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Seismic Bearing Capacity of Shallow Foundations Placed on an Anisotropic and Nonhomogeneous Inclined Ground

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Abstract

In the present study, the limit equilibrium method combined with the pseudo-static seismic loading approach and applying the simplified Coulomb failure mechanism were employed for calculating the bearing capacity of nonhomogeneous and anisotropic soils on slopes. It was assumed that the cohesion coefficient was nonhomogeneous and anisotropic and anisotropy effect was ignored for the friction angle. For estimating optimal bearing capacity values, the particle swarm optimization algorithm was used. Comparing the results of previous researchers with those of the present study for isotropic and homogeneous soils indicated that the present solution provided acceptable values for the bearing capacity of shallow foundations. The effect of anisotropy ratio and the nonhomogeneous coefficient on the seismic bearing capacity was evaluated and found that decreasing the anisotropy ratio and increasing the nonhomogeneous coefficient cause an increase in the seismic bearing capacity. Furthermore, the results showed that the depth of the failure zone decreases with increasing the nonhomogeneous coefficient, the anisotropy ratio, and the seismic acceleration coefficient, while the depth of the failure zone increases with an increase in the slope inclination.

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Abbreviations

c i :

The cohesion corresponding to an inclination i

c v :

The cohesion coefficients in the vertical direction

c h :

The cohesion coefficients in the horizontal direction

K :

The anisotropy coefficient

υ :

The nonhomogeneous coefficient

i :

The angle between the horizontal direction and the maximum principal stress

N c :

Bearing capacity factor

N γ :

Bearing capacity factor

γ :

The unit weight of the soil

c h0 :

The cohesion coefficient in the horizontal direction in h = 0

λ :

The variation of the cohesion coefficient with depth

B 0 :

Width of the footing

P U :

The ultimate vertical load on the foundation

β :

The slope inclination

q ult :

The ultimate bearing capacity of the strip footing

α A :

Slip surface angle in the active zone

α B :

Slip surface angle in the passive zone

φ :

Angle of internal friction of soil

δ :

The friction angle along the surface between the active and passive zones

P a :

The active thrust

P p :

The passive resistance

k v :

The vertical seismic acceleration coefficient

k h :

The horizontal seismic acceleration coefficient

h :

The depth of failure zone

W A :

Weight of triangular wedge AEC

W B :

Weight of triangular wedge BEC

C AE, C EB and C CE :

The cohesion coefficient components

References

  1. Budhu M, Al-Karni A (1993) Seismic bearing capacity of soils. Geotechnique 43(1):181–187

    Article  Google Scholar 

  2. Dormieux L, Pecker A (1995) Seismic bearing capacity of foundation on cohesionless soil. J Geotech Eng 121(3):300–303

    Article  Google Scholar 

  3. Ghosh S, Debnath L (2017) Seismic bearing capacity of shallow strip footing with coulomb failure mechanism using limit equilibrium method. Geotech Geol Eng 35(6):2647–2661

    Article  Google Scholar 

  4. Kumar J (2003) N γ for rough strip footing using the method of characteristics. Can Geotech J 40(3):669–674

    Article  Google Scholar 

  5. Paolucci R, Pecker A (1997) Seismic bearing capacity of shallow strip foundations on dry soils. Soils Found 37(3):95–105

    Article  Google Scholar 

  6. Richards R Jr, Elms D, Budhu M (1993) Seismic bearing capacity and settlements of foundations. J Geotech Eng 119(4):662–674

    Article  Google Scholar 

  7. Sarma S, Iossifelis I (1990) Seismic bearing capacity factors of shallow strip footings. Geotechnique 40(2):265–273

    Article  Google Scholar 

  8. Soubra A-H (1997) Seismic bearing capacity of shallow strip footings in seismic conditions. Proc Inst Civ Eng Geotech Eng 125(4):230–241

    Article  Google Scholar 

  9. Soubra A-H (1999) Upper-bound solutions for bearing capacity of foundations. J Geotech Geoenviron Eng 125(1):59–68

    Article  Google Scholar 

  10. Zhu D (2000) The least upper-bound solutions for bearing capacity factor Nγ. Soils Found 40(1):123–129

    Article  Google Scholar 

  11. Halder K, Chakraborty D, Kumar Dash S (2019) Bearing capacity of a strip footing situated on soil slope using a non-associated flow rule in lower bound limit analysis. Int J Geotech Eng 13(2):103–111

    Article  Google Scholar 

  12. Hansen JB (1970) A revised and extended formula for bearing capacity

  13. Meyerhof G (1957) The ultimate bearing capacity of foundations on slopes. In: Proc., 4th int. conf. on soil mechanics and foundation engineering, pp 384–386

  14. Azzouz AS, Baligh MM (1983) Loaded areas on cohesive slopes. J Geotech Eng 109(5):724–729

    Article  Google Scholar 

  15. Castelli F, Motta E (2010) Bearing capacity of strip footings near slopes. Geotech Geol Eng 28(2):187–198

    Article  Google Scholar 

  16. Chen C-f, Dong W-z, Tang Y-z (2007) Seismic ultimate bearing capacity of strip footings on slope. J Cent South Univ Technol 14(5):730

    Article  Google Scholar 

  17. Choudhury D, Subba Rao K (2006) Seismic bearing capacity of shallow strip footings embedded in slope. Int J Geomech 6(3):176–184

    Article  Google Scholar 

  18. Kovalev I (1964) De la resistance ultime des fondations limitees par un talus. Traduction du russe Extrait du recueil des travaux de LIIZLT, fascicule 225

  19. Kumar J, Kumar N (2003) Seismic bearing capacity of rough footings on slopes using limit equilibrium. Geotechnique 53(3):363–369

    Article  Google Scholar 

  20. Mizuno T, TOKUMITSU Y, Kawakami H, (1960) On the bearing capacity of a slope of cohesionless soil. Soils Found 1(2):30–37

    Article  Google Scholar 

  21. Narita K, Yamaguchi H (1990) Bearing capacity analysis of foudations on slopes by use of log-spiral sliding surfaces. Soils Found 30(3):144–152

    Article  Google Scholar 

  22. Sarma S, Chen Y (1996) Bearing capacity of strip footings near sloping ground during earthquakes. In: Proceedings of the 11th world conference on earthquake engineering, Acapulco

  23. Kumar J, Mohan Rao V (2003) Seismic bearing capacity of foundations on slopes. Geotechnique 53(3):347–361

    Article  Google Scholar 

  24. Graham J, Andrews M, Shields D (1988) Stress characteristics for shallow footings in cohesionless slopes. Can Geotech J 25(2):238–249

    Article  Google Scholar 

  25. Giroud J (1971) Force portante d'une fondation sur une pente. ANN ITBTP-SERIE: THEORIES ET METHODES DE CALCUL NO 142 (283/284)

  26. Andrews M (1986) Computation of bearing capacity coefficients for shallow footings on cohesionless slopes using stress characteristics

  27. Kumar J, Chakraborty D (2013) Seismic bearing capacity of foundations on cohesionless slopes. J Geotech Geoenviron Eng 139(11):1986–1993

    Article  Google Scholar 

  28. Chakraborty D, Kumar J (2015) Seismic bearing capacity of shallow embedded foundations on a sloping ground surface. Int J Geomech 15(1):04014035

    Article  Google Scholar 

  29. Chakraborty D, Kumar J (2013) Bearing capacity of foundations on slopes. Geomech Geoeng 8(4):274–285

    Article  Google Scholar 

  30. Mofidi Rouchi J, Farzaneh O, Askari F (2014) Bearing capacity of strip footings near slopes using lower bound limit analysis. Civ Eng Infrastruct J 47(1):89–109

    Google Scholar 

  31. Shiau J, Merifield R, Lyamin A, Sloan S (2011) Undrained stability of footings on slopes. Int J Geomech 11(5):381–390

    Article  Google Scholar 

  32. Chakraborty D, Mahesh Y (2016) Seismic bearing capacity factors for strip footings on an embankment by using lower-bound limit analysis. Int J Geomech 16(3):06015008

    Article  Google Scholar 

  33. Askari F, Farzaneh O (2003) Upper-bound solution for seismic bearing capacity of shallow foundations near slopes. Geotechnique 53(8):697–702

    Article  Google Scholar 

  34. Georgiadis K (2010) Undrained bearing capacity of strip footings on slopes. J Geotech Geoenviron Eng 136(5):677–685

    Article  Google Scholar 

  35. Kumar J, Ghosh P (2006) Seismic bearing capacity for embedded footings on sloping ground. Geotechnique 56(2):133–140

    Article  Google Scholar 

  36. KuSakabe O, Kimura T, Yamaguchi H (1981) Bearing capacity of slopes under strip loads on the top surfaces. Soils Found 21(4):29–40

    Article  Google Scholar 

  37. Leshchinsky B, Xie Y (2017) Bearing capacity for spread footings placed near c′-ϕ′ slopes. J Geotech Geoenviron Eng 143(1):06016020

    Article  Google Scholar 

  38. Sawada T, NomachiChen SGW-F (1994) Seismic bearing capacity of a mounded foundation near a down-hill slope by pseudo-static analysis. Soils Found 34(1):11–17

    Article  Google Scholar 

  39. Yamamoto K (2010) Seismic bearing capacity of shallow foundations near slopes using the upper-bound method. Int J Geotech Eng 4(2):255–267

    Article  Google Scholar 

  40. Bishop AW (1966) The strength of soils as engineering materials. Geotechnique 16(2):91–130

    Article  Google Scholar 

  41. Casagrande A (1944) Shear failure of anisotropic materials. Proc Boston Soc Civ Eng 31:74–87

    Google Scholar 

  42. Livneh M, Komornik A (1967) Anisotropic strength of compacted clays. In: Asian conf soil mech and fdn e proc/is

  43. Skempton A (1948) Vane tests in the alluvial plain of the River Forth near Grangemouth. Geotechnique 1(2):111–124

    Article  Google Scholar 

  44. Pakdel P, Jamshidi Chenari R, Veiskarami M (2019) Seismic bearing capacity of shallow foundations rested on anisotropic deposits. Int J Geotech Eng 15:1–12

    Google Scholar 

  45. Al-Shamrani MA, Moghal AAB (2012) Upper bound solutions for bearing capacity of footings on anisotropic cohesive soils. In: GeoCongress 2012: state of the art and practice in geotechnical engineering, pp 1066–1075

  46. Al-Shamrani MA (2005) Upper-bound solutions for bearing capacity of strip footings over anisotropic nonhomogeneous clays. J Jpn Geotech Soc Soils Found 45(1):109–124

    Google Scholar 

  47. Davis E, Booker J (1973) The effect of increasing strength with depth on the bearing capacity of clays. Geotechnique 23(4):551–563

    Article  Google Scholar 

  48. Davis EH, Christian JT (1970) Bearing capacity of anisotropic cohesive soil

  49. Gourvenec S, Randolph M (2003) Effect of strength non-homogeneity on the shape of failure envelopes for combined loading of strip and circular foundations on clay. Géotechnique 53(6):575–586

    Article  Google Scholar 

  50. Izadi A, Nazemi Sabet Soumehsaraei M, Jamshidi Chenari R, Ghorbani A (2019) Pseudo-static bearing capacity of shallow foundations on heterogeneous marine deposits using limit equilibrium method. Mar Georesour Geotechnol 37(10):1163–1174

    Article  Google Scholar 

  51. Menzies B (1976) An approximate correction for the influence of strength anisotropy on conventional shear vane measurements used to predict field bearing capacity. Geotechnique 26(4):631–634

    Article  Google Scholar 

  52. Raymond GP (1967) The bearing capacity of large footings and embankments on clays. Geotechnique 17(1):1–10

    Article  Google Scholar 

  53. Reddy AS, Rao KV (1981) Bearing capacity of strip footing on anisotropic and nonhomogeneous clays. Soils Found 21(1):1–6

    Article  Google Scholar 

  54. Reddy AS, Sriniuasan R (1970) Bearing capacity of footings on anisotropic soils. J Soil Mech Found Div 96(6):1967–1986

    Article  Google Scholar 

  55. Reddy AS, Srinivasan R (1967) Bearing capacity of footings on layered clays. J Soil Mech Found Div 93(2):83–99

    Article  Google Scholar 

  56. Reddy AS, Srinivasan R (1971) Bearing capacity of footings on clays. Soils Found 11(3):51–64

    Article  Google Scholar 

  57. Salencon J (1974) Bearing capacity of a footing on a φ= 0 soil with linearly varying shear strength. Geotechnique 24(3):443–446

    Article  Google Scholar 

  58. Yang X-L, Du D-C (2016) Upper bound analysis for bearing capacity of nonhomogeneous and anisotropic clay foundation. KSCE J Civ Eng 20(7):2702–2710

    Article  Google Scholar 

  59. Chen W-F (2013) Limit analysis and soil plasticity. Elsevier

    Google Scholar 

  60. Skempton A (1951) The bearing capacity of clays. In: Selected papers on soil mechanics, pp 50–59

  61. Sreenivasulu V, Ranganatham B (1971) Bearing Capacity of anisotripy nonhomogenous medium under φ= 0 condition. Soils Found 11(2):17–27

    Article  Google Scholar 

  62. Meyerhof G (1978) Bearing capacity of anisotropic cohesionless soils. Can Geotech J 15(4):592–595

    Article  Google Scholar 

  63. Reddy AS, Rao KV (1982) Bearing capacity of strip footing on c-ψ soils exhibiting anisotripy and nonhomogeneity in cohesion. Soils Found 22(1):49–60

    Article  Google Scholar 

  64. Halder K, Chakraborty D (2019) Seismic bearing capacity of a strip footing over an embankment of anisotropic clay. Front Built Environ 5:134

    Article  Google Scholar 

  65. Ghazavi M, Eghbali AH (2008) A simple limit equilibrium approach for calculation of ultimate bearing capacity of shallow foundations on two-layered granular soils. Geotech Geol Eng 26(5):535–542

    Article  Google Scholar 

  66. Livneh M, Greenstein J (1972) The bearing capacity of footings on nonhomogeneous clays. Bruner Institute of Transportation, Technion Research and Development Foundation Limited

    Google Scholar 

  67. Tant K, Craig W (1995) Bearing capacity of circular foundations on soft clay of strength increasing with depth. Soils Found 35(4):21–35

    Article  Google Scholar 

  68. Wood DM (2003) Geotechnical modelling, vol 1. CRC Press

    Google Scholar 

  69. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN'95-international conference on neural networks. IEEE, pp 1942–1948

  70. Debnath L, Ghosh S (2018) Pseudostatic analysis of shallow strip footing resting on two-layered soil. Int J Geomech 18(3):04017161

    Article  Google Scholar 

  71. Debnath L, Ghosh S (2019) Pseudo-static bearing capacity analysis of shallow strip footing over two-layered soil considering punching shear failure. Geotech Geol Eng 37(5):3749–3770

    Article  Google Scholar 

  72. Vesic AS (1973) Analysis of ultimate loads of shallow foundations. J Soil Mech Found Div 99(1):45–73

    Article  Google Scholar 

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Appendix: Analytical Functions of Eqs. (18 and 19)

Appendix: Analytical Functions of Eqs. (18 and 19)

$$ a = \left( {\frac{{\left( {1 - k_{{\text{v}}} } \right)\sin \left( {\alpha_{{\text{A}}} - \varphi } \right) + k_{{\text{h}}} \cos \left( {\alpha_{{\text{A}}} - \varphi } \right)}}{{\cos \left( {\alpha_{{\text{A}}} - \varphi - \delta } \right)}}} \right) $$
(23)
$$ \begin{aligned} b & = \tan \alpha_{{\text{A}}} \left[ {\left( {\frac{{\tan \alpha_{{\text{A}}} }}{{\tan \alpha_{{\text{B}}} + \tan \beta }}} \right)\left( {\frac{{\left( {1 - k_{{\text{v}}} } \right)\sin \left( {\alpha_{{\text{B}}} + \varphi } \right) - k_{{\text{h}}} \cos \left( {\alpha_{{\text{B}}} + \varphi } \right)}}{{\cos \left( {\alpha_{{\text{B}}} + \varphi + \delta } \right)}}} \right)} \right. \\ & \quad \left. { - \left( {\frac{{\left( {1 - k_{{\text{v}}} } \right)\sin \left( {\alpha_{{\text{A}}} - \varphi } \right) + k_{{\text{h}}} \cos \left( {\alpha_{{\text{A}}} - \varphi } \right)}}{{\cos \left( {\alpha_{{\text{A}}} - \varphi - \delta } \right)}}} \right)} \right] \\ \end{aligned} $$
(24)
$$ \begin{aligned} & e = \frac{1}{K}\left( {1 + \left( {K - 1} \right)\sin^{2} \alpha_{{\text{B}}} } \right)\left( {\frac{{\tan \alpha_{{\text{A}}} \tan \alpha_{{\text{B}}} }}{{\tan \alpha_{{\text{B}}} + \tan \beta }}} \right)\left( {\frac{{\sin \left( {\alpha_{{\text{B}}} + \varphi } \right) + \cot \alpha_{{\text{B}}} \cos \left( {\alpha_{{\text{B}}} + \varphi } \right)}}{{\cos \left( {\alpha_{{\text{B}}} + \varphi + \delta } \right)}}} \right) \\ & \quad + \left( {\tan \alpha_{{\text{A}}} } \right)\frac{{\sin \left( {\alpha_{{\text{B}}} + \varphi } \right)}}{{\cos \left( {\alpha_{{\text{B}}} + \varphi + \delta } \right)}} + \left( {\tan \alpha_{{\text{A}}} } \right)\frac{{\sin \left( {\alpha_{{\text{A}}} - \varphi } \right)}}{{\cos \left( {\alpha_{{\text{A}}} - \varphi - \delta } \right)}} \\ & \quad + \frac{1}{K}\left( {1 + \left( {K - 1} \right)\sin^{2} \alpha_{{\text{A}}} } \right)\left( {\tan \alpha_{{\text{A}}} } \right)\left( {\frac{{\sin \left( {\alpha_{{\text{A}}} - \varphi } \right) + \cot \alpha_{{\text{A}}} \cos \left( {\alpha_{{\text{A}}} - \varphi } \right)}}{{\cos \left( {\alpha_{{\text{A}}} - \varphi - \delta } \right)}}} \right) \\ \end{aligned} $$
(25)
$$ \begin{aligned} f & = \left( \frac{1}{K} \right)\left[ {\left( {\tan \alpha_{{\text{A}}} } \right)^{2} \left( {\frac{{\sin \left( {\alpha_{{\text{A}}} - \varphi } \right) + 0.5\cot \alpha_{{\text{A}}} \cos \left( {\alpha_{{\text{A}}} - \varphi } \right)}}{{\cos \left( {\alpha_{{\text{A}}} - \varphi - \delta } \right)}}} \right)} \right. \\ & \quad + \left( {\frac{{\tan \alpha_{{\text{A}}} \tan \alpha_{{\text{B}}} }}{{\tan \alpha_{{\text{B}}} + \tan \beta }}} \right)^{2} \left( {\frac{{0.5\sin \left( {\alpha_{{\text{B}}} + \varphi } \right) + 0.5\cot \alpha_{{\text{B}}} \cos \left( {\alpha_{{\text{B}}} + \varphi } \right)}}{{\cos \left( {\alpha_{{\text{B}}} + \varphi + \delta } \right)}}} \right) \\ & \quad \left. { + \left( {\tan \alpha_{{\text{A}}} } \right)^{2} \left( {\frac{{0.5\sin \left( {\alpha_{{\text{B}}} + \varphi } \right)}}{{\cos \left( {\alpha_{{\text{B}}} + \varphi + \delta } \right)}}} \right)} \right] \\ \end{aligned} $$
(26)

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Haghsheno, H., Arabani, M. Seismic Bearing Capacity of Shallow Foundations Placed on an Anisotropic and Nonhomogeneous Inclined Ground. Indian Geotech J 51, 1319–1337 (2021). https://doi.org/10.1007/s40098-021-00534-7

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