Attenuation and Damping of Multireflected Transient Elastic Waves in a Pile
Publication: Journal of Engineering Mechanics
Volume 137, Issue 8
Abstract
Early time transient waves multireflected in a finite pile, governed by a damped wave equation, are analyzed by a reverberation-ray matrix. The pile is surrounded by compacted soil, and the composite is modeled by elastic springs and viscous dampers distributed along the length and at the tip of an elastic rod. Steady-state waves with complex frequencies and wave numbers that are generated by a source of harmonic time function at the top and reverberated between the top and bottom surface of the pile are sorted in matrix form into ray-groups, arriving at a receiver in successive orders of reflections from the bottom. The steady-state ray groups are synthesized into a series of nonsingular Fourier integrals that can be evaluated accurately with a fast Fourier-transform algorithm. The first integral (zeroth-order) has also been reduced by complex contour integration to the well-known closed-form solution in Bessel functions for a semi-infinite pile. Detailed time records of velocity response received at the top after three reflections are calculated to illustrate attenuation and damping; arrival times and amplitude-phase change on each reflection for various lateral and base supports. The calculated records resemble ultrasonic nondestructive testing data.
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Acknowledgments
Professor Y. H. Pao would like to acknowledge the financial support provided by the National Science Foundation of P. R. China, Grant No. NSF10472103. Professor Y. Y. Yu would like to acknowledge the financial support provided by the Chinese Development Fund for the visiting study at the National Taiwan University in April to July 2003, by the Key Project of Education Ministry (UNSPECIFIED208150), and by Qing Lan Talent Engineering Funds (UNSPECIFIEDQL0507A), Lanzhou Jiaotong University. They also acknowledge the supply of reference (Smith 1960) by Professor Jien-Hisn Yuan, Research Institute of Soil and Rock Mechanics, Chinese Academy of Science, Wuhan, P. R. China, and the help of Dr. H.M. Peng, National Taiwan University.
References
Baranov, V. A. (1967). “On the calculation of excited vibrations of an embedded foundation.” Voprosy Dynamiki i Prochnocti, No. 14, Polytechnical Institute of Riga, 195–209 (in Russian).
Chen, J. F., and Pao, Y. H. (2003). “Effects of causality and joint conditions on method of reverberation-ray matrix.” AIAA J., 41(6), 1138–1141.
Howard, S. M., and Pao, Y. H. (1998). “Analysis and experiments on stress waves in planar trusses.” J. Eng. Mech., 124(8), 884–891.
Isaacs, D. V. (1931). “Reinforced concrete pile formulae.” Trans. Inst. Eng Aust. Civ. Eng., 305–323.
Mclachlan, N. W. (1955). Bessel functions for engineers, 2nd Ed., Oxford, London.
Mylonakis, G., and Gazetas, G. (1998). “Settlement and additional internal forces of grouped piles in layered soil.” Geotech., 48(1), 55–72.
Novak, M. (1974). “Dynamic stiffness and damping of piles.” Can. Geotech. J., 11(4), 574–598.
Novak, M., and Aboul-Ella, F. (1978). “Impedance function of piles in layered media.” J. Eng. Mech. Div., 104(EM6), 643–661.
Nogami, T., and Konagai, K. (1986). “Time domain axial response of dynamically loaded single piles.” J. Eng. Mech., 112(11), 1241–1252.
Nogami, T., and Novak, M. (1980). “Coefficients of soil reaction to pile vibration.” J. Geotech. Eng. Div., 106(GT5), 565–570.
Pao, Y. H., Keh, D. C., and Howard, S. M. (1999). “Dynamic response and wave propagation in plane trusses and frames.” AIAA J., 37(5), 594–603.
Pao, Y. H., and Sun, G. J. (2003). “Dynamic bending strain in planar trusses with pinned or rigid joints.” J. Eng. Mech., 129(3), 324–332.
Pao, Y. H., Chen, W. Q., and Su, X. Y. (2007). “The reverberation-ray matrix and transfer matrix analysis of unidirectional wave motion.” Wave Motion, 44(6), 419–438.
Smith, E. A. L. (1960). “Pile-driving analysis by the wave equation.” J. Soil Mech. and Found. Div., 86(4), 35–61.
Timoshenko, Stephen P., and Goodier, J. N. (1951). Theory of elasticity, 2nd Ed., McGraw-Hill, New York.
Von Koten, H., Middendorp, P., and Van Brederode, P. (1980). “An analysis of dissipative wave propagation in a pile.” Proc., Int. Seminar on the Application of Stress-Wave Theory on Piles, H. Bredenberg, ed., Stockholm, 22–40.
Wang, K. H., Xie, K. H., and Zeng, G. X. (1997). “Analytical solution to vibration of finite length pile under exciting force and its application.” Chin. J. Geotech. Eng., 19(6), 27–35.
Wang, T., Wang, K. H., and Xie, K. H. (2001). “An analytical solution to longitudinal vibration of a pile of arbitrary segments with variable modulus.” Acta Mechanica Solida Sinica, 14(1), 67–73.
Yu, Y. Y. (2004). “Transient response analysis of frame structures embedded partially in soil by the method reverberation-ray matrix.” Ph.D. thesis, Zhejiang Univ., Zhejiang, P. R. China.
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Information
Published In
Journal of Engineering Mechanics
Volume 137 • Issue 8 • August 2011
Pages: 571 - 579
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Jun 27, 2009
Accepted: Mar 9, 2011
Published online: Mar 10, 2011
Published in print: Aug 1, 2011
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