A Review on Bearing Capacity Factor Nγ of Shallow Foundations with Different Shapes
Authors: S. Taghvamanesh, R. Ziaie Moayed
Abstract:
There are several methods for calculating the bearing capacity factors of foundations and retaining walls. In this paper, the bearing capacity factor Nγ (shape factor) for different types of foundation have been investigated. The formula for bearing capacity on c–φ–γ soil can still be expressed by Terzaghi’s equation except that the bearing capacity factor Nγ depends on the surcharge ratio, and friction angle φ. It is apparent that the value of Nγ increases irregularly with the friction angle of the subsoil, which leads to an excessive increment in Nγ of foundations with larger width. Also, the bearing capacity factor Nγ will significantly decrease with an increase in foundation`s width. It also should be highlighted that the effect of shape and dimension will be less noticeable with a decrease in the relative density of the soil. Hence, the bearing capacity factor Nγ relatively depends on foundation`s width, surcharge and roughness ratio. This paper presents the results of various studies conducted on the bearing capacity factor Nγ of: different types of shallow foundation and foundations with irregular geometry (ring footing, triangular footing, shell foundations and etc.) Further studies on the effect of bearing capacity factor Nγ on mat foundations and the characteristics of this factor with or without consideration for the presence of friction between soil and foundation are recommended.
Keywords: Bearing capacity, Bearing capacity factor, irregular foundation, shallow foundation.
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[1] B. M. Das, “Soil mechanics,” The Engineering Handbook, Second Edition. pp. 81-1-81–7, 2004, doi: 10.1201/9781315137322-4.
[2] M. Jahanandish, M. Veiskarami, and A. Ghahramani, “Effect of Stress Level on the Bearing Capacity Factor, Nγ, by the ZEL Method,” KSCE Journal of Civil Engineering, vol. 14, no. 5, pp. 709–723, 2010, doi: 10.1007/s12205-010-0866-1.
[3] D. Y. Zhu, C. F. Lee, and H. D. Jiang, “A numerical study of the bearing capacity factor Nγ,” Canadian Geotechnical Journal, vol. 38, no. 5, pp. 1090–1096, 2001, doi: 10.1139/cgj-38-5-1090.
[4] K. Terzaghi, “Theoretical Soil Mechanics,” Theoretical Soil Mechanics, 1943, doi: 10.1002/9780470172766.
[5] B. S. Frydman, “N-y,” vol. 123, no. January, pp. 20–29, 1997.
[6] P. Mrunal, J. N. Mandal, and D. M. Dewaikar, “Computation of bearing capacity factor Nγ,” International Journal of Geotechnical Engineering, vol. 8, no. 4, pp. 372–382, 2014, doi: 10.1179/1939787913Y.0000000021.
[7] E. G. Diaz-Segura, “Assessment of the range of variation of Nγ from 60 estimation methods for footings on sand,” Canadian Geotechnical Journal, vol. 50, no. 7, pp. 793–800, 2013, doi: 10.1139/cgj-2012-0426.
[8] D. Han, X. Xie, L. Zheng, and L. Huang, “The bearing capacity factor N γ of strip footings on c–ϕ–γ soil using the method of characteristics,” SpringerPlus, vol. 5, no. 1, 2016, doi: 10.1186/s40064-016-3084-6.
[9] J. Magar, A. Kudtarkar, J. Pachpohe, and P. Nagargoje, “Study and Analysis of Types of Foundation and Design Construction,” International Research Journal of Engineering and Technology, vol. 7, no. 8, pp. 3301–3307, 2020, doi: 10.5281/zenodo.3995061.
[10] F. Of, I. On, E. By, F. Zhu, J. I. Clark, and R. Phillips, “S c d s f r,” no. July, pp. 613–621, 2001.
[11] O. Sargazi and E. Seyedi Hosseininia, “Bearing capacity of ring footings on cohesionless soil under eccentric load,” Computers and Geotechnics, vol. 92, pp. 169–178, 2017, doi: 10.1016/j.compgeo.2017.08.003.
[12] H. Gholami and E. S. Hosseininia, “Bearing Capacity Factors of Ring Footings by Using the Method of Characteristics,” Geotechnical and Geological Engineering, vol. 35, no. 5, pp. 2137–2146, 2017, doi: 10.1007/s10706-017-0233-9.
[13] M. R. Taha and E. B. Altalhe, “Numerical and experimental evaluation of bearing capacity factor N of strip footing on sand slopes,” International Journal of Physical Sciences, vol. 8, no. 36, pp. 1807–1823, 2013, doi: 10.5897/IJPS2013.4000.
[14] S. Benmebarek, M. S. Remadna, N. Benmebarek, and L. Belounar, “Numerical evaluation of the bearing capacity factor Nγ’ of ring footings,” Computers and Geotechnics, vol. 44, pp. 132–138, 2012, doi: 10.1016/j.compgeo.2012.04.004.
[15] K. J. L. Stone and S. Kozman, “An Investigation of the Bearing Capacity of Irregular Shaped (Triangular) Footings,” no. January, 2007.
[16] J. Kumar, “Nγ for rough strip footing using the method of characteristics,” Canadian Geotechnical Journal, vol. 40, no. 3, pp. 669–674, 2003, doi: 10.1139/t03-009.
[17] J. Kumar and V. N. Khatri, “Effect of footing width on Ng,” Canadian Geotechnical Journal, vol. 45, no. 12, pp. 1673–1684, 2008, doi: 10.1139/T08-113.
[18] A, Naeim Abadi and S. Hanifi, "a numerical evaluation of bearing capacity factor for ring footings on layered soil."
[19] O. Clay, C. Yang, Z. Zhu, and Y. Xiao, “applied sciences Bearing Capacity of Ring Foundations on Sand,” 2020.
[20] A. B. Cerato and A. J. Lutenegger, “Bearing Capacity of Square and Circular Footings on a Finite Layer of Granular Soil Underlain by a Rigid Base,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 132, no. 11, pp. 1496–1501, 2006, doi: 10.1061/(asce)1090-0241(2006)132:11(1496).
[21] F. Yang and J. S. Yang, “A revised failure mechanism of strip footings for upper bound solution,” Electronic Journal of Geotechnical Engineering, vol. 13 F, 2008.
[22] Taylor, D.W. 1948. Fundamentals of soil mechanics. John Wiley and Sons, Inc., New York.
[23] Caquot, A., and Kérisel, J. 1953. Sur le terme de surface dans le calcul des fondations en milieu pulverulent. In Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, Zürich, Switzerland. Vol. 1, pp. 336–337.
[24] Biarez, J., Burel, M., and Wack, B. 1961. Contributions a l'etude de la force portante des foundations. In Proceedings of the 5th International Conference on Soil Mechanics and Foundation Engineering, Paris. Vol. 1, pp. 603–609.
[25] Feda, J. 1961. Research on the bearing capacity of loose soil. In Proceedings of the 5th International Conference on Soil Mechanics and Foundation Engineering, Paris. Vol. 1, pp. 635–642.
[26] Meyerhof, G.G. 1963. Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, 1(1): 16–26. doi:10.1139/t63-003.
[27] Hu, G.G.Y. 1964. Variable-factors theory of bearing capacity. Journal of the Soil Mechanics and Foundations Division, ASCE, 90(4): 85–95.
[28] Krizek, R.J. 1965. Approximation for Terzaghi's bearing capacity. Journal of the Soil Mechanics and Foundations Division, ASCE, 91(2): 1–3.
[29] Booker, J.R. 1969. Applications of the theory of plasticity to cohesive-frictional soils. Ph.D. thesis, University of Sydney, Sydney.
[30] Hansen, B., and Christensen, N. 1969. Discussion of “Theoretical bearing capacity of very shallow footings” by A.L. Larkin. Journal of the Soil Mechanics and Foundations Division, ASCE, 95(6): 1568–1572
[31] Muhs, H., and Weiss, K. 1969. The Influence of the load inclination on the bearing capacity of shallow footings. In Proceedings of the 7th International Conference on Soil Mechanics and Foundation Engineering, México City. Vol. 2, pp. 187–194.
[32] Abdul-Baki, A., and Beik, L.A. 1970. Bearing capacity of foundations on sand. Journal of the Soil Mechanics and Foundations Division, ASCE, 96(SM2): 545–559.
[33] Brinch-Hansen, J. 1970. A revised extended formula for bearing capacity. Danish Geotechnical Institute Bulletin, 28: 5–11.
[34] Davis, E.H., and Booker, J.R. 1971. The bearing capacity of strip footings from the standpoint of plasticity theory. In Proceedings of the 1st Australia – New Zealand Conference on Geomechanics, Melbourne. pp. 276–282.
[35] Chummar, A.V. 1972. Bearing capacity theory from experimental results. Journal of the Soil Mechanics and Foundations Division, ASCE, 98(12): 1311–1324.
[36] Vesic, A.S. 1973. Analysis of ultimate loads of shallow foundations. Journal of the Soil Mechanics and Foundations Division, ASCE, 99(1):4573. doi:10.1016/0148- 9062(74)90598-1.
[37] Chen, W.F. 1975. Limit analysis and soil plasticity. Elsevier, Amsterdam.
[38] Salençon, J., Florentin, P., and Gabriel, Y. 1976. Capacité portante globale de une fondation sur un sol nonhomogene. Géotechnique, 26(2): 351–370. doi:10. 1680/geot.1976.26.2.351.
[39] Steenfelt, J.S. 1977. Scale effect on bearing capacity factor. In Proceedings of the 9th International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Vol. 1, pp. 749–752.
[40] Lundgren, H., and Mortensen, K. 1953. Determination by the theory of plasticity on the bearing capacity of continuous footings on sand. In Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, Zürich, Switzerland. Vol. 1, pp. 409–412.
[41] Craig, R.F., and Pariti, L.N.M. 1978. Limiting equilibrium analysis of strip footings. Journal of the Geotechnical Engineering Division, ASCE, 104(3): 357–368
[42] Spangler, M.G., and Handy, R.L. 1982. Soil engineering. 4th ed. Harper and Row Publishers, New York.
[43] Ingra, T.S., and Baecher, G.B. 1983. Uncertainty in bearing capacity of sands. Journal of Geotechnical Engineering,109(7):899–914. doi:10.1061/(ASCE)0733- 9410(1983)109:7(899).
[44] Simone, P., and Restaino, M. 1984. Il coefficiente di carico limite Nγ. Revista italiana di geotecnica, 3: 137–147.
[45] Hettler, A., and Gudehus, G. 1988. Influence of the foundation width on the bearing capacity factor. Soils and Foundations, 28(4): 81–92. doi:10.3208/sandf1972.28.4_81.
[46] Saran, S., and Agarwal, R.K. 1991. Bearing capacity of eccentrically obliquely loaded footing. Journal of Geotechnical Engineering, 117(11): 1669–1690. doi:10.1061/(ASCE)0733-9410(1991)117:11(1669).
[47] Bolton, M.D., and Lau, C.K. 1993. Vertical bearing capacity factors for circular and strip footings on Mohr–Coulomb soil. Canadian Geotechnical Journal, 30(6): 1024–1033. doi:10.1139/t93-099
[48] Kumbhojkar, A.S. 1993. Numerical evaluation of Terzaghi's Nγ. Journal of Geotechnical Engineering, 119(3): 598–607. doi:10.1061/(ASCE)0733-9410(1993) 119:3(598).
[49] Zadroga, B. 1994. Bearing capacity of shallow foundations on noncohesive soils. Journal of Geotechnical Engineering, 120(11): 1991–2008. doi:10.1061/(ASCE) 0733-9410(1994)120:11(1991).
[50] Manoharan, N., and Dasgupta, S.P. 1995. Bearing capacity of surface footings by finite elements. Computers & Structures, 54(4): 563–586. doi:10.1016/0045-7949(94)00381-C.
[51] Bowles, J.E. 1996. Foundation analysis and design. 5th ed. McGraw Hill, New York.
[52] Frydman, S., and Burd, H. 1997. Numerical studies of bearing-capacity factor, Nγ. Journal of Geotechnical and Geoenvironmental Engineering, 123(1): 20–29. doi:10.1061/(ASCE)1090-0241(1997)123:1(20)
[53] Michalowski, R. 1997. An estimate of the influence of soil weight on bearing capacity using limit analysis. Soils and Foundations, 37(4): 57–64. doi:10.3208/sandf.37.4_57.
[54] Paolucci, R., and Pecker, A. 1997. Seismic bearing capacity of shallow strip foundations on dry soils. Soils and Foundations, 37(3): 95–105. doi:10.3208/sandf.37.3_95
[55] Danish Standards Association. 1998. Norm for fundering (code of practice for foundation engineering). Danish standard DS415. 4th ed. Danish Standards Association, Copenhagen.
[56] Soubra, A.H. 1999. Upper-bound solutions for bearing capacity of foundations. Journal of Geotechnical and Geoenvironmental Engineering, 125(1): 59–68. doi:10.1061/(ASCE)1090-0241(1999)125:1(59).
[57] Coduto, D.P. 2001. Foundation design: Principles and practices. 2nd ed. Prentice–Hall, Inc., Upper Saddle River, N.J.
[58] Perkins, S.W., and Madson, C.R. 2000. Bearing capacity of shallow foundations on sand: A relative density approach. Journal of Geotechnical and Geoenvironmental Engineering, 126(6): 521–530. doi:10.1061/(ASCE)1090-0241(2000)126:6(521).
[59] Poulos, H.G., Carter, P.J., and Small, J.C. 2001. Foundations and retaining structures—Research and practice. In Proceedings of the 15th International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul. Vol. 4, pp. 2527–2606.
[60] Ueno, K., Miura, K., Kusakabe, O., and Nishimura, M. 2001. Reappraisal of size effect of bearing capacity from plastic solution. Journal of Geotechnical and Geoenvironmental Engineering, 127(3): 275–281. doi:10.1061/(ASCE)1090-0241(2001)127:3(275).
[61] Wang, Y.-J., Yin, J.-H., and Chen, Z.-Y. 2001. Calculation of bearing capacity of a strip footing using an upper bound method. International Journal for Numerical and Analytical Methods in Geomechanics, 25(8): 841–851. doi:10.1002/nag.151.
[62] Zhu, F., Lee, C.F., and Jiang, H.D. 2001. A numerical study of the bearing capacity factor. Canadian Geotechnical Journal, 38(5): 1090–1096. doi:10.1139/t01-023.
[63] Cassidy, M.J., and Houlsby, G.T. 2002. Vertical bearing capacity factors for conical footings on sand. Géotechnique,52(9):687–692. doi:10.1680/geot.2002.52.9.687
[64] Dewaikar, D.M., and Mohapatra, B.G. 2003. Computation of bearing capacity factor -Prandtl's mechanism soils and foundations. Soils and Foundations, 43(3): 1–10. doi:10.3208/sandf.43.3_1.
[65] Kumar, J. 2003. Nγ for rough strip footing using the method of characteristics. Canadian Geotechnical Journal, 40(3): 669–674. doi:10.1139/t03-009.
[66] Ukritchon, B., Whittle, A.J., and Klangvijit, C. 2003. Calculations of bearing capacity factor Nγ using numerical limit analyses. Journal of Geotechnical and Geoenvironmental Engineering, 129(5): 468–474. doi:10.1061/(ASCE)1090- 0241(2003)129:6(468).
[67] Hjiaj, M., Lyamin, A., and Sloan, S. 2005. Numerical limit analysis solutions for the bearing capacity factor Nγ. International Journal of Solids and Structures, 42(5-6): 1681–1704.doi:10.1016/j.ijsolstr.2004.08.002
[68] Martin, C.M. 2005. Exact bearing capacity calculations using the method of characteristics. In 11th of the Proceedings of the International Conference on Analytical and Computational Methods in Geomechanics, Turin. Vol. 4, pp. 441–450.
[69] Smith, C.C. 2005. Complete limiting stress solutions for the bearing capacity of strip footings on a Mohr–Coulomb soil. Géotechnique, 55(8): 607–612. doi:10.1680/geot.2005.55.8.607.
[70] Kumar, J., and Kouzer, K.M. 2007. Effect of footing roughness on bearing capacity factor Nγ. Journal of Geotechnical and Geoenvironmental Engineering, 133(5): 502–511. doi:10.1061/(ASCE)1090-0241(2007)133:5(502).
[71] Lyamin, A., Salgado, R., Sloan, S.W., and Prezzi, M. 2007. Two- and threedimensional bearing capacity of footings in sand. Géotechnique, 57(8): 647– 662. doi:10.1680/geot.2007.57.8.647.
[72] Kumar, J., and Khatri, V.N. 2011. Bearing capacity factors of circular foundations for a general c– soil using lower bound finite elements limit analysis. International Journal for Numerical and Analytical Methods in Geomechanics, 35(3): 393–405. doi:10.1002/nag.900.
[73] Salgado, R. 2008. The engineering of foundations. McGraw-Hill, New York.
[74] Yang, F., and Yang, J.S. 2008. A revised failure mechanism of strip footings for upper bound solution. The Electronic Journal of Geotechnical Engineering, 13(F): 1–17.
[75] Zhu, D. (2000) “The least upper-bound solution for the bearing capacity factor Nγ.” Soils and Foundations, 40(1), 123–129
[76] De Simone, P. Bearing capacity of a circular footing on a Coulomb medium. In Proceedings of the 5th International Conference Numerical Methods Geomechanics, Nayoya, Japan, 1–5 April 1985; Volume 2, pp. 829–836.
[77] Erickson, H.L.; Drescher, A. Bearing capacity of circular footings. J. Geotech. Geoenviron. Eng. 2002, 128, 38–43
[78] Martin, C.M. ABC-Analysis of Bearing Capacity. 2004. Available online: www-civil.eng.ox.ac.uk/people/ cmm/software/abc.
[79] Loukidis, D.; Salgado, R. Bearing capacity of strip and circular footings in sand using finite elements. Comput. Geotech. 2009, 36, 871–879.
[80] Kumar, J.; Chakraborty, M. Bearing capacity of a circular foundation on layered sand–clay media. Soils Found. 2015, 55, 1058–106
[81] Seyedi hoseininia E, Bearing capacity factorss of ring footings. Iran J Sci Technol-Trans Civ Eng 2016;40(2):121-32
[82] Zhao L, Wang JH. Vertical bearing capacity for ring footings. Comput Geotech 2008;35(2):292-304
[83] Lundgren, H., and Mortensen, K. 1953. Determination by the theory of plasticity on the bearing capacity of continuous footings on sand. In Proceedings of the3rd International Conference on Soil Mechanics and Foundation Engineering, Zürich, Switzerland. Vol. 1, pp. 409–412.
[84] Vogel, C., and Barcos, A. (1973). Discussion of "Analysis of Ultimate load of shallow foundations" (January 1973)
[85] Chen, W. F., and Davidson H. T. (1973). Bearing Capacity Determination by Limit Analysis. ASCE Soil Mechanics and Foundation Division Journal 99(6).
[86] Sokolovski, V. V., (1960), “Statics of Soil Media”, Butterworth, London.
[87] Meyerhof, G. G., (1951), “The Ultimate Bearing Capacity of Foundations”. doi: 10.1680/geot.1951.2.4.301
[88] Sokolovskii (Sokolovsky), V. D. (ed.). Military Strategy. With an introduction by Raymond L. Garthoff. New York: Frederick A. Praeger, 1963.