Response analysis of soil deposit considering both frequency and strain amplitude dependencies using nonlinear causal hysteretic damping model

  • Received : 2010.10.23
  • Accepted : 2012.03.27
  • Published : 2013.02.25


It is well known that the properties of the soil deposits, especially the damping, depend on both frequency and strain amplitude. Therefore it is important to consider both dependencies to calculate the soil response against earthquakes in order to estimate input motions to buildings. However, it has been difficult to calculate the seismic response of the soil considering both dependencies directly. The author has studied the time domain evaluation of the frequency dependent dynamic stiffness, and proposed a simple hysteretic damping model that satisfies the causality condition. In this paper, this model was applied to nonlinear analyses considering the effects of the strain amplitude dependency of the soil. The basic characteristics of the proposed method were studied using a two layered soil model. The response behavior was compared with the conventional model e.g. the Ramberg-Osgood model and the SHAKE model. The characteristics of the proposed model were studied with regard to the effects of element divisions and the frequency dependency that is a key feature of the model. The efficiency of the model was confirmed by these studies.


  1. Hardin, B.O. and Drnevich, V.P. (1972), "Shear modulus and damping in soils: Design equations and curves", J. SMFD, proc., ASCE, 98(7), 667-692.
  2. Jennings, P.C. (1963), "Periodic response of a general yielding structure", Proc. ASCE, 90(2), 131-166.
  3. Kausel, E. and Assimaki, D. (2002), "Seismic simulation of inelastic soils via frequency-dependent moduli and damping", J. Eng. Mech.-ASCE, 128(1), 34-47.
  4. Kumazaki, I. (1998), "Hysteresis model considering shear-strain dependency of fractal dimension and momentary deformation modulus", Proceedings of International Association for Mathematical Geology, 602-607.
  5. Masing, G. (1926), "Eigenspannungen und verfestigung being messing", Proc. 2nd int. Congress of Applied Mechanics, Zurich, Switzerland, 332-335.
  6. Nakamura, N. (2006a), "A practical method to transform frequency dependent impedance to time domain", Earthq. Eng. Struct. D., 35(2), 217-234.
  7. Nakamura, N. (2006b), "Improved methods to transform frequency dependent complex stiffness to time domain", Earthq. Eng. Struct. D., 35(8), 1037-1050.
  8. Nakamura, N. (2007), "Practical causal hysteretic damping", Earthq. Eng. Struct. D., 36(5), 597-617.
  9. Nakamura, N. (2008a), "Transform methods for frequency dependent complex stiffness to time domain using real or imaginary data only", Earthq. Eng. Struct. D., 37(4), 495-515.
  10. Nakamura, N. (2008b), "Nonlinear response analysis considering dynamic stiffness with both frequency and strain dependencies", J. Eng. Mech.-ASCE, 134(4), 530-541.
  11. Sato, T., Fushimi, M. and Tatsumi, Y. (2001), "Inversion of strain-dependent nonlinear characteristics of soil using weak and strong motions observed by borehole sites in japan", B. Seismol. Soc. Am., 91(2), 365-380.
  12. Schnabel, P.B., Lysmer, J. and Seed, H.B. (1972), "SHAKE A computer program for earthquake response analysis of horizontally layered sites", Report No.EERC72-12, University of California, Berkeley.
  13. Yoshida, N., Kobayashi, S., Suetomi, I. and Miura, K. (2002), "Equivalent linear method considering frequency dependent characteristics of stiffness and damping", Soil Dyn. Earthq. Eng., 22, 205-222.

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