Abstract
The undrained ultimate bearing capacity of rigid strip footings near slopes is one of important problems in practical foundation engineering in soft clay. To solve the bearing capacity conveniently and easily, a calculation method for the undrained bearing capacity is proposed using the upper bound limit analysis. A bilateral failure mechanism is provided and involved in the analysis, and two failure modes including slope face failure and below-toe failure are pointed out considering the location relationship between the critical slip surface and the slope toe. The upper bound solution of the undrained bearing capacity based on the failure mechanism is carried specifically out via the optimization algorithm. Analysis of some typical examples shows that the results calculated by the proposed method agree well with those obtained using other analytical and numerical methods. The maximum error is about 10%. Cohesion of slope soil, slope angle, horizontal setback distance of strip footing from the slope crest, slope height, surcharge on the slope top level, and the footing depth have great influence on the undrained bearing capacity.
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Abbreviations
- b :
-
Width of the strip footing
- c :
-
Cohesion of the soil, where the subscript u denotes undrained shear strength
- φ :
-
Internal friction angle of the soil
- D :
-
Internal energy dissipation rate of the slope soil, where the subscript i denotes the total energy dissipation rate of admissible failure mechanism
- W :
-
Work rates of external forces, where the subscripts g, qu, and σ denote gravity of the slope soil, locally compressive stress on the bottom of the footing, and surcharge on the top surface of the slope, respectively
- H :
-
Height of the slope
- h :
-
Vertical depth of the intersection K between critical slip surface and slope face to the slope crest
- λ :
-
Ratio of the distance from the footing to slope crest over the footing width
- α 1 :
-
Angle between lines AC and AB in the zone ABC under the footing
- α 2 :
-
Angle between lines BC and AB in the zone ABC under the footing
- β :
-
Dip angle of the slope face
- γ :
-
Unit weight of the soil
- θ :
-
Rotation angle of any point on log-spiral curves ACD or BCF from lines AC or BC
- S :
-
Horizontal distance from the toe of the slope P to the intersection K
- v :
-
Velocity of the kinematic soil body, where the subscripts D, F, x, y, C1 and C2 denote point D, point F, horizontal and vertical direction at point C, and direction perpendicular to line AC and BC at point C, respectively
- q u :
-
Ultimate bearing capacity of the footing near the slope
- σ :
-
Surcharge on the slope top level
- d :
-
Depth of the footing with respect to the top surface of the slope
- N c :
-
Undrained bearing capacity factor
References
Shields D, Chandler N, Garnier J (1990) Bearing capacity of foundations in slopes. J Geotech Eng 116(3):528–537. https://doi.org/10.1061/(ASCE)0733-9410(1990)116:3(528)
Meyerhof GG (1957) The ultimate bearing capacity of foundations on slopes. In: Proc. 4th int. conf. on soil mechanics and foundation engineering, London
Keskin MS, Laman M (2013) Model studies of bearing capacity of strip footing on sand slope. KSCE J Civ Eng 17(4):699–711. https://doi.org/10.1007/s12205-013-0406-x
Sokolovski VV (1960) Statics of granular media. Butterworth Scientific Publications, London
Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York
de Buhan P, Garnier D (1998) Three dimensional bearing capacity analysis of a foundation near a slope. Soils Found 38(3):153–163. https://doi.org/10.3208/sandf.38.3_153
Chen WF (1975) Limit analysis and soil plasticity. Elsevier, Amsterdam
Davis EH, Booker JR (1973) Some adaptations of classical plasticity theory for soil stability problems. In: Proceedings of the symposium on the role of plasticity in soil mechanics, Cambridge
Drucker DC, Prager W, Greenberg HJ (1952) Extended limit design theorems for continuous media. Q Appl Math 9:381–389. https://doi.org/10.1090/qam/45573
Hansen JB (1961) A general formula for bearing capacity. Danish Geotech Instit Bull 11:38–46
Meyerhof GG (1963) Some recent research on the bearing capacity of foundations. Can Geotech J 1(1):16–26
Vesic AS (1975) Bearing capacity of shallow foundations, foundation engineering handbook. Van Nostrand Reinhold, New York
Kusakabe O, Kimura T, Yamaguchi H (1981) Bearing capacity of slopes under strip loads on the top surfaces. Soils Found 21(4):29–40. https://doi.org/10.3208/sandf1972.21.4_29
Michalowski RL (1989) Three-dimensional analysis of locally loaded slopes. Géotechnique 39(1):27–38. https://doi.org/10.1680/geot.1989.39.1.27
Azzouz AS, Baligh MM (1983) Loaded areas on cohesive slopes. J Geotech Eng 109(5):724–729. https://doi.org/10.1061/(ASCE)0733-9410(1983)109:5(724)
Narita K, Yamaguchi H (1990) Bearing capacity analysis of foundations on slopes by use of log-spiral sliding surfaces. Soils Found 30(3):144–152. https://doi.org/10.3208/sandf1972.30.3_144
Kumar J, Mohan Rao VBK (2003) Seismic bearing capacity of foundations on slopes. Geotechnique 53(3):347–361. https://doi.org/10.1680/geot.2003.53.3.347
Kumar J, Ghosh P (2006) Seismic bearing capacity for embedded footings on sloping ground. Geotechnique 56(2):133–140. https://doi.org/10.1680/geot.2006.56.2.133
Castelli F, Motta E (2008) Bearing capacity of shallow foundations near slopes: static analysis. In: Proceedings of the second international british geotechnical association conference on foundations, ICOF 2008, Watford
Georgiadis K (2010) An upper bound solution for the undrained bearing capacity of strip footings at the top of a slope. Géotechnique 60(10):801–806. https://doi.org/10.1680/geot.09.T.016
Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 12:61–67. https://doi.org/10.1002/nag.1610120105
Sloan SW (1989) Upper bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 13:263–282. https://doi.org/10.1002/nag.1610130304
Sloan SW, Kleeman PW (1995) Upper bound limit analysis using discontinuous velocity fields. Comput Methods Appl Mech Eng 127(1):293–314. https://doi.org/10.1016/0045-7825(95)00868-1
Lyamin AV, Sloan SW (2002) Lower bound limit analysis using non-linear programming. Int J Numer Meth Eng 55(55):573–611. https://doi.org/10.1002/nme.511
Lyamin AV, Sloan SW (2002) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Meth Geomech 26:181–216. https://doi.org/10.1002/nag.198
Krabbenhoft K, Lyamin AV, Hjiaj M, Sloan SW (2005) A new discontinuous upper bound limit analysis formulation. Int J Numer Meth Eng 63(7):1069–1088. https://doi.org/10.1002/nme.1314
Georgiadis K (2010) The influence of load inclination on the undrained bearing capacity of strip footings on slopes. Comput Geotech 37(3):311–322. https://doi.org/10.1016/j.compgeo.2009.11.004
Georgiadis K (2010) Undrained bearing capacity of strip footings on slopes. J Geotech Geoenviron Eng 136(5):677–685. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000269
Chakraborty D, Kumar J (2013) Bearing capacity of foundations on slopes. Geomech Geoeng 8(4):274–285. https://doi.org/10.1080/17486025.2013.770172
Shiau JS, Merifield RS, Lyamin AV, Sloan SW (2011) Undrained stability of footings on slopes. Int J Geomech 11(5):381–390. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000092
Mofidi RJ, Farzaneh O, Askari F (2014) Undrained bearing capacity of strip footings near slopes using lower bound limit analysis. Civ Eng Infrastruct J 47(1):89–109. https://doi.org/10.7508/ceij.2014.01.007
Koushik H, Debarghya C, Sujit K (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. https://doi.org/10.1080/19386362.2017.1325119
Leshchinsky B (2015) Bearing capacity of footings placed adjacent to c′–ϕ′ slopes. J Geotech Geoenviron Eng 141(6):04015022. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001306
Leshchinsky B, Xie Y (2017) Bearing capacity for spread footings placed near c′–ϕ′ slopes. J Geotech Geoenviron Eng 143(1):06016020. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001578
Zhou H, Zheng G, Yin X, Jia X, Yang X (2018) The bearing capacity and failure mechanism of a vertically loaded strip footing placed on the top of slopes. Comput Geotech 94:12–21. https://doi.org/10.1016/j.compgeo.2017.08.009
Deutsche N (2006) DIN 4017–soil: calculation of design bearing capacity of soil beneath shallow foundations, Deutsche Norm, Berlin
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant Nos. 51578466 and 51278430) and the Program for New Century Excellent Talents in University (NCET-13-0976).
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Chen, T., Xiao, S. An Upper Bound Solution to Undrained Bearing Capacity of Rigid Strip Footings Near Slopes. Int J Civ Eng 18, 475–485 (2020). https://doi.org/10.1007/s40999-019-00463-w
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DOI: https://doi.org/10.1007/s40999-019-00463-w