 1987. Zbl0607.73076 MR906535 · Zbl 0607.73076 · doi:10.1016/0045-7825(87)90168-X |

[2] | F. BREZZIOn the Existence, Uniqueness and Approximation of Saddle-point Problems Arising From Lagrange Multipliers, RAIRO Modél. Math. Anal. Numér., 8, R-2, 129-151, 1974. Zbl0338.90047 MR365287 · Zbl 0338.90047 |

[3] | D. N. ARNOLD, Discretization by Finite Elements of a Model Parameter Dependent Problem, Numer. Math., 37, 129-151, 1974. Zbl0446.73066 MR627113 · Zbl 0446.73066 · doi:10.1007/BF01400318 |

[4] | L. P. FRANCIA, New Mixed Finite Element Methods, PhD Dissertation, Stanford University, 1987. |

[5] | P. G. CIARLET, The Finite Element Method for Elliptc Problems, North-Holland Publishing Company, Amsterdam, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058 |

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