Efficient Dynamic Analysis of Foundation via a Coupled Axisymmetric SBFEM-3D FEM

Abstract

An axisymmetric scaled boundary finite element method (SBFEM) for the elastodynamic analysis of 3D layered continua is derived in frequency domain. The general three-dimensional unbounded domain can be divided into a number of independent two-dimensional problems and this analysis offers considerable savings in storage and computation times as compared to a full three-dimensional analysis. A new coupled formulation has been established to match the axi-symmetrically modeled unbounded far field to a general 3D finite element of the near field. The derived formulation will be used for frequency domain analysis of foundation embedded in or resting on a layered elastic soil supported by the surface of rigid bedrock. The dynamic response calculated using the proposed method is compared with analytical or numerical solutions. Numerical examples demonstrate the accuracy and high efficiency of the proposed new method.

Keywords

• axisymmetric scaled boundary finite element method,layered soil,soil-structure interaction,Fourier series,dynamic stiffness

References

1. Kausel E., “Local transmitting boundaries”, Journal of engineering mechanics, 114(6), 1011–1027, 1988.
2. Givoli D., “Non-reflecting boundary conditions: a review”, Journal of Computational Physics, 8, 1–29, 1991.
3. Givoli D., Numerical methods for problems in infinite domains. vol. 33, Elsevier, 2013.
4. Tsynkov S.V., “Numerical solution of problems on unbounded domains. a review”, Applied Numerical Mathematics, 27(4), 465–532, 1998.
5. Givoli D., “High-order local non-reflecting boundary conditions: a review”, Wave motion, 39(4), 319–326, 2004.
6. Astley R.J., “Infinite elements for wave problems: a review of current formulations and an assessment of accuracy”, International Journal for Numerical Methods in Engineering, 49, 951–976, 2000.
7. Karabalis D.L., and Mohammadi M., “3-D dynamic foundation-soil-foundation interaction on layered soil”, Soil Dynamics and Earthquake Engineering, 17(3), 139 – 152, 1998.
8. Bouchon M., “A simple method to calculate Green’s functions for elastic layered media”, Bulletin of the Seismological Society of America, 71(4), 959 – 971, 1981.

9. Schmid G., Willms G., Huh Y., and Gibhardt M., “SSI 2D/3D soil structure interaction: Ein Programmsystem zur Berechnung von Bauwerk-Boden-Wechselwirkungsproblemen mit der Randelementmethode”, Wissenschaftliche Mitteilungen, Nr. 12, SFB 151, Ruhruniversität Bochum, 1988.
10. Lysmer J., and Waas G., “Shear waves in plane infinite structures”, Journal of the Engineering Mechanics Division, 98(EM1), 85–105, 1972.
11. Kausel E., and Roesset J.M., “Dynamic stiffness of circular foundations”, Journal of the Engineering Mechanics Division, 101, 771–785, 1975.
12. Kausel E., and Peek R., “Dynamic loads in the interior of a layered stratum: An explicit solution”, Bulletin of the Seismological Society of America, 72, 1459–1480, 1982.
13. Kausel E., “Wave propagation in anisotropic layered media”, International Journal for Numerical Methods in Engineering, 23, 1567–1578, 1986.
14. Seale S.H., and Kausel E., “Point loads in cross–anisotropic layered half spaces”, Journal of Engineering Mechanics, 115, 509– 524, 1989.
15. Kausel E., “Thin–layer method: formulation in the time domain”, International Journal for Numerical Methods in Engineering, 37, 927–941, 1994.
16. Meek J.W., and Wolf J.P., “Cone models for soil layer on rigid rock. II”, Journal of Geotechnical Engineering, 118, 686–703, 1992.
17. Wolf J.P., and Paronesso A., “Lumped–parameter model for a rigid cylindrical foundation embedded in a soil layer on rigid rock”, Earthquake Engineering and Structural Dynamics, 21, 1021–1038, 1992.
18. Wolf J.P., and Preisig M., “Dynamic stiffness of foundation embedded in layered halfspace based on wave propagation in cones”, Earthquake Engineering and Structural Dynamics, 32(7), 1075 – 1098, 2003.
19. Pradhan P.K., Baidya D.K., and Ghosh D.P., “Dynamic response of foundations resting on layered soil by cone model”, Soil Dynamics and Earthquake Engineering, 24(6), 425 – 434, 2004.
20. Baidya D.K., Muralikrishna G., and Pradhan P.K., “Investigation of foundation vibrations resting on a layered soil system”, Journal of Geotechnical and Geoenvironmental Engineering, 132(1), 116 – 123, 2006.
21. Anam I., and Roësset J.M., “Dynamic stiffness of surface foundations: An explicit solution”, International Journal of Geomechanics, 4(3), 216–223, 2004.
22. Nogami T., Mahbub A.A., and Chen S.H., “A new method for formulation of dynamic responses of shallow foundations in simple general form”, Soil Dynamics and Earthquake Engineering, 25(7-10), 679 – 688, 2005.
23. Gazetas G., “Static and dynamic displacements of foundations on heterogeneous multilayered soils”, Géotechnique, 30(2), 159 – 177, 1980.
24. Andersen L., and Clausen J., “Impedance of surface footings on layered ground”, Computers and Structures, 86(1-2), 72 – 87, 2008.
25. Bouchon M., “A simple method to calculate Green’s functions for elastic layered media”, Bulletin of the Seismological Society of America, 71(4), 959-971, 1981.
26. Luco J.E., and Apsel R.J., “On the Green’s functions for a layered half-space. Part I”, Bulletin of the Seismological Society of America, 73(4), 909–929, 1983.
27. Apsel R.J., and Luco J.E., “On the Green’s functions for a layered half–space. Part II”, Bulletin of the Seismological Society of America, 73(4), 931–951, 1983.
28. Birk C., and Behnke R., “A modified scaled boundary finite element method for three-dimensional dynamic soil-structure interaction in layered soil”, International Journal for Numerical Methods in Engineering, 89(3), 371–402, 2012.
29. Wolf J.P., Foundation Vibration Analysis Using Simple Physical Models, Prentice-Hall, Englewood Cliffs NJ, 1994.
30. Gazetas G., “Analysis of machine foundation vibrations: State of the art”, Soil Dynamics and Earthquake Engineering, 2(1), 2 – 42, 1983.
31. Wolf J.P., The Scaled Boundary Finite Element Method, Wiley & Sons: Chichester, 2003.
32. Genes M.C., and Kocak S., “Dynamic soil-structure interaction analysis of layered unbounded media via a coupled finite element/ boundary element/scaled boundary finite element model”, International Journal for Numerical Methods in Engineering, 62(6), 798–823, 2005.
33. Genes M., and Kocak S., “A combined finite element based soil-structure interaction model for large-scale systems and applications on parallel platforms”, Engineering Structures, 24(9), 1119–1131, 2002.
34. Genes M.C., “Dynamic analysis of large-scale SSI systems for layered unbounded media via a parallelized coupled finite element/ boundary-element/scaled boundary finite-element model”, Engineering Analysis with Boundary Elements, 36(5), 845–857, 2012.
35. Yaseri A., Bazyar M., and Hataf N., “3D coupled scaled boundary finite-element/finite-element analysis of ground vibrations induced by underground train movement”, Computers and Geotechnics, 60, 1–8, 2014.
36. Syed N.M., and Maheshwari B.K., Improvement in the computational efficiency of the coupled FEM-SBFEM approach for 3D seismic SSI analysis in the time domain. Computers and Geotechnics, 67, 204–212, 2015.
37. Rahnema H., Mohasseb S., and JavidSharifi B., “2-D soil-structure interaction in time domain by the SBFEM and two non-linear soil models”, Soil Dynamics and Earthquake Engineering, 88,152–175, 2016.
38. Li B., Cheng L., Deeks A.J., and Teng B., “A modified scaled boundary finite-element method for problems with parallel side-faces. Part I. Theoretical developments”, Applied Ocean Research, 27(4 - 5), 216 – 223, 2005.
39. Lin G., Du J., and Hu Z., “Dynamic dam-reservoir interaction analysis including effect of reservoir boundary absorption”, Science in China Series E: Technological Sciences, 50, 1 – 10, 2007.
40. Doherty J.P, and Deeks A.J., “Scaled boundary finite-element analysis of a non-homogeneous elastic half-space”, International journal for numerical methods in engineering, 57(7), 955–973, 2003.
41. Aslmand M., Kani I.M., Birk C., Gravenkamp H., Krome F., Ghadi M.E., “Dynamic soil-structure interaction in a 3D layered medium treated by coupling a semi-analytical axisymmetric far field formulation and a 3D finite element model” Soil Dynamics and Earthquake Engineering, under production, 2018.
42. Sommerfeld A. Partial Differential Equations in Physics, Academic Press: New York, 1949.
43. Guzina B.B., Fata S.N., and Bonnet M., “On the stress-wave imaging of cavities in a semi-infinite solid”, International Journal of Solids and Structures, 40, 1505–1523, 2003.
44. Madyarov A.I., and Guzina B.B., “A radiation condition for layered elastic media”, Journal of Elasticity, 82, 73–98, 2006.
45. Song C., and Wolf J.P., “The scaled boundary finite-element method – alias consistent infinitesimal finite-element cell method – for elastodynamics”, Computer Methods in Applied Mechanics and Engineering, 147, 329–355, 1997.
46. Laub AJ., “A Schur method for solving algebraic Riccati equations”, IEEE Transactions on Automatic Control 1979, 24(6):913–921.
47. Bartels R.H., and Stewart G.W., “Solution of the equation AX+XB=C”, Communications of the ACM 15, 9, 820-826, 1972.
48. Ansys computer program, Ansys Inc. Version 18.0, 2016.
49. Sun E.Q. Shear locking and hourglassing in msc nastran, abaqus, and ansys, Msc software users meeting, 2006.
50. Emperador J.M., and Dominguez J., “Dynamic response of axisymmetric embedded foundations”, Earthquake Engineering and Structural Dynamics, 18(8), 1105 – 1117, 1989.
51. Chow Y., “Vertical vibration of three-dimensional rigid foundations on layered media”, Earthquake engineering & structural dynamics, 15(5), 585–594, 1987.
52. Seo C.G., Yun C.B., and Kim J.M., “Three-dimensional frequency-dependent infinite elements for soil–structure interaction”, Engineering Structures, 29(11), 3106–3120, 2007.