Document Type : Article

**Authors**

School of Engineering, Shiraz University, Shiraz, Iran

**Abstract**

An extension of the slip line theory to unsaturated soils is presented and applied to the problem of stability of slopes in unsaturated soils. The matric suction is then introduced and implemented in governing equations by the well-known Bishop effective stress concept. In this regard, the van Genuchten model is utilized to estimate the effective stress parameter required for the computation of the effective stress. The influence of the soil matric suction on the stability of slopes has been investigated for a variety of soils under steady state evaporation and infiltration, i.e., the distribution of the matric suction was assumed to remain constant with time. In addition, a measure of stability in terms of some factor of stability has been introduced.

**Keywords**

References:

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3. Brinch Hansen, J. "A revised and extended formula for bearing capacity", Danish Geotechnical Institute, Bulletin No. 28, Copenhagen (1970).

4. Vesic, A.S. "Analysis of ultimate loads of shallow foundations", Journal of the Soil Mechanics and Foundations, 99(1), pp. 45-73 (1973).

5. Michalowski, R.L. "An estimate of the influence of the soil weight on the bearing capacity using limit analysis", Soils and Foundations, 37(4), pp. 57-64 (1997).

6. Kumar, J. "N for rough strip footing using the method of characteristics", Canadian Geotechnical Journal, 40(3), pp. 669-674 (2003).

7. Kumar, J. "The variation of N with footing roughness using the method of characteristics", International Journal for Numerical and Analytical Methods in Geomechanics, 33(2), pp. 275-284 (2009).

8. Jahanandish, J., Habibagahi, G., and Veiskarami, M. "Bearing capacity factor, N , for unsaturated soils by ZEL method", Acta Geotechnica, 5(3), pp. 177-188 (2010).

9. Veiskarami, M. and Habibagahi, G. "Foundations bearing capacity subjected to seepage by the kinematic approach of the limit analysis", Frontiers of Structural and Civil Engineering, 7(3), pp. 446-455 (2013).

10. Veiskarami, M., Kumar, J., and Valikhah, F. "Effect of the flow rule on the bearing capacity of strip foundations on sand by the upper-bound limit analysis and slip lines", International Journal of Geomechanics, 14(3), 04014008 (2014).

11. Veiskarami, M., Jamshidi Chenari, R., and Jameei, A.A. "Bearing capacity of strip footings on anisotropic soils by the finite elements and linear programming", International Journal of Geomechanics, 17(12), 04017119 (2017).

12. Coulomb, C.A. "Essay on maximums and minimums of rules to some static problems relating to architecture", al Academie Royale des Sciences Memoires de Mathematique et de Physique, Presentes, par Divers Scavans, 7, pp. 343-382 (1973).

13. Rankine, W.J.W. "On the stability of loose earth", Philosophical Transactions of the Royal Society of London, 147, pp. 9-27 (1857).

14. Rochette, P.A. "Earth pressures on structures and mobilized Shear resistance", 15th Can. Soil Mech. Conf, Tech. Mem., 73(3) p. 59 (1961).

15. Rowe, P.W. and Peaker, K. "Passive earth pressure measurements", Geotechnique, 15(1), p. 57 (1965).

16. James, R.G. and Bransby, D.L. "Experimental and theoretical investigations of a passive earth pressure problem", Geotechnique, 20(1), pp. 17-37 (1970).

17. Veiskarami, M., Jamshidi Chenari, R., and Jameei, A.A. "A study on the static and seismic earth pressure problems in anisotropic granular media", Geotechnical and Geological Engineering, 37(3), pp. 1987-2005 (2019).

18. Jamshidi Chenari, R., Kamyab, H., and Izadi, A. "Continuous slip surface method for stability analysis of heterogeneous vertical trenches", Scientia Iranica, 27(6), pp. 2657-2668 (2020).

19. Bishop, A.W. "The use of slip circle in the stability analysis of earth slopes", Geotechnique, 5(1), pp. 7-17 (1955).

20. Fang, H.Y. and Hirst, T.J. "Application of plasticity theory to slope stability problems", Highway Research Record, 323, pp. 26-38 (1970).

21. Lo, K.Y. and Lee, C.F. "Stress analysis and slope stability in strain-softening materials", Geotechnique, 23(1), pp. 1-11 (1973).

22. Lehner, F.K. and Schopfer, M.P.J. "Slope stability and exact solutions for cohesive critical Coulomb wedges from Mohr diagrams", Journal of Structural Geology, 116, pp. 234-240 (2018).

23. Xu, H., Ren, X., Chen, J.N., Liu, C.N., Xia, L., and Liu, Y.W. "Centrifuge model tests of geogrid-reinforced slope supporting a high embankment", Geosynthetics International, 26(6), pp. 629- 640 (2019).

24. Fredlund, D.G. "The stability of slopes with negative pore-water pressures", The Ian Boyd Donald Symposium on Modern Developments in Geomechanics, 3168 (1995).

25. Johari, A., Hooshmand Nejad, A., and Mousavi, S. "Probabilistic model of unsaturated slope stability considering the uncertainties of soil-water characteristic curve", Scientia Iranica, 25(4), pp. 2039-2050 (2018).

26. Rahardjo, H., Satyanaga, A., Wang, C.L., Wong, J.L.H., and Lim, V.H. "Effects of unsaturated properties on stability of slope covered with Caesalpinia crista in Singapore", Environmental Geotechnics, 7(6), pp. 393-403 (2018).

27. Satyanaga, A. and Rahardjo, H. "Stability of unsaturated soil slopes covered with Melastoma malabathricum in Singapore", Proceedings of the Institution of Civil Engineers-Geotechnical Engineering, 172(6), pp. 530-540 (2019).

28. Li, Z., He, Y., Li, H., and Wang, Y. "Antecedent rainfall induced shallow landslide-A case study of Yunnan landslide, China", Scientia Iranica, 26(1), pp. 202-212 (2019).

29. Ching, R.K.H., Sweeney, D.J., and Fredlund, D.G. "Increase in factor of safety due to soil suction for two Hong Kong slopes", Proc., 4th Int. Symp. on Landslides (1984).

30. Fellenius, W. "Calculation of the stability of earth dams", Trans. 2nd Congr. Large Dams, Washington (1936).

31. Janbu, N. "Earth pressure and bearing capacity calculations by generalized procedure of slices", Proc. 4 ICSMFE, 2, London, UK, pp. 207-212 (1957).

32. Morgenstern, N.R. and Price, V.E. "The analysis of the stability of general slip surfaces", Geotechnique, 15(1), pp. 79-93 (1965).

33. Gvozdev, A.A. "Determination of the collapse load for statically indeterminate structures subjected to plastic deformations", Proc. Conf. Plastic Deformations, Moscow, Russia (1938).

34. Drucker, D.C., Prager, W., and Greenberg, H.J. "Extended limit design theorems for continuous media", Quarterly of Applied Mathematics, 9(4) pp. 381-389 (1952).

35. Chen, W.F., Limit Analysis and Soil Plasticity, Amsterdam, the Netherlands, Elsevier (1975).

36. Michalowski, R.L., "Stability charts for uniform slopes", Journal of Geotechnical and Geoenvironmental Engineering, 128(4), pp. 351-355 (2002).

37. Yuan, S. and Du, J. "A lower-bound formulation for unsaturated soils", Geotechnique, 70(2), pp. 123-137 (2020).

38. Sokolovskii, V.V., Statics of Soil Media, Butterworths Scientific Publications, London, UK (1960).

39. Harr, M.E., Foundations of Theoretical Soil Mechanics, McGraw-Hill (1966).

40. Bolton, M.D. and Lau, C.K. "Vertical bearing capacity factors for circular and strip footings on Mohr- Coulomb soil", Canadian Geotechnical Journal, 30(6), pp. 1024-1033 (1993).

41. Kumar, J. and Mohan Rao, V.B.K. "Seismic bearing capacity of foundations on slopes", Geotechnique, 53(3), pp. 347-361 (2003).

42. Martin, C.M. "Analysis of bearing capacity", OUEL Report, 2261(03), Oxford, UK (2003).

43. Martin, C.M. "Exact bearing capacity calculations using the method of characteristics", Proc. IACMAG. Turin, 11, Turin, Italy, pp. 441-450 (2005).

44. Thanh, Vo. and Russell, A.R. "Slip line theory applied to a retaining wall-unsaturated soil interaction problem", Computers and Geotechnics, 55, pp. 416- 428 (2014).

45. Veiskarami, M. and Zanj, A. "Stability of sheet-pile walls subjected to seepage flow by slip lines and finite elements", Geotechnique, 64(10), pp. 759-775 (2014).

46. Johari, A., Hosseini, S.M., and Keshavarz, A. "Reliability analysis of seismic bearing capacity of strip footing by stochastic slip lines method", Computers and Geotechnics, 91, pp. 203-217 (2017).

47. Bishop, A.W. "The principle of effective stress", Teknisk Ukeblad, 106(39), pp. 859-863 (1959).

48. Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E., and Clifton, A.W. "Model for the prediction of shear strength with respect to soil suction", Canadian Geotechnical Journal, 33(3), pp. 379-392 (1996).

49. Khalili, N. and Khabbaz M.H. "A unique relationship for for the determination of the shear strength of unsaturated soils", Geotechnique, 48(5), pp. 681-687 (1998).

50. Khalili, N., Geiser, F., and Blight, G.E. "Effective stress in unsaturated soils: Review with new evidence", International Journal of Geomechanics, 4(2), pp. 115-126 (2004).

51. Vaunat, J. and Casini, F. "A procedure for the direct determination of Bishop's parameter from changes in pore size distribution", Geotechnique, 67(7), pp. 631- 636 (2017).

52. Van Genuchten, M. Th. "A closed-form equation for predicting the hydraulic conductivity of unsaturated soils", Soil Science Society of America Journal, 44(5), pp. 892-898 (1980).

53. Anvar, S.A. and Ghahramani, A. "Equilibrium equations, on zero extension lines and their application to soil engineering", Iranian Journal of Science and Technology, 21(1), pp. 11-34 (1997).

54. Fredlund, D.G. and Rahardjo, H., Soil Mechanics for Unsaturated Soils, John Wiley, New York, US (1993).

55. Johari, A. and Amjadi, A.H., Stochastic Analysis of Settlement Rate in Unsaturated Soils, Geo-Risk, 2017, pp. 631-639 (2013).

56. Jahanandish, M. "Development of a zero extension line method for axially symmetric problems in soil mechanics", Scientia Iranica, 10(2), pp. 203-210 (2003).

57. Veiskarami, M., Jahanandish, M., and Ghahramani, A. "Stress level based bearing capacity of foundations: Verification of results with 131 case studies", KSCE Journal of Civil Engineering, 16(5), pp. 723-732 (2012).

2. Meyerhof, G.G. "Some recent research on the bearing capacity of foundations", Canadian Geotechnical Journal, 1(1), pp. 16-26 (1963).

3. Brinch Hansen, J. "A revised and extended formula for bearing capacity", Danish Geotechnical Institute, Bulletin No. 28, Copenhagen (1970).

4. Vesic, A.S. "Analysis of ultimate loads of shallow foundations", Journal of the Soil Mechanics and Foundations, 99(1), pp. 45-73 (1973).

5. Michalowski, R.L. "An estimate of the influence of the soil weight on the bearing capacity using limit analysis", Soils and Foundations, 37(4), pp. 57-64 (1997).

6. Kumar, J. "N for rough strip footing using the method of characteristics", Canadian Geotechnical Journal, 40(3), pp. 669-674 (2003).

7. Kumar, J. "The variation of N with footing roughness using the method of characteristics", International Journal for Numerical and Analytical Methods in Geomechanics, 33(2), pp. 275-284 (2009).

8. Jahanandish, J., Habibagahi, G., and Veiskarami, M. "Bearing capacity factor, N , for unsaturated soils by ZEL method", Acta Geotechnica, 5(3), pp. 177-188 (2010).

9. Veiskarami, M. and Habibagahi, G. "Foundations bearing capacity subjected to seepage by the kinematic approach of the limit analysis", Frontiers of Structural and Civil Engineering, 7(3), pp. 446-455 (2013).

10. Veiskarami, M., Kumar, J., and Valikhah, F. "Effect of the flow rule on the bearing capacity of strip foundations on sand by the upper-bound limit analysis and slip lines", International Journal of Geomechanics, 14(3), 04014008 (2014).

11. Veiskarami, M., Jamshidi Chenari, R., and Jameei, A.A. "Bearing capacity of strip footings on anisotropic soils by the finite elements and linear programming", International Journal of Geomechanics, 17(12), 04017119 (2017).

12. Coulomb, C.A. "Essay on maximums and minimums of rules to some static problems relating to architecture", al Academie Royale des Sciences Memoires de Mathematique et de Physique, Presentes, par Divers Scavans, 7, pp. 343-382 (1973).

13. Rankine, W.J.W. "On the stability of loose earth", Philosophical Transactions of the Royal Society of London, 147, pp. 9-27 (1857).

14. Rochette, P.A. "Earth pressures on structures and mobilized Shear resistance", 15th Can. Soil Mech. Conf, Tech. Mem., 73(3) p. 59 (1961).

15. Rowe, P.W. and Peaker, K. "Passive earth pressure measurements", Geotechnique, 15(1), p. 57 (1965).

16. James, R.G. and Bransby, D.L. "Experimental and theoretical investigations of a passive earth pressure problem", Geotechnique, 20(1), pp. 17-37 (1970).

17. Veiskarami, M., Jamshidi Chenari, R., and Jameei, A.A. "A study on the static and seismic earth pressure problems in anisotropic granular media", Geotechnical and Geological Engineering, 37(3), pp. 1987-2005 (2019).

18. Jamshidi Chenari, R., Kamyab, H., and Izadi, A. "Continuous slip surface method for stability analysis of heterogeneous vertical trenches", Scientia Iranica, 27(6), pp. 2657-2668 (2020).

19. Bishop, A.W. "The use of slip circle in the stability analysis of earth slopes", Geotechnique, 5(1), pp. 7-17 (1955).

20. Fang, H.Y. and Hirst, T.J. "Application of plasticity theory to slope stability problems", Highway Research Record, 323, pp. 26-38 (1970).

21. Lo, K.Y. and Lee, C.F. "Stress analysis and slope stability in strain-softening materials", Geotechnique, 23(1), pp. 1-11 (1973).

22. Lehner, F.K. and Schopfer, M.P.J. "Slope stability and exact solutions for cohesive critical Coulomb wedges from Mohr diagrams", Journal of Structural Geology, 116, pp. 234-240 (2018).

23. Xu, H., Ren, X., Chen, J.N., Liu, C.N., Xia, L., and Liu, Y.W. "Centrifuge model tests of geogrid-reinforced slope supporting a high embankment", Geosynthetics International, 26(6), pp. 629- 640 (2019).

24. Fredlund, D.G. "The stability of slopes with negative pore-water pressures", The Ian Boyd Donald Symposium on Modern Developments in Geomechanics, 3168 (1995).

25. Johari, A., Hooshmand Nejad, A., and Mousavi, S. "Probabilistic model of unsaturated slope stability considering the uncertainties of soil-water characteristic curve", Scientia Iranica, 25(4), pp. 2039-2050 (2018).

26. Rahardjo, H., Satyanaga, A., Wang, C.L., Wong, J.L.H., and Lim, V.H. "Effects of unsaturated properties on stability of slope covered with Caesalpinia crista in Singapore", Environmental Geotechnics, 7(6), pp. 393-403 (2018).

27. Satyanaga, A. and Rahardjo, H. "Stability of unsaturated soil slopes covered with Melastoma malabathricum in Singapore", Proceedings of the Institution of Civil Engineers-Geotechnical Engineering, 172(6), pp. 530-540 (2019).

28. Li, Z., He, Y., Li, H., and Wang, Y. "Antecedent rainfall induced shallow landslide-A case study of Yunnan landslide, China", Scientia Iranica, 26(1), pp. 202-212 (2019).

29. Ching, R.K.H., Sweeney, D.J., and Fredlund, D.G. "Increase in factor of safety due to soil suction for two Hong Kong slopes", Proc., 4th Int. Symp. on Landslides (1984).

30. Fellenius, W. "Calculation of the stability of earth dams", Trans. 2nd Congr. Large Dams, Washington (1936).

31. Janbu, N. "Earth pressure and bearing capacity calculations by generalized procedure of slices", Proc. 4 ICSMFE, 2, London, UK, pp. 207-212 (1957).

32. Morgenstern, N.R. and Price, V.E. "The analysis of the stability of general slip surfaces", Geotechnique, 15(1), pp. 79-93 (1965).

33. Gvozdev, A.A. "Determination of the collapse load for statically indeterminate structures subjected to plastic deformations", Proc. Conf. Plastic Deformations, Moscow, Russia (1938).

34. Drucker, D.C., Prager, W., and Greenberg, H.J. "Extended limit design theorems for continuous media", Quarterly of Applied Mathematics, 9(4) pp. 381-389 (1952).

35. Chen, W.F., Limit Analysis and Soil Plasticity, Amsterdam, the Netherlands, Elsevier (1975).

36. Michalowski, R.L., "Stability charts for uniform slopes", Journal of Geotechnical and Geoenvironmental Engineering, 128(4), pp. 351-355 (2002).

37. Yuan, S. and Du, J. "A lower-bound formulation for unsaturated soils", Geotechnique, 70(2), pp. 123-137 (2020).

38. Sokolovskii, V.V., Statics of Soil Media, Butterworths Scientific Publications, London, UK (1960).

39. Harr, M.E., Foundations of Theoretical Soil Mechanics, McGraw-Hill (1966).

40. Bolton, M.D. and Lau, C.K. "Vertical bearing capacity factors for circular and strip footings on Mohr- Coulomb soil", Canadian Geotechnical Journal, 30(6), pp. 1024-1033 (1993).

41. Kumar, J. and Mohan Rao, V.B.K. "Seismic bearing capacity of foundations on slopes", Geotechnique, 53(3), pp. 347-361 (2003).

42. Martin, C.M. "Analysis of bearing capacity", OUEL Report, 2261(03), Oxford, UK (2003).

43. Martin, C.M. "Exact bearing capacity calculations using the method of characteristics", Proc. IACMAG. Turin, 11, Turin, Italy, pp. 441-450 (2005).

44. Thanh, Vo. and Russell, A.R. "Slip line theory applied to a retaining wall-unsaturated soil interaction problem", Computers and Geotechnics, 55, pp. 416- 428 (2014).

45. Veiskarami, M. and Zanj, A. "Stability of sheet-pile walls subjected to seepage flow by slip lines and finite elements", Geotechnique, 64(10), pp. 759-775 (2014).

46. Johari, A., Hosseini, S.M., and Keshavarz, A. "Reliability analysis of seismic bearing capacity of strip footing by stochastic slip lines method", Computers and Geotechnics, 91, pp. 203-217 (2017).

47. Bishop, A.W. "The principle of effective stress", Teknisk Ukeblad, 106(39), pp. 859-863 (1959).

48. Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E., and Clifton, A.W. "Model for the prediction of shear strength with respect to soil suction", Canadian Geotechnical Journal, 33(3), pp. 379-392 (1996).

49. Khalili, N. and Khabbaz M.H. "A unique relationship for for the determination of the shear strength of unsaturated soils", Geotechnique, 48(5), pp. 681-687 (1998).

50. Khalili, N., Geiser, F., and Blight, G.E. "Effective stress in unsaturated soils: Review with new evidence", International Journal of Geomechanics, 4(2), pp. 115-126 (2004).

51. Vaunat, J. and Casini, F. "A procedure for the direct determination of Bishop's parameter from changes in pore size distribution", Geotechnique, 67(7), pp. 631- 636 (2017).

52. Van Genuchten, M. Th. "A closed-form equation for predicting the hydraulic conductivity of unsaturated soils", Soil Science Society of America Journal, 44(5), pp. 892-898 (1980).

53. Anvar, S.A. and Ghahramani, A. "Equilibrium equations, on zero extension lines and their application to soil engineering", Iranian Journal of Science and Technology, 21(1), pp. 11-34 (1997).

54. Fredlund, D.G. and Rahardjo, H., Soil Mechanics for Unsaturated Soils, John Wiley, New York, US (1993).

55. Johari, A. and Amjadi, A.H., Stochastic Analysis of Settlement Rate in Unsaturated Soils, Geo-Risk, 2017, pp. 631-639 (2013).

56. Jahanandish, M. "Development of a zero extension line method for axially symmetric problems in soil mechanics", Scientia Iranica, 10(2), pp. 203-210 (2003).

57. Veiskarami, M., Jahanandish, M., and Ghahramani, A. "Stress level based bearing capacity of foundations: Verification of results with 131 case studies", KSCE Journal of Civil Engineering, 16(5), pp. 723-732 (2012).

Transactions on Civil Engineering (A)

November and December 2021Pages 3058-3071