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

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Abstract

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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References

  1. Amara, M., Capatina-Papaghiuc, D., Chatti, A.: New locking-free mixed method for the Reissner-Mindlin thin plate model. SIAM J. Numer. Anal. 40, 1561–1582 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N.: Discretization by finite elements of a model parameter dependent problem. Numer. Math. 37, 405–421 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 641–787. North-Holland, Amsterdam (1991)

  4. Behrens, E.M., Guzmán, J.: A new family of mixed methods for the Reissner-Mindlin plate model based on a system of first-order equations. J. Sci. Comput. 49, 137–166 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beirão da Veiga, L., Lovadina, C., Reali, A.: Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods. Comput. Methods Appl. Mech. Eng. 241–244, 38–51 (2012)

    Article  Google Scholar 

  6. Beirão da Veiga, L., Mora, D., Rodríguez, R.: Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner-Mindlin plate model. Numer. Methods PDE. 29, 40–63 (2013)

    Article  MATH  Google Scholar 

  7. Boffi, D., Brezzi, F., Gastaldi, L.: On the convergence of eigenvalues for mixed formulations. Ann. Sc. Normale Super. Pisa Cl. Sci. 25, 131–154 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Boffi, D., Brezzi, F., Gastaldi, L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69, 121–140 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bouclier, R., Elguedj, T., Combescure, A.: Locking free isogeometric formulations of curved thick beams. Comput. Methods Appl. Mech. Eng. 245–246, 144–162 (2012)

    Article  MathSciNet  Google Scholar 

  10. Calo, V.M., Collier, N.O., Niemi, A.H.: Analysis of the discontinuous Petrov-Galerkin method with optimal test functions for the Reissner-Mindlin plate bending model. Comput. Math. Appl. 66, 2570–2586 (2014)

    Article  MathSciNet  Google Scholar 

  11. Carstensen, C., Xie, X., Yu, G., Zhou, T.: A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng. 200, 1161–1175 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Celiker, F., Cockburn, B., Shi, K.: Hybridizable discontinuous Galerkin methods for Timoshenko beams. J. Sci. Comput. 44, 1–37 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. Part 1: the problem of convergence. RAIRO Anal. Numér. 12, 97–112 (1978)

    MathSciNet  MATH  Google Scholar 

  14. Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. Part 2: error estimates for the Galerkin method. RAIRO Anal. Numér. 12, 113–119 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Durán, R., Hernández, E., Liberman, E., Rodríguez, R., Solomin, J.: Error estimates for low-order isoparametric quadrilateral finite elements for plates. SIAM J. Numer. Anal. 41, 1751–1772 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Durán, R., Hervella-Nieto, L., Liberman, E., Rodríguez, R., Solomin, J.: Approximation of the vibration modes of a plate by Reissner-Mindlin equations. Math. Comput. 68, 1447–1463 (1999)

    Article  MATH  Google Scholar 

  17. Durán, R., Rodríguez, R., Sanhueza, F.: Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate. Math. Comput. 80, 1239–1264 (2011)

    Article  MATH  Google Scholar 

  18. Ferreira, A.J.M., Fasshauer, G.E.: Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method. Comput. Methods Appl. Mech. Eng. 196, 134–146 (2006)

    Article  MATH  Google Scholar 

  19. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  20. Hernández, H., Otárola, E.: A locking-free FEM in active vibration control of a Timoshenko beam. SIAM J. Numer. Anal. 47, 2432–2454 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hernández, H., Otárola, E., Rodríguez, R., Sanhueza, F.: Approximation of the vibration modes of Timoshenko curved rod of arbitrary geometry. IMA J. Numer. Anal. 29, 180–207 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  23. Lepe, F., Mora, D., Rodriguez, R.: Locking-free finite element method for a bending moment formulation of Timoshenko beams. Comput. Math. Appl. 68, 118–131 (2014)

    Article  MathSciNet  Google Scholar 

  24. Lovadina, C., Mora, D., Rodríguez, R.: Approximation of the buckling problem for Reissner-Mindlin plates. SIAM J. Numer. Anal. 48, 603–632 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lovadina, C., Mora, D., Rodríguez, R.: A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam. ESAIM Math. Model. Numer. Anal. 45, 603–626 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Niemi, A.H., Bramwell, J.A., Demkowicz, L.F.: Discontinuous Petrov-Galerkin method with optimal test functions for thin-body problems in solid mechanics. Comput. Methods Appl. Mech. Eng. 200, 1291–1300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Reddy, J.N.: An Introduction to the Finite Element Method. McGraw-Hill, New York (1993)

    Google Scholar 

  28. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)

    Article  Google Scholar 

  29. Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43, 125–131 (1922)

    Article  Google Scholar 

  30. Vadillo, G., Loya, J.A., Fernández-Sáez, J.: First order solutions for the buckling loads of weakened Timoshenko columns. Comput. Math. Appl. 64, 2395–2407 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

F. Lepe was supported by a CONICYT fellowship (Chile). D. Mora was partially supported by CONICYT-Chile through FONDECYT project 1140791, by DIUBB through Project 151408 GI/VC and by Anillo ANANUM, ACT1118, CONICYT (Chile). R. Rodríguez was partially supported by BASAL project CMM, Universidad de Chile, by Anillo ANANUM, ACT1118, CONICYT (Chile) and by Red Doctoral REDOC.CTA, MINEDUC Project UCO1202 at Universidad de Concepción (Chile).

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    F. Lepe & R. Rodríguez

  2. Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Casilla 160-C, Concepción, Chile

    F. Lepe, D. Mora & R. Rodríguez

  3. GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    D. Mora

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  1. F. Lepe
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Correspondence to D. Mora.

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Lepe, F., Mora, D. & Rodríguez, R. Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam. J Sci Comput 66, 825–848 (2016). https://doi.org/10.1007/s10915-015-0046-z

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  • DOI: https://doi.org/10.1007/s10915-015-0046-z

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