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Wave Propagation Analysis in Beams Using Shear Deformable Beam Theories Considering Second Spectrum

Published online by Cambridge University Press:  15 May 2017

U. Gul  and
M. Aydogdu
U. Gul*
Affiliation:
Department of Mechanical EngineeringTrakya UniversityEdirne, Turkey
M. Aydogdu
Affiliation:
Department of Mechanical EngineeringTrakya UniversityEdirne, Turkey
*
*Corresponding author (ufukgul@trakya.edu.tr)
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Abstract

In this study, wave propagation in beams is studied using different beam theories like Euler-Bernoulli, Timoshenko and Reddy beam theories. Dispersion curves obtained for these beam theories are compared with the exact plane elasticity solutions. It is obtained that, there are two branches for Reddy beam theory similar to the Timoshenko beam theory. However, one branch is obtained for Euler-Bernoulli beam theory. The effects of in-plane load on Timoshenko and Reddy beam theories are examined and dispersion curves of the Timoshenko and Reddy beams are compared with exact plane elasticity solution. In Timoshenko beam theory, qualitative difference between the two spectrums has been lost with in-plane loads for some wave numbers.

Keywords

Wave propagationTimoshenko beam theoryReddy beam theorySecond spectrum
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 3 , June 2018 , pp. 279 - 289
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

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References

1. Traill-Nash, R. W. and Collar, A. R., “The Effect of Shear Flexibility and Rotary Inertia on the Bending Vibrations of Beams,” Quarterly Journal of Mechanics and Applied Mathematics, 6, pp. 186–222 (1953).CrossRefGoogle Scholar
2. Stephen, N. G., “The Second Frequency Spectrum of Timoshenko Beams,” Journal of Sound and Vibration, 80, pp. 578–582 (1982).CrossRefGoogle Scholar
3. Stephen, N. G., “The Second Spectrum of Timoshenko Beam Theory-Further Assessment,” Journal of Sound and Vibration, 292, pp. 372–389 (2006).Google Scholar
4. Renton, J. D., “A Check on the Accuracy of Timoshenko's Beam Theory,” Journal of Sound and Vibration, 245, pp. 559–561 (2001).Google Scholar
5. Bhaskar, A., “Elastic Waves in Timoshenko Beams: the ‘Lost and Found’ of an Eigenmode,” Proceedings of the Royal Society A, 465, pp. 239–255 (2009).Google Scholar
6. Elishakoff, I. and Lubliner, E., Random Vibration of a Structure Via Classical and Nonclassical Theories, in Probabilistic Methods in Mechanics and Structures, S. Eggwertz, ed., Springer Verlag, Berlin, pp. 455–468 (1985).Google Scholar
7. Elishakoff, I. and Livshits, D., “Some Closed form Solutions in Random Vibrations of Timoshenko Beams,” Journal of Probabilistic Engineering Mechanics, 4, pp. 49–54 (1989).CrossRefGoogle Scholar
8. Elishakoff, I., An Equation Both More Consistent and Simpler than Bresse-Timoshenko Equation, in Advances in Mathematical Modeling and Experimental Methods for Materials and Structures, R. Gilat and L. Sills-Banks, eds., Springer Verlag, Berlin, pp. 249–254 (2009).Google Scholar
9. Bhashyam, G. R. and Prathap, G., “The Second Frequency Spectrum of Timoshenko Beams,” Journal of Sound and Vibration, 76, pp. 407–420 (1981).CrossRefGoogle Scholar
10. Abbas, B. A. H. and Thomas, J., “The Second Frequency Spectrum of Timoshenko Beams,” Journal of Sound and Vibration, 51, pp. 123–137 (1977).Google Scholar
11. Manevich, A. I., “Dynamics of Timoshenko Beam on Linear and Nonlinear Foundation: Phase Relations, Significance of the Second Spectrum, Stability,” Journal of Sound and Vibration, 344, pp. 209–220 (2015).Google Scholar
12. Elishakoff, I., Kaplunov, J. and Nolde, E., “Celebrating the Centenary of Timoshenko's Study of Effects of Shear Deformation and Rotary Inertia,” Applied Mechanics Reviews, 67, 060802 (2015).Google Scholar
13. Elishakoff, I. and Soret, C., “A Consistent Set of Nonlocal Bresse-Timoshenko Equations for Nonlocal Nano-Beams with Surface Effects,” Journal of Applied Mechanics, 80, 061001 (2013).CrossRefGoogle Scholar
14. Elishakoff, I., Ghyselinck, G. and Bucas, S., “Virus Sensor Based on Single-Walled Carbon Nanotube treated as Bresse-Timoshenko beam,” Journal of Applied Mechanics, 79, 064502 (2012).Google Scholar
15. Elishakoff, I. and Pentaras, D., “Natural Frequencies of Carbon Nanotubes Based on Simplified Bresse-Timoshenko Theory,” Journal of Computational and Theoretical Nanoscience, 6, pp. 1527–1531 (2009).Google Scholar
16. Reddy, J. N., “A Simple Higher-Order Theory for Laminated Composite Plates,” Journal of Applied Mechanics, 51, pp. 745–752 (1984).Google Scholar
17. Aydogdu, M., “A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration,” Physica E, 41, pp. 1651–1655 (2009).Google Scholar
18. Aydogdu, M., “Vibration of Multi-Walled Carbon Nanotubes by Generalized Shear Deformation Theory,” International Journal of Mechanical Sciences, 50, pp. 837–844 (2008).CrossRefGoogle Scholar
19. Soldatos, K. P. and Sophocleous, C., “On Shear Deformable Beam Theories: The Frequency and Normal Mode Equations of the Homogeneous Orthotropic Bickford Beam,” Journal of Sound and Vibration, 242, pp. 215–245 (2001).Google Scholar
20. Chan, K. T., Lai, K. F., Stephen, N. G. and Young, K., “A New Method to Determine the Shear Coefficient of Timoshenko Beam Theory,” Journal of Sound and Vibration, 330, pp. 3488–3497 (2011).CrossRefGoogle Scholar
21. Cowper, G. R., “On the Accuracy of Timoshenko's Beam Theory,” Proceedings ASCE Journal of the Engineering Mechanics Division, 94, pp. 1447–1453 (1968).Google Scholar
22. Cowper, G. R., “The Shear Coefficient in Timoshenko Beam Theory,” Journal of Applied Mechanics, 33, pp. 335–340 (1966).Google Scholar
23. Hutchinson, J. R., “Shear Coefficients for Timoshenko Beam Theory,” Journal of Applied Mechanics, 68, pp. 87–92 (2001).Google Scholar
24. Kaneko, T., “On Timoshenko's Correction for Shear in Vibrating Beams,” Journal of Physics D: Applied Physics, 8, pp. 1927–1936 (1975).CrossRefGoogle Scholar
25. Corradi Dell'Acqua, L., Meccanica delle Strutture, McGraw-Hill Inc., New York, 1, pp. 346–350 (1992).Google Scholar
26. Franco-Villafañe, J. A. and Méndez-Sánchez, R. A., “On the Accuracy of the Timoshenko Beam Theory Above the Critical Frequency: Best Shear Coefficient,” Journal of Mechanics, 32, pp. 515–518 (2016).Google Scholar