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Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method

Year 2016, Volume: 16 Issue: 3, 693 - 710, 31.12.2016

Abdurrahman Sahin

Abstract

The behaviors of structural systems are generally described with ordinary or partial differential equations. Finite Difference Method (FDM) mainly replaces the derivatives in the differential equations by finite difference approximations. It can be said that finite difference formulation offers a more direct approach to the numerical solution of partial differential equations. In this study, matrix approach is proposed for structural analysis with FDM. The system analysis procedure including stiffness matrix development, applying boundary and loading conditions on a structural element is proposed. The interacting points group is determined depending on the differential equations of the structural element and system rigidity matrix is generated by using this dynamic points group. The proposed algorithms are developed for Euler Bernoulli beams in this study because of its simplicity and may be enhanced for any other structural system in future studies by using same steps.

Keywords

Finite difference Method Matrix Methods Structural Analysis Euler Bernoulli Beams Matlab

References

  • Chapel, R.E. and Smith, H.W., 1968. Finite-difference solutions for plane stresses, The American Institute of Aeronautics and Astronautics (AIAA) Journal, 6(6), 1156–7.
  • Chawla, M.M. and Katti, C.P. 1982. Finite difference methods and their convergence for a class of singular two-point boundary value problems, Numerische Mathematik, 39, 341–350.
  • Cocchi, G.M., 2000. The finite difference method with arbitrary grids in the Elastic-static analysis of three-dimensional continua. Computers and Structures, 75, 187-208
  • Cocchi, G.M., Cappello, F., 1990. Convergence in elastic-static analysis of three-dimensional continua using the finite difference method with arbitrary grids. Computer and Structures, 36(3), 389-400.
  • D’Amico, B., Zhang , H., Kermani, A., 2016. A finite-difference formulation of elastic rod for the design of actively bent structures. Engineering Structures, 117, 518–527
  • Forsythe, G.E. and Wasow, W.R., 1960. Finite Difference Methods for Partial Differential Equations, John Wiley & Sons, Inc., New York-London. Jones, J., Wu, C., Oehlers, D. J., Whittaker, A. S., Sun, W., Marks, S., Coppol, R., 2009. Finite difference analysis of simply supported RC slabs for blast loadings. Engineering Structures, 31(12), 2825–2832.
  • Jovanovic, B.S. and Popovic, B.Z., 2001. Convergence of a finite difference scheme for the third boundary-value problem for an elliptic equation with variable coefficients. Computational Methods in Applied Mathematics. 1 (4), 356–366.
  • Jovanovic, B.S., Lemeshevsky, S.V., Matus, P.P., Vabishchevich, P.N., 2006. Stability of solutions of differential-operator and operator-difference equations in the sense of perturbations of operators. Computational Methods in Applied Mathematics, 6 (3), 269–290.
  • Jovanovic, B.S., Vulkov, L.G., 2001. On the convergence of finite difference schemes for the heat equation with concentrated capacity. Numerische Mathematik, 89 (4), 715–734.
  • Kalyani, V.K., Pallavika, Chakraborty, S.K., 2014. Finite-difference time-domain method for modelling of seismic wave propagation in viscoelastic media. Applied Mathematics and Computation, 237, 133–145
  • Liu, Y., Yin, C., 2014. 3D anisotropic modeling for airborne EM systems using finite-difference method. Journal of Applied Geophysics, 109, 186–194.
  • MATLAB (2009). The MathWorks, Natick, MA. Moreno-García, P., Lopes, H. , Araújo dos Santos, J.V., 2015. Application of higher order finite differences to damage localization in laminated composite plates, Composite Structures.
  • Strikwerda, J.C., 1990. Finite Difference Schemes and Partial Differential Equations. Chapman & Hall, New York.
  • Thomee, V., 1990. Finite difference methods for linear parabolic equations. P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, Vol. I. Finite Difference Methods (Part 1), North-Holland, Amsterdam, 5–196
  • . Zienkiewicz, O.C., 1971. The Finite Element Method in Engineering Science, London: McGraw-Hill.

Year 2016, Volume: 16 Issue: 3, 693 - 710, 31.12.2016

Abdurrahman Sahin

Abstract

References

  • Chapel, R.E. and Smith, H.W., 1968. Finite-difference solutions for plane stresses, The American Institute of Aeronautics and Astronautics (AIAA) Journal, 6(6), 1156–7.
  • Chawla, M.M. and Katti, C.P. 1982. Finite difference methods and their convergence for a class of singular two-point boundary value problems, Numerische Mathematik, 39, 341–350.
  • Cocchi, G.M., 2000. The finite difference method with arbitrary grids in the Elastic-static analysis of three-dimensional continua. Computers and Structures, 75, 187-208
  • Cocchi, G.M., Cappello, F., 1990. Convergence in elastic-static analysis of three-dimensional continua using the finite difference method with arbitrary grids. Computer and Structures, 36(3), 389-400.
  • D’Amico, B., Zhang , H., Kermani, A., 2016. A finite-difference formulation of elastic rod for the design of actively bent structures. Engineering Structures, 117, 518–527
  • Forsythe, G.E. and Wasow, W.R., 1960. Finite Difference Methods for Partial Differential Equations, John Wiley & Sons, Inc., New York-London. Jones, J., Wu, C., Oehlers, D. J., Whittaker, A. S., Sun, W., Marks, S., Coppol, R., 2009. Finite difference analysis of simply supported RC slabs for blast loadings. Engineering Structures, 31(12), 2825–2832.
  • Jovanovic, B.S. and Popovic, B.Z., 2001. Convergence of a finite difference scheme for the third boundary-value problem for an elliptic equation with variable coefficients. Computational Methods in Applied Mathematics. 1 (4), 356–366.
  • Jovanovic, B.S., Lemeshevsky, S.V., Matus, P.P., Vabishchevich, P.N., 2006. Stability of solutions of differential-operator and operator-difference equations in the sense of perturbations of operators. Computational Methods in Applied Mathematics, 6 (3), 269–290.
  • Jovanovic, B.S., Vulkov, L.G., 2001. On the convergence of finite difference schemes for the heat equation with concentrated capacity. Numerische Mathematik, 89 (4), 715–734.
  • Kalyani, V.K., Pallavika, Chakraborty, S.K., 2014. Finite-difference time-domain method for modelling of seismic wave propagation in viscoelastic media. Applied Mathematics and Computation, 237, 133–145
  • Liu, Y., Yin, C., 2014. 3D anisotropic modeling for airborne EM systems using finite-difference method. Journal of Applied Geophysics, 109, 186–194.
  • MATLAB (2009). The MathWorks, Natick, MA. Moreno-García, P., Lopes, H. , Araújo dos Santos, J.V., 2015. Application of higher order finite differences to damage localization in laminated composite plates, Composite Structures.
  • Strikwerda, J.C., 1990. Finite Difference Schemes and Partial Differential Equations. Chapman & Hall, New York.
  • Thomee, V., 1990. Finite difference methods for linear parabolic equations. P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, Vol. I. Finite Difference Methods (Part 1), North-Holland, Amsterdam, 5–196
  • . Zienkiewicz, O.C., 1971. The Finite Element Method in Engineering Science, London: McGraw-Hill.
There are 15 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Abdurrahman Sahin

Publication Date December 31, 2016
Submission Date April 13, 2016
Published in Issue Year 2016 Volume: 16 Issue: 3

Cite

APA Sahin, A. (2016). Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 16(3), 693-710.
AMA Sahin A. Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. December 2016;16(3):693-710.
Chicago Sahin, Abdurrahman. “Matrix Method Development for Structural Analysis of Euler Bernoulli Beams With Finite Difference Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 16, no. 3 (December 2016): 693-710.
EndNote Sahin A (December 1, 2016) Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 16 3 693–710.
IEEE A. Sahin, “Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 16, no. 3, pp. 693–710, 2016.
ISNAD Sahin, Abdurrahman. “Matrix Method Development for Structural Analysis of Euler Bernoulli Beams With Finite Difference Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 16/3 (December 2016), 693-710.
JAMA Sahin A. Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2016;16:693–710.
MLA Sahin, Abdurrahman. “Matrix Method Development for Structural Analysis of Euler Bernoulli Beams With Finite Difference Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 16, no. 3, 2016, pp. 693-10.
Vancouver Sahin A. Matrix Method Development for Structural Analysis of Euler Bernoulli Beams with Finite Difference Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2016;16(3):693-710.