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On Approximate Solution of the Euler-Bernoulli Beam Equation via Galerkin Method

Year 2018, Volume: 11 Issue: 2, 341 - 346, 31.08.2018
https://doi.org/10.18185/erzifbed.396146

Abstract

In this paper, numerical
solution of a boundary value problem for a fourth-order ordinary differential
equation, known as the Euler-Bernoulli beam equation, is presented. The related
equation is used extensively in engineering areas such as huge buildings, long
bridges across big rivers, planes and cars. The approximate solution of the
problem considered is obtained by using the Galerkin method with basis
functions that satisfy the boundary conditions given. The accuracy of the
proposed method is given through two numerical examples with the help of
Maple®program.

References

  • Al-Omari, A., Schüttler, H.B., Arnold, J. Taha, T. 2013. Solving nonlinear systems of first order ordinary differential equations using a Galerkin finite element method. IEEE Access, 1, 408-417.
  • Biot, M.A. 1937. Bending of an infinite beam on an elastic foundation. Journal of Applied Mechanics, 2, 165-184.
  • Dubeau, F., Ouansafi, A., Sakat, A. 2003. Galerkin methods for nonlinear ordinary differential equation with impulses. Numerical Algorithms, 33, 215-225.
  • Gunakala, S.R., Comissiong, D.M.G., Jordan, K. Sankar, A. 2012. A finite element solution of the beam equation via Matlab. International Journal of Applied Science and Technology, 2(8), 80-88.
  • Han, S.M., Benaroya, H., Wei, T. 1999. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225 (5), 935-988.
  • Hunter, J.K., Narchtergaele, B. 2000. Applied Analysis, World Scientific, Singapore.
  • Kostadinova, S., Buralieva, J.V., Hadzi, K., Saneva, V. 2013. Wavelet-Galerkin solution of some ordinary differential equations, Proceedings of the XI International Conferences ETAI, Ohrid, Republic of Macedonia.
  • Ladyzhenskaya, O.A. 1985. Boundary Value Problems in Mathematical Physics, Springer, New York.
  • Lesnic, D. 2006. Determination of the flexural rigidity of a beam from limited boundary measurements. J. Appl. Math. And Computing, 20(1-2), 17-34.
  • Musa, A.E.S. 2017. Galerkin method for bending analysis of beams on non-homogeneous foundation. Journal of Applied Mathematics and Computational Mechanics, 16(3), 61-72.
  • Peradze, J. 2009. A numerical algorithm for a Kirchhoff-type nonlinear static beam. Journal of Applied Mathematics, Article ID 818269, 12 p.
  • Peradze, J. 2016. On the approximate solution of Kirchhoff type static beam equation. Transactions of A. Razmadze Mathematical Institute, 170, 266-271.
  • Smith, R.C., Bowers, K.L., Lund, J. 1992. A fully Sinc-Galerkin method for Euler-Bernoulli beam models. Numerical Methods for Partial Differential Equations, 8(2), 171-202.
  • Subaşı, M., Şener, S.Ş., Saraç, Y. 2011. A procedure for the Galerkin Method for a vibrating system. Computers and Mathematics with Applications, 61, 2854-2862.
  • Şener, S.Ş., Saraç, Y., Subaşı, M. 2013. Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions. Applied Mathematical Modelling, 37, 2623-2629.
  • Younesian, D., Saadatnia, Z., Askari, H. 2012. Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method. Journal of Theoretical and Applied Mechanics, 50 (2), 639-652.
  • Thankane, K.S., Stys, T. 2009. Finite difference method for beam equation with free ends using Mathematica. Southern Africa Journal of Pure and Applied Mathematics, 4, 61-78.

On Approximate Solution of the Euler-Bernoulli Beam Equation via Galerkin Method

Year 2018, Volume: 11 Issue: 2, 341 - 346, 31.08.2018
https://doi.org/10.18185/erzifbed.396146

Abstract

Bu çalışmada Euler-Bernoulli
kiriş denklemi olarak bilinen dördüncü mertebeden bir adi diferansiyel denklem
için sınır değer probleminin sayısal çözümü sunulmuştur. İlgili denklem büyük
binalar, büyük nehirler arasındaki uzun köprüler, uçaklar ve arabalar gibi
mühendislik alanlarında yaygın olarak kullanılmaktadır. Ele alınan problemin
yaklaşık çözümü, sınır koşullarını sağlayan temel fonksiyonlar ile Galerkin
metodu kullanılarak elde edilmektedir. Önerilen yöntemin doğruluğu Maple®
programı yardımıyla iki nümerik örnek üzerinden gösterilmektedir.

References

  • Al-Omari, A., Schüttler, H.B., Arnold, J. Taha, T. 2013. Solving nonlinear systems of first order ordinary differential equations using a Galerkin finite element method. IEEE Access, 1, 408-417.
  • Biot, M.A. 1937. Bending of an infinite beam on an elastic foundation. Journal of Applied Mechanics, 2, 165-184.
  • Dubeau, F., Ouansafi, A., Sakat, A. 2003. Galerkin methods for nonlinear ordinary differential equation with impulses. Numerical Algorithms, 33, 215-225.
  • Gunakala, S.R., Comissiong, D.M.G., Jordan, K. Sankar, A. 2012. A finite element solution of the beam equation via Matlab. International Journal of Applied Science and Technology, 2(8), 80-88.
  • Han, S.M., Benaroya, H., Wei, T. 1999. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225 (5), 935-988.
  • Hunter, J.K., Narchtergaele, B. 2000. Applied Analysis, World Scientific, Singapore.
  • Kostadinova, S., Buralieva, J.V., Hadzi, K., Saneva, V. 2013. Wavelet-Galerkin solution of some ordinary differential equations, Proceedings of the XI International Conferences ETAI, Ohrid, Republic of Macedonia.
  • Ladyzhenskaya, O.A. 1985. Boundary Value Problems in Mathematical Physics, Springer, New York.
  • Lesnic, D. 2006. Determination of the flexural rigidity of a beam from limited boundary measurements. J. Appl. Math. And Computing, 20(1-2), 17-34.
  • Musa, A.E.S. 2017. Galerkin method for bending analysis of beams on non-homogeneous foundation. Journal of Applied Mathematics and Computational Mechanics, 16(3), 61-72.
  • Peradze, J. 2009. A numerical algorithm for a Kirchhoff-type nonlinear static beam. Journal of Applied Mathematics, Article ID 818269, 12 p.
  • Peradze, J. 2016. On the approximate solution of Kirchhoff type static beam equation. Transactions of A. Razmadze Mathematical Institute, 170, 266-271.
  • Smith, R.C., Bowers, K.L., Lund, J. 1992. A fully Sinc-Galerkin method for Euler-Bernoulli beam models. Numerical Methods for Partial Differential Equations, 8(2), 171-202.
  • Subaşı, M., Şener, S.Ş., Saraç, Y. 2011. A procedure for the Galerkin Method for a vibrating system. Computers and Mathematics with Applications, 61, 2854-2862.
  • Şener, S.Ş., Saraç, Y., Subaşı, M. 2013. Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions. Applied Mathematical Modelling, 37, 2623-2629.
  • Younesian, D., Saadatnia, Z., Askari, H. 2012. Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method. Journal of Theoretical and Applied Mechanics, 50 (2), 639-652.
  • Thankane, K.S., Stys, T. 2009. Finite difference method for beam equation with free ends using Mathematica. Southern Africa Journal of Pure and Applied Mathematics, 4, 61-78.
There are 17 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Yeşim Saraç

Publication Date August 31, 2018
Published in Issue Year 2018 Volume: 11 Issue: 2

Cite

APA Saraç, Y. (2018). On Approximate Solution of the Euler-Bernoulli Beam Equation via Galerkin Method. Erzincan University Journal of Science and Technology, 11(2), 341-346. https://doi.org/10.18185/erzifbed.396146