### Vibration of Timoshenko Beam-Soil Foundation Interaction by Using the Spectral Element Method

Document Type: Research Paper

Authors

1 Department of Civil Engineering, University of Jijel, Jijel, Algeria

2 Department of Mechanical Engineering, National Polytechnic School, Constantine, Algeria

3 Civil Engineering Department, University of Constantine, Constantine, Algeria

10.22034/jsm.2020.1879476.1503

Abstract

This article presents an analysis of free vibration of elastically supported Timoshenko beams by using the spectral element method. The governing partial differential equation is elaborated to formulate the spectral stiffness matrix. Effectively, the non classical end boundary conditions of the beam are the primordial task to calibrate the phenomenon of the Timoshenko beam-soil foundation interaction. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Upon solving the eigenvalue problem, non-dimensional frequencies are computed for the first three modes of vibration. Obtained results of this study are intended to describe multiple objects, such as: (1) the establishment of the modal analysis with and without elastic springs, (2) the quantification of the influence of the beam soil foundation interaction, (3) the influence of soil foundation stiffness’ on free vibration characteristics of Timoshenko beam. For this propose, the first three eigenvalues of Timoshenko beam are calculated and plotted for various stiffness of translational and rotational springs.

Keywords

[1] Tabatabaiefar H.R., Clifton T., 2016, Significance of considering soil-structure interaction effects on seismic design of unbraced building frames resting on soft soils, Australian Geomechanics 5(1): 55-66.
[2] Mohod M.V., Dhadse G.D., 2014, Importance of soil structure interaction for framed structure, International Conference on Advances in Civil and Mechanical Engineering Systems, Surat, India.
[3] Lee U., 2009, Spectral element analysis method, In Spectral Element Method in Structural Dynamics, Chichester, UK.
[4] Hamioud S., Khalfallah S., 2016, Free-vibration of Bernoulli-Euler Beam by the spectral element method, Technical Journal 10(3-4): 106-112.
[5] Hamioud S., Khalfallah S., 2018, Free-vibration of Timoshenko Beam using the spectral element method, International Journal for Engineering Modelling 31(1-2): 61-76.
[6] Kocaturk T., Şimşek M., 2005, Free vibration analysis of Timoshenko beams under various boundary conditions, Sigma Journal of Engineering and Natural Sciences 3: 79-93.
[7] Lee U., Cho J., 2008, FFT-based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo-force method, International Journal for Numerical Methods in Engineering 74: 159-174.
[8] Gopalakrishnan S., Chakraborty A., Roy Mahapatra D., 2008, Spectral Finite Element Method: Wave Propagation, Diagnostics and Control in Anisotropic and Inhomogenous Structures, Springer, London.
[9] Abbas B.A.H., 1984, Vibrations of beams with elastically restrained end, Journal of Sound and Vibration 97: 541-548.
[10] Hernandez E., Otrola E., Rodriguez R., Sahueza F., 2008, Finite element approximation of the vibration problem for a Timoshenko curved rod, Revista de La Union Matemtica 49: 15-28.
[11] Azevedo A.S.D.C, Vasconcelos A.C.A., Hoefel S.D.S., 2016, Dynamic analysis of elastically supported Timoshenko beam, XXXVII Iberian Latin American Congress on Computational Methods in Engineering, Brazilia, Brazil.
[12] Lee J., Schultz W.W., 2004, Eigenvalue analysis of Timoshenko beams and axi-symmetric Mindlin plates by the pseudospectral method, Journal of Sound and Vibration 269: 609-621.
[13] Zhou D., 2001, Free vibration of multi-span Timoshenko beams using static Timoshenko beam functions, Journal of Sound and Vibration 241: 725-734.
[14] Farghaly S.H., 1994, Vibration and stability analysis Timoshenko beams with discontinuities in cross-section, Journal of Sound and Vibration 174: 591-605.
[15] Banerjee J.R., 1998, Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method, Computers & Structures 69: 197-208.
[16] Lee Y.Y., Wang C.M., Kitipornchai S., 2003, Vibration of Timoshenko beams with internal hinge, Journal of Engineering Mechanics 129(3): 293-301.
[17] Auciello N.M., Ercolano A., 2004, A general solution for dynamic response of axially loaded non-uniform Timoshenko beams, Journal of Solids and Structures 41(18-19): 4861-4874.
[18] Thom T.T., Kien N.D., 2018, Free vibration of two-directional FGM beams using a higher-order Timoshenko beam element, Vietnam Journal of Science and Technology 56(3): 380-396.
[19] Shali S., Jafarali P., Nagaraja S.R., 2018, Identification of second spectrum of a Timoshenko beam using differential transform method, Journal of Engineering Science and Technology 13(4): 893-908.
[20] Magdalena P., 2018, Spectral methods for modeling of wave propagation in structure in terms of damage detection- A review, Applied Sciences Journal 8(7): 1-25.
[21] Sarigul M., 2018, Effect of elastically supports on nonlinear vibrations of a slightly curved beam, Uludag University Journal of the Faculty of Engineering 23(2): 255-274.
[22] Hamioud S., Khalfallah S., 2017, Dynamic analysis of rods using the spectral element method, Algerian & Equipment Journal 57: 49-55.
[23] Boudaa S., Khalfallah S., Hamioud S., 2019, Dynamic analysis of soil-structure interaction by the spectral element method, Innovative Infrastructure Solutions 4(1): 40.
[24] Ruta P., 1999, Application of Chebyshev series to solution of non-prismatic beam vibration problems, Journal of Sound and Vibration 227(2): 449-467.
[25] Chen G., Qian L., Yin Q., 2014, Dynamic analysis of a Timoshenko beam subjected to an accelerating mass using spectral element method, Shock and Vibration 2014: 12.