KSCE Journal of Civil Engineering

, Volume 22, Issue 4, pp 1101–1108 | Cite as

Spatially Varying Small-strain Stiffness in Soils Subjected to K0 Loading

  • Hyun-Ki KimEmail author
  • J. Carlos Santamarina
Geotechnical Engineering


Grain-scale characteristics and formation history determine spatial variability in granular masses. We investigate the effect of spatially varying stiffness on the load-deformation response under zero-lateral strain conditions using numerical simulations of correlated random fields, where the granular medium is represented by a non-linear stress-dependent meso-scale model. Results show that stiffness heterogeneity results in higher global compressibility as compared to the homogeneous medium with the same arithmetic mean stiffness. Furthermore, the non-homogeneous stress field that develops inside the granular mass is characterized by focused load transfer along columnar regions, higher stress anisotropy and lower horizontal-to-vertical stress ratio K0 than in a granular medium of homogenous stiffness. As the applied stress increases, the inherent stress-dependent response of the granular material leads to a more homogenous stress field. While greater variance in stiffness causes lower global stiffness, a longer correlation length results in greater variance in global mechanical response among multiple realizations.


spatial variability effective stiffness K0 coefficient uniaxial compaction zero-lateral strain loading 


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  1. ABAQUS/Standard User’s Manual, Version 6.6 (2007). Habbitt, Karlsson & Sorenson, Inc., Pawtucke.Google Scholar
  2. Ang, A. H. S. and Tang, W. H. (1975). Probability concept in engineering planning and design, Vol. 1. New York: Wiley and Sons.Google Scholar
  3. Antonellini, M. A., Aydin, A., and Pollard, D. D. (1994). “Microstructure of deformation bands in porous sandstones at Arches National Park, Utah.” Journal of Structural Geology, Vol. 16, pp. 941–959.CrossRefGoogle Scholar
  4. Antonellini, M. A., Aydin, A., Pollard, D. D., and D’Onfro, P. (1994). “Petrophysical study of faults in sandstones using petrographic image analysis and X-ray computerized tomography.” Pure and Applied Geophysics, Vol. 143, pp. 181–201.CrossRefGoogle Scholar
  5. Arévalo, R., Zuriguel, I., and Maza, D. (2009) “Topological properties of the contact network of granular materials.” International Journal of Bifurcation and Chaos, Vol. 19, pp. 695–702.CrossRefzbMATHGoogle Scholar
  6. Barreto, D. and O’Sullivan, C. (2012). “The influence of interparticle friction and the intermediate stress ratio on soil response under generalised stress conditions.” Granular Matter, Vol. 14, No. 4, pp. 505–521.CrossRefGoogle Scholar
  7. Beacher, G. B. and Ingra, T. S. (1981). “Stochastic FEM in settlement predictions.” ASCE Journal of Soil Mechanics and Foundation Division, Vol. 107, No. 4, pp. 449–463.Google Scholar
  8. Behringer, R., Daniels, K. E., Majmudar, T. S., and Sperl, M. (2008). “Fluctuations, correlations, and transitions in granular materials: Statistical mechanics for a non-conventional system.” Philosophical Transactions of the Royal Society A, Vol. 366, No. 1865, pp. 493–504.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cambou, B. (1975). “Applications of first-order uncertainty analysis in the finite elements method in linear elasticity.” Proc. Applications of Statistics and Probability in Soil and Structure Engineering, 2nd International Conference, Aachen, Germany, pp. 117–122.Google Scholar
  10. Ching, J. and Phoon, K. K. (2013). “Effect of element sizes in random field finite element simulations of soil shear strength.” Computers and Structures, Vol. 126, No. 15, pp. 120–134.CrossRefGoogle Scholar
  11. Ching, J. and Phoon, K. K. (2013). “Mobilized shear strength of spatially variable soils under simple stress states.” Structural Safety, Vol. 41, pp. 20–28.CrossRefGoogle Scholar
  12. Cho, G. C., Lee, J. S., and Santamarina, J. C. (2004). “Spatial variability in soils: high resolution assessment with electrical needle probe.” ASCE Journal of Geotechnical and Geoenvironmental Engineering. Vol. 130, No. 8, pp. 843–850.CrossRefGoogle Scholar
  13. Cundall, P. A. and Strack, O. D. L. (1979). “A discrete numerical model for granular assemblies.” Geotechnique, Vol. 29, pp. 47–65.CrossRefGoogle Scholar
  14. DeGroot, D. J. (1996). “Analyzing spatial variability of in-situ soil properties.” Proc. Uncertainty’ 96, Madison, pp. 210–238.Google Scholar
  15. DeGroot, D. J. and Beacher, G. B. (1993). “Estimating autocovariance of in-situ soil properties.” ASCE Journal of Geotechnical Engineering, Vol. 119, No. 1, pp. 147–166.CrossRefGoogle Scholar
  16. Díaz-Rodríguez, J. A. and Santamarina, J. C. (1999). “Thixotropy: The Case of Mexico City Soils.” XI Panamerican Conference on Soil Mechanics and Geotechnical Engineering, Iguazu Falls, Brazil, Vol. 1, pp. 441–448.Google Scholar
  17. Duncan, J. M. and Chang, C. Y. (1970). “Nonlinear analysis of stress and strain in soils.” ASCE Journal of the Soil Mechanics and Foundations Division, Vol. 96, No. 5, pp. 1629–1653.Google Scholar
  18. El-Kadi, A. I. and Williams, S. A. (2000). “Generating two-dimensional fields of auto-correlated, normally distributed parameters by the matrix decomposition technique.” Ground Water, Vol. 38, No. 4, pp. 530–532.CrossRefGoogle Scholar
  19. Fenton, G. A. (1994). “Error evaluation of three random field generators.” ASCE Journal of Engineering Mechanics, Vol. 120, No. 12, pp. 2478–2497.CrossRefGoogle Scholar
  20. Fenton, G. A. and Griffiths, D. V. (2005). “Three-dimensional probabilistic foundation settlement.” ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 131, No. 2, pp. 232–239.CrossRefGoogle Scholar
  21. Fenton, G. A. and Griffiths, D. V. (2008). Risk assessment in geotechnical engineering, John Wiley and Sons, New York.CrossRefGoogle Scholar
  22. Fernandez, A. L. (2000). Tomographic imaging the state of stress, PhD Thesis, Georgia Institute of Technology.Google Scholar
  23. Garzón, L. X., Caicedo, B., Sánchez-Silva, M., and Phoon, K. K. (2015). “Physical modelling of soil uncertainty.” International Journal of Physical Modelling in Geotechnics, Vol. 15, No. 1, pp. 19–34.CrossRefGoogle Scholar
  24. Griffiths, D. V. and Fenton, G. A. (2009). “Probabilistic settlement analysis by stochastic and random finite-element methods.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 11, pp. 1629–1637.CrossRefGoogle Scholar
  25. Grigoriu, M. (1984). “Crossing of non-Gaussian translation process.” ASCE Journal of Engineering Mechanics, Vol. 110, No. 4, pp. 610–620CrossRefGoogle Scholar
  26. Harr, M. E. (1987). Reliability based design in civil engineering, McGraw Hill, London.Google Scholar
  27. Hegazy, A. H., Mayne, P. M., and Rouhani, S. (1996). “Geostatistical assessment of spatial variability in piezocone tests.” Proc. Uncertainty’ 96, Madison, pp. 254–268.Google Scholar
  28. Huang, J. and Griffiths, D. V. (2015). “Determining an appropriate finite element size for modelling the strength of undrained random soils.” Computers and Geotechnics, Vol. 69, pp. 506–513.CrossRefGoogle Scholar
  29. Huang, J., Griffiths, D. V., and Fenton G. A. (2010). “Probabilistic Analysis of Coupled Soil Consolidation.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 3, pp. 417–430.CrossRefGoogle Scholar
  30. Hurley, R., Marteau, E., Ravichandran, G., and Andrade, J. E. (2014). “Extracting inter-particle forces in opaque granular materials: Beyond photoelasticity.” Journal of the Mechanics and Physics of Solids, Vol. 63, pp. 154–166.CrossRefGoogle Scholar
  31. Jang, D. J., Frost, J. D., and Park, J. Y. (1999). “Preparation of epoxy impregnated sand coupons for image analysis.” ASTM Geotechnical Testing Journal, Vol. 22, No. 2, pp. 147–158.Google Scholar
  32. Jimenez, R. and Sitar, N. (2009). “The importance of distribution types on finite element analyses of foundation settlement.” Computers and Geotechnics, Vol. 36, pp. 474–483.CrossRefGoogle Scholar
  33. Jones, A. L., Kramer, S. L., and Arduino, P. (2002). Estimation of uncertainty in geotechnical properties for performance-based earthquake engineering, PEER report 2002/16.Google Scholar
  34. Kulhawy, F. H. (1992). “On evaluation of static soil properties.” Stability and Performance of Slopes and Embankments II (GSP 31) ASCE, New York, pp. 95–115.Google Scholar
  35. Lacasse, S. and Nadim, F. (1996). “Uncertainties in characterizing soil properties.” Proc. Uncertainty’ 96, Madison, pp. 49–75.Google Scholar
  36. Majmudar, T. S. and Behringer, R. P. (2005). “Contact force measurements and stress-induced anisotropy in granular materials.” Nature, Vol. 435, pp. 1079–1082.CrossRefGoogle Scholar
  37. Muthuswamy, M. and Tordesillas, A. (2006). “How do interparticle friction, packing density and degree of polydispersity affect force propagation in particulate assemblies?” Journal of Statistical Mechanics: Theory and Experiment ( Scholar
  38. Niemunis, A., Wichtmann, T., Petryna, Y., and Triantafyllidis, T. (2005). “Stochastic modelling of settlements due to cyclic loading for soilstructure interaction.” Proc. the 9th International Conference on Structural Safety and Reliability, ICOSSAR’05, Rome, Italy. Rotterdam, Millpress.Google Scholar
  39. Oda, M., Takemura, T., and Takahashi, M. (2004) “Microstructure in shear band observed by microfocus X-ray computed tomography.” Geotechnique, Vol. 54, No. 8, pp. 539–542.CrossRefGoogle Scholar
  40. Paice, G. M., Griffiths, D. V., and Fenton, G. A. (1996). “Finite element modeling of settlements on spatially random soil.” ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 122, No. 9, pp. 777–779.CrossRefGoogle Scholar
  41. Peña, A. A., Hermann, H. J., and Lind, P. G. (2009). “Force chains in sheared granular media of irregular particles.” Powders and Grains 2009: Proceedings of the Sixth International Conference Micromechanics of Granular Media, Colorado, USA., pp. 321–324.Google Scholar
  42. Phoon, K. K. and Kulhawy, F. H. (1999). “Characterization of geotechnical variability.” Canadian Geotechnical Journal, Vol. 36, pp. 612–624.CrossRefGoogle Scholar
  43. Radjai, F., Wolf, D. E., Jean, M., and Moreau, J. J. (1998). “Bimodal character of stress transmission in granular packings,” Physical Review Letters, Vol. 80, pp. 61–64.CrossRefGoogle Scholar
  44. Ravi, V. (1992). “Statistical modeling of spatial variability of undrained strength.” Canadian Geotechnical Journal, Vol. 29, pp. 721–729.CrossRefGoogle Scholar
  45. Resendiz, D. and Herrera, I. (1969). “A probabilistic formulation of settlement control design.” Proc. 7 th ICSMFE, Vol. 2, Mexico, pp. 217–225.Google Scholar
  46. Rothenburg, L. and Bathurst, R. J. (1989). “Analytical study of induced anisotropy in idealized granular materials.” Geotechnique, Vol. 39, No. 4, pp. 601–614.CrossRefGoogle Scholar
  47. Santamarina, J. C., Klein, K. A., and Fam, M. A. (2001). Soils and waves, John Wiley and Sons, New York.Google Scholar
  48. Santoso, A. M., Phoon, K. K., and Quek, S.-T. (2011). “Effects of soil spatial variability in rainfall-induced landslides.” Computers and Structures, Vol. 89, Nos. 11-12, pp. 893–900.CrossRefGoogle Scholar
  49. Sheng, Y., Lawrence, C. J., Briscoe, B. J., and Thorton, C. (2004). “Numerical studies of uniaxial powder compaction process by 3D DEM.” Engineering Computations, Vol. 21, pp. 304–317.CrossRefzbMATHGoogle Scholar
  50. Song, K.-I., Cho, G. C., and Lee, S. W. (2011). “Effects of spatially variable weathered rock properties on tunnel behavior.” Probabilistic Engineering Mechanics Vol. 26, No. 3, pp. 413–426.CrossRefGoogle Scholar
  51. Suchomel, R. and Masin, D. (2011). “Probabilistic analyses of a strip footing on horizontally stratified sandy deposit using advanced constitutive model.” Computers and Geotechnics, Vol. 38, pp. 363–374.CrossRefGoogle Scholar
  52. Tang, W. (1979). “Probabilistic evaluation of penetration resistance.” ASCE Journal of Geotechnical Engineering, Vol. 105, No. 10, pp. 1173–1191.Google Scholar
  53. Tordesillas, A., O’Sullivan, P., and Walker, D. M. (2010). “Paramitha Evolution of functional connectivity in contact and force chain networks: Feature vectors, k-cores and minimal cycles.” Comptes Rendus Mécanique, Vol. 338, pp. 556–569.CrossRefGoogle Scholar
  54. Vanmarcke, E. H. (1977). “Probabilistic modeling of soil profiles.” Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 103, No. 11, pp. 1227–1246.Google Scholar
  55. Vio, R., Andreani, P., and Wamsteker, W. (2001). “Numerical simulation of non-Gaussian random fields with prescribed correlation structure.” The Astronomical Society of the Pacific, Vol. 113, pp. 1009–1020.CrossRefGoogle Scholar
  56. Wu, T. H., Gale, S. M., Zhou, S. Z., and Geiger, E. C. (2011). “Reliability of settlement prediction—case history.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 137, No. 4, pp. 312–322.CrossRefGoogle Scholar
  57. Yamazaki, F. and Shinozuka, M. (1988). “Digital generation of non-Gaussian stochastic fields.” ASCE Journal of Engineering Mechanics, Vol. 114, No. 7, pp. 1183–1197.CrossRefGoogle Scholar
  58. Zeitoun, D. G. and Baker, R. (1992). “A stochastic approach for settlement predictions of shallow foundations.” Geotechnique, Vol. 42, No. 4, pp. 617–629.CrossRefGoogle Scholar

Copyright information

© Korean Society of Civil Engineers 2018

Authors and Affiliations

  1. 1.School of Civil & Environmental EngineeringKookmin UniversitySeoulKorea
  2. 2.Dept. of Earth Science and EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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