Skip to main content
SpringerLink
Log in

Influence of the superposition and flow rule assumptions on the bearing capacity of shallow strip footings

  • Research Paper
  • Published:
Acta Geotechnica Aims and scope Submit manuscript

Abstract

A rigorous solution aimed at accounting for the superposition and flow rule assumptions in the assessment of the bearing capacity of shallow strip footings under plane strain conditions has been developed using the method of characteristics and the finite element method. With reference to the Hill and Prandtl failure mechanisms, a bearing capacity factor Nγq and a superposition corrective coefficient μ that allow to directly obtain the actual bearing capacity of shallow strip foundations have been proposed. The geometry of the plastic volume involved in the failure mechanism has been also investigated, highlighting the influence of the superposition and flow rule assumptions. Starting from the numerical results obtained using the method of characteristics, several empirical relationships have been proposed for an accurate estimate of Nγq and μ as well as for the lateral extension and depth of the plastic volume. The accuracy of the numerical results has been checked through finite element analyses under the assumption of associated and non-associated flow rule showing that the proposed superposition corrective coefficient μ is not influenced by the flow rule assumption. Thus, a general bearing capacity equation for shallow strip footings has been provided which accounts for superposition approximation and non-associated flow rule.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

Δε s :

Incremental deviatoric strain

γ :

Soil unit weight

λ :

Surcharge ratio

μ :

Superposition corrective coefficient

σ′1, σ′3 :

Maximum and minimum principal effective stresses

φ′:

Soil friction angle

ψ :

Dilatancy angle

ω :

Angle formed by the maximum principal effective stress with the horizontal axis

B :

Foundation width

C′:

Soil cohesion

d :

Depth of the plastic volume

N c, N q, N γ :

Bearing capacity factors

N q N - A, N γ N - A :

Bearing capacity factors accounting for non-associated flow rule

N γ q :

Generalized bearing capacity factor for the general case c′ ≠ 0, φ′ ≠ 0, q ≠ 0

q :

Surface surcharge

q ult :

Ultimate load according to the superposition assumption

\(\overline{q}_{{{\text{ult}}}}\) :

Actual ultimate load

s′:

Effective stress invariant

S α, S β :

Curvilinear coordinates

x :

Width of the plastic volume

References

  1. Bolton MD, Lau CK (1993) Vertical bearing capacity factors for circular and strip footings on Mohr–Coulomb soil. Can Geotech J 30(6):1024–1033. https://doi.org/10.1139/t93-099

    Article  Google Scholar 

  2. De Borst R, Vermeer PA (1984) Possibilities and limitations of finite elements for limit analysis. Géotechnique 34(2):199–210. https://doi.org/10.1680/geot.1984.34.2.199

    Article  Google Scholar 

  3. Cascone E, Biondi G, Casablanca O (2019) Influence of earthquake-induced excess pore water pressures on seismic bearing capacity of shallow foundations. In: Earthquake geotechnical engineering for protection and development of environment and constructions. Proceedings of the 7th international conference on earthquake geotechnical engineering, pp 566–581

  4. Cascone E, Biondi G, Casablanca O (2021) Groundwater effect on bearing capacity of shallow strip footings. Comput Geotech 139:104417. https://doi.org/10.1016/J.COMPGEO.2021.104417

    Article  Google Scholar 

  5. Cox AD (1962) Axially-symmetric plastic deformation in soils—II. Indentation of ponderable soils. Int J Mech Sci 4(5):371–380. https://doi.org/10.1016/S0020-7403(62)80024-1

    Article  Google Scholar 

  6. Davis EH, Booker JR (1971) The bearing capacity of strip footings from the standpoint of plasticity theory. In: Proceedings 1st Australia–New Zealand conference on geomechanics. Melbourne, Australia, pp 276–282

  7. De Simone P (1985) Bearing capacity of a circular footing on a Coulomb medium. In: Proceedings of the 5th international conference on numerical methods in geomechanics, Nagoya, pp 829–836

  8. Erickson HL, Drescher A (2002) Bearing capacity of circular footings. J Geotech Geoenviron Eng 128(1):38–43. https://doi.org/10.1061/(ASCE)1090-0241(2002)128:1(38)

    Article  Google Scholar 

  9. Griffiths DV (1982) Computation of bearing capacity factors using finite elements. Géotechnique 32(3):195–202. https://doi.org/10.1680/geot.1982.32.3.195

    Article  Google Scholar 

  10. Han DD (2021) An approximation solution to estimate the ultimate bearing capacity of smooth strip footings on ponderable soil. IOP Conf Ser Earth Environ Sci 787(1):012032. https://doi.org/10.1088/1755-1315/787/1/012032

    Article  Google Scholar 

  11. Han D, Xie X, Zheng L, Huang L (2016) The bearing capacity factor Nγ of strip footings on c–ϕ–γ soil using the method of characteristics. Springerplus 5(1):1482. https://doi.org/10.1186/s40064-016-3084-6

    Article  PubMed  PubMed Central  Google Scholar 

  12. Loukidis D, Chakraborty T, Salgado R (2008) Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads. Can Geotech J 45(6):768–787. https://doi.org/10.1139/T08-015

    Article  Google Scholar 

  13. Loukidis D, Salgado R (2009) Bearing capacity of strip and circular footings in sand using finite elements. Comput Geotech 36(5):871–879. https://doi.org/10.1016/J.COMPGEO.2009.01.012

    Article  Google Scholar 

  14. Manoharan N, Dasgupta SP (1995) Bearing capacity of surface footings by finite elements. Comput Struct 54(4):563–586. https://doi.org/10.1016/0045-7949(94)00381-C

    Article  ADS  Google Scholar 

  15. Martin C (2005) Exact bearing capacity calculations using the method of characteristics. In: 11th International conference of the international association for computer methods and advances in geomechanics, pp 441–450

  16. Nordal S (2008) Can we trust numerical collapse load simulations using non-associated flow rules?. In: Proceedings of the 12th international conference on IACMAG, pp 755–762

  17. Nova R (2004) The role of non-normality in soil mechanics and some of its mathematical consequences. Comput Geotech 31(3):185–191. https://doi.org/10.1016/J.COMPGEO.2004.01.006

    Article  Google Scholar 

  18. Plaxis 2D V21 (2021) Reference manual. Bentley Systems International Limited, Dublin, p 576

    Google Scholar 

  19. Prandtl L (1920) Über die Härte Plastischer Körper. Nachrichten der königlichen Gesellshaft der Wissenschaften Göttingen Mathematisch-Physikalische Klasse Languages, pp 74–85

  20. Reissner H (1924) Zum Erddruckproblem. In: Proceedings of the 1st international conference applied mech. Delft, pp 295–311

  21. Salençon J (1977) Applications of the theory of plasticity in soil mechanics. Wiley, Chichester, p 158

    Google Scholar 

  22. Sun JP, Zhao ZY, Cheng YP (2013) Bearing capacity analysis using the method of characteristics. Acta Mech Sin Xuebao 29:179–188. https://doi.org/10.1007/s10409-013-0018-1

    Article  ADS  MathSciNet  Google Scholar 

  23. Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York. https://doi.org/10.1002/9780470172766

    Book  Google Scholar 

  24. Vermeer PA (1990) The orientation of shear bands in biaxial tests. Géotechnique 40(2):223–236. https://doi.org/10.1680/geot.1990.40.2.223

    Article  Google Scholar 

  25. Yin JH, Wang YJ, Selvadurai APS (2001) Influence of nonassociativity on the bearing capacity of a strip footing. J Geotech Geoenviron Eng 127(11):985–989. https://doi.org/10.1061/(ASCE)1090-0241(2001)127:11(985)

    Article  Google Scholar 

  26. Zhu DY, Lee CF, Law KT (2003) Determination of bearing capacity of shallow foundations without using superposition approximation. Can Geotech J 40(2):450–459. https://doi.org/10.1139/t02-105

    Article  Google Scholar 

Download references

Acknowledgements

This work is part of the research activities carried out by the Messina Research Unit in the framework of the work package WP16—Contributi normativi—Geotecnica (Task 16.3) of a Research Project funded by the ReLuis (Network of Seismic Engineering University Laboratories) consortium (Accordo Quadro DPC/ReLUIS 2022-2024).

Author information

Authors and Affiliations

  1. Department of Engineering, University of Messina, Messina, Italy

    O. Casablanca, E. Cascone & G. Biondi

Authors
  1. O. Casablanca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Cascone
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Biondi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Casablanca.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Casablanca, O., Cascone, E. & Biondi, G. Influence of the superposition and flow rule assumptions on the bearing capacity of shallow strip footings. Acta Geotech. (2024). https://doi.org/10.1007/s11440-024-02284-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11440-024-02284-1

Keywords

Search

Navigation