Study on the Dynamic Response of Single and Multi-Spans Beam Subjected to the Base Excitation Using Time Weighted Residual Method
Authors
Abstract
Accurate determination of the response of structures under dynamic loads such as earthquake loads plays an important role in the safe and economical design of structures. The purpose of this paper is to utilize a novel solution method based on the use of exponential basis functions for dynamic analysis of Bernoulli beam subjected to different types of base excitations. This method was firstly introduced for solving scalar wave propagation problems, named as stepwise time-weighted residual method. The proposed method considers the solution as a series of exponential basis functions with unknown constant coefficients; and the problem is solved in time without the need for spatial discretization of the beam and by using an appropriate recursive relation to correct the coefficients of the exponential bases. In order to apply the earthquake excitation, first by using the central finite difference relation, the earthquake acceleration history is converted to displacement history. Moreover, the displacement history is applied to the beam as a time-varying boundary condition. In this study, the capabilities of the proposed method in solving several sample problems of vibration of single and multi-span beams under various stimuli such as earthquake acceleration variations are compared with the results of other existing methods.
Keywords
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2. Kane, T., Ryan, R., and Banerjee, A. ,“Dynamics of a Cantilever Beam Attached to a Moving Base”, Journal of Guidance, Control, and Dynamics, Vol. 10, No. 2, pp. 139-151, 1987.
3. Du, H., Hitchings, D., and Davies, G., “A Finite Element Structural Dynamics Model of a Beam with an Arbitrary Moving Base—Part I: Formulations”, Finite Elements in Analysis and Design, Vol. 12, No. 2, pp. 117-131, 1992.
4. Du, H., Hitchings, D., and Davies, G., “A Finite Element Structural Dynamics Model of a Beam with an Arbitrary Moving Base—Part II: Numerical Examples and Solutions”, Finite Elements in Analysis and Design, Vol. 12, No. 2, pp. 133-150, 1992.
5. Tan, T., Lee, H., and Leng, G., “Dynamic Stability of a Radially Rotating Beam Subjected to Base Excitation”, Computer Methods in Applied Mechanics and Engineering, Vol. 146, No. 3-4, pp. 265-279, 1997.
6. Ş. Yuksel and T. Aksoy, “Flexural Vibrations of a Rotating Beam Subjected to Different Base Excitations”, Gazi University Journal of Science, Vol. 22, No. 1, pp. 33-40, 2009.
7. M. Li, “Analytical Study on the Dynamic Response of a Beam with Axial Force Subjected to Generalized Support Excitations”, Journal of Sound and Vibration, Vol. 338, pp. 199-216, 2015.
8. Beskos D., and Boley, B., “Use of Dynamic Influence Coefficients in Forced Vibration Problems with the aid of Laplace Transform”, Computers & Structures, Vol. 5, No. 5-6, pp. 263-269, 1975.
9. Manolis G., and Beskos, D., “Thermally Induced Vibrations of Beam Structures”, Computer Methods in Applied Mechanics and Engineering, Vol. 21, No. 3, pp. 337-355, 1980.
10. Manolis G., and Beskos, D., “Dynamic Response of Beam Structures with the Aid of Numerical Laplace Transform”, Midwestern Mechanics Conference, pp. 85-89, 1979
11. Beskos D., and Narayanan, G., “Dynamic Response of Frameworks by Numerical Laplace Transform”, Computer Methods in Applied Mechanics and Engineering, Vol. 37, No. 3, pp. 289-307, 1983.
12. Narayanan G., and Beskos, D., “Use of Dynamic Influence Coefficients in Forced Vibration Problems with the aid of Fast Fourier Transform”, Computers & Structures, Vol. 9, No. 2, pp. 145-150, 1978.
13. Movahedian, B., Boroomand B., and Mansouri, S., “A Robust Time-Space Formulation for Large‐Scale Scalar Wave Problems Using Exponential Basis Functions”, International Journal for Numerical Methods in Engineering, Vol. 114, No. 7, pp. 719-748, 2018.
14. Borji, A., Movahedian B., and Boroomand, B., “Using the Time-Weighted Residual Method in Forced Vibration Analysis of Timoshenko Beam under Moving Load”, Amirkabir Journal of Civil Engineering, doi: 10.22060/ceej.2019.16867.6381, (in Persian)
15. Pao Y.-H. , and Sun, G., “Dynamic Bending Strains in Planar Trusses with Pinned or Rigid Joints”, Journal of Engineering Mechanics, Vol. 129, No. 3, pp. 324-332, 2003.
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