State space solution to 3D multilayered elastic soils based on order reduction method

doi:10.1007/s10483-012-1629-8

Abstract

Abstract: Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.

Key words: state space solution, multilayered elastic soil, double Fourier transform, order reduction method, adaptive structure, active member, state vector approach, frequency, damping ratio

2010 MSC Number:

TU43
O343

Zhi-yong AI;Yi-chong CHENG;Peng LIU . State space solution to 3D multilayered elastic soils based on order reduction method. Applied Mathematics and Mechanics (English Edition), 2012, 33(11): 1371-1380.

TrendMD

References

[1] Mindlin, R. D. Force at a point in the interior of a semi-infinite solid. Physics, 7(5), 195-202(1936)
[2] Muki, R. Asymmetric problems of the theory of elasticity for a semi-infinite solid and thick plate.Progress in Soild Mechanics, North-Holland, Amsterdam, 399-439 (1960)
[3] Sneddon, I. N. On Muki’s solution of the equations of linear elasticity. International Journal ofEngineering Science, 30(10), 1237-1246 (1992)
[4] Pan, E. Static Green’s functions in multilayered half spaces. Applied Mathematical Modelling,21(8), 509-521 (1997)
[5] Bufler, H. Theory of elasticity of a multilayered medium. Journal of Elasticity, 1(2), 125-143(1971)
[6] Bahar, L. Y. Transfer matrix approach to layered systems. Journal of the Engineering MechanicsDivision, 98(5), 1159-1172 (1972)
[7] Kausel, E. and Seak, S. H. Static loads in layered half-spaces. Journal of Applied Mechanics,54(2), 403-408 (1987)
[8] Small, J. C. and Booker, J. R. Finite layer analysis of layered elastic material using a flexibilityapproach, part 2—circular and rectangular loadings. International Journal for Numerical Methodsin Engineering, 23(5), 959-978 (1986)
[9] Ai, Z. Y., Yue, Z. Q., Tham, L. G., and Yang, M. Extended Sneddon and Muki solutions formultilayered elastic materials. International Journal of Engineering Science, 40(13), 1453-1483(2002)
[10] Wang, L. S. The initial parameter method for solving three-dimensional axisymmetrical problemsof layered foundation (in Chinese). Acta Mechanica Sinica, 18(6), 528-537 (1986)
[11] Zhong, Y., Wang, Z. R., Guo, D. Z., and Wang, Z. Y. Transfer matrix method for solving nonaxisymmetricalproblems in multilayered elastic half space (in Chinese). China Civil EngineeringJournal, 28(1), 66-72 (1995)
[12] Ai, Z. Y. and Wu, C. Transfer matrix solution for multilayered soils in rectangular coordinatesystem (in Chinese). Journal of Chongqing Jianzhu University, 30(2), 43-46 (2008)
[13] Yue, Z. Q. and Wang, R. Static solution for transversely isotropic elastic N-layered systems (inChinese). Acta Scientiarum Naturalium Universitatis Pekinensis, 24(2), 202-211 (1988)
[14] Wang, Y. K. and Gong, Y. Q. Analytical solution of transversely isotropic elastic multilayeredsubgrade under arbitrary loading in rectangular coordinates (in Chinese). Engineering Mechanics,23(5), 9-13 (2006)
[15] Timoshenko, S. P. and Good, J. N. Theory of Elasticity, McGraw-Hill, New York (1970)
[16] Sneddon, I. N.The Use of Integral Transform, McGraw-Hill, New York (1972)
[17] Pastel, E. C. and Leckie, F. A. Matrix Methods in Elasto-Mechanics, McGraw-Hill, New York(1963)