Semianalytical Solution for Evaluating Bearing Capacity of a Footing Adjacent to a Slope
Publication: International Journal of Geomechanics
Volume 21, Issue 2
Abstract
The study of the influence of foundations, placed adjacent to slopes, on the bearing capacity is of great importance in geotechnical engineering. The available analytical and empirical methods have some limitations in considering the soil properties and the location of the footings on the top surface of the slope. To overcome these limitations, finite element analysis (FEA) is usually performed. However, FEA requires significant computational effort and, therefore, the need for an analytical solution that can consider all the effective parameters on bearing capacity of the foundation still exists. Hence, a new analysis is proposed. In this regard, the stress distribution within a slope, due to the placement of a footing on the upper surface of the slope, is evaluated in this study. Following that, the minimum value of the vertical footing load required to destabilize the slope is computed. The strength properties and unit weight of the soil, the width and location of the footing, and the slope geometry are all taken into account in the analysis. The results show good agreement with the literature, including numerical, analytical, and empirical approaches. Thus, the proposed method that is more comprehensive compared with previous solutions can help both researchers and engineers to verify the FEM model or conducting a parametric study of the ultimate bearing capacity of the footing near the slope.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Notation
The following symbols are used in this paper:
- a
- end coordinate of the footing;
- c
- cohesion;
- c
- path of complex linear integration (real part);
- E
- Young’s modulus;
- F(z)
- function of loading in complex space;
- f(r)
- function of loading in real space;
- g
- gravity;
- H
- height of the slope;
- I1, I2, I3
- first, second, and third stress invariants;
- i
- imaginary number;
- J2, J3
- second and third deviatoric stress invariants;
- p
- footing load;
- r
- radial in polar coordinate;
- u
- step function;
- X
- x-axis of Cartesian coordinate;
- X′
- local Cartesian coordinate;
- x
- footing size;
- Y
- y-axis of Cartesian coordinate;
- Z
- depth direction of slope in Goodman and Brown’s model;
- z
- complex number;
- λ
- normalized footing distance;
- θ
- angle that measured from the bisector of the wedge;
- θ
- lode angle;
- α
- half of the wedge angle;
- α
- slope angle in Goodman and Brown’s model;
- β
- polar coordinate angle in Goodman and Brown’s model;
- ϕ(r, θ)
- real Airy stress function;
- Φ(z, θ)
- complex Airy stress function;
- φ
- friction angle;
- ρ
- density;
- polar stress component;
- σij
- stress tensor;
- δij
- Kronecker delta;
- ψ
- slope angle;
- ν
- Poisson’s ratio; and
- π
- pi number.
References
Bowles, L. 1996. Foundation analysis and design. New York: McGraw-Hill.
de Souza Neto, E. A., D. Peric, and D. R. Owen. 2011. Computational methods for plasticity: Theory and applications. Hoboken, NJ: John Wiley & Sons.
Filon, L. N. G. 1930. “III.—On a quadrature formula for trigonometric integrals.” Proc. R. Soc. Edinburgh 49: 38–47. https://doi.org/10.1017/S0370164600026262.
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.
Goodman, L., and C. Brown. 1963. “Dead load stresses and the instability of slopes.” J. Soil Mech. Found. Div. 89 (3): 103–136.
Haghgouei, H., A. R. Kargar, M. Amini, and K. Esmaeili. 2020. “An analytical solution for analysis of toppling-slumping failure in rock slopes.” Eng. Geol. 265: 105396. https://doi.org/10.1016/j.enggeo.2019.105396.
Hansen, B. 1961. “A general formula for bearing capacity.” Danish Geotech. Inst. Bull. 11: 38–46.
Kargar, A. R. 2019. “An analytical solution for circular tunnels excavated in rock masses exhibiting viscous elastic-plastic behavior.” Int. J. Rock Mech. Min. Sci. 124: 104128.
Kargar, A. R., and H. Haghgouei. 2020. “An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure.” Appl. Math. Modell. 77: 1625–1642.
Kargar, A. R., H. Haghgouei, and N. Babanouri. 2020. “Time-dependent analysis of stress components around lined tunnels with circular configuration considering tunnel advancing rate effects.” Int. J. Rock Mech. Min. Sci. 133: 104422.
Krahn, J. 2003. “The 2001 R.M. Hardy Lecture: The limits of limit equilibrium analyses.” Can. Geotech. J. 40 (3): 643–660. https://doi.org/10.1139/t03-024.
Kusakabe, O., T. Kimura, and H. Yamaguchi. 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.
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.
Meyerhof, G. 1957. “The ultimate bearing capacity of foundations on slopes.” In Proc., 4th Int. Conf. on Soil Mechanics and Foundation Engineering, 384–386. London: Butterworths Scientific Publications.
Meyerhof, G. G. 1963. “Some recent research on the bearing capacity of foundations.” Can. Geotech. J. 1 (1): 16–26. https://doi.org/10.1139/t63-003.
Narita, K., and H. Yamaguchi. 1990. “Bearing capacity analysis of foudations on slopes by use of log-spiral sliding surfaces.” Soils Found. 30 (3): 144–152. https://doi.org/10.3208/sandf1972.30.3_144.
Sadd, M. H. 2009. Elasticity: Theory, applications, and numerics. Cambridge, MA: Academic Press.
Stianson, J. R., D. Chan, and D. Fredlund. 2004. “Comparing slope stability analysis based on linear elastic or elastoplastic stresses using dynamic programming techniques.” In Proc., 57th Canadian Geotechnical Conf., 23–30. Canada: BiTech Publishers.
Terzaghi, K. 1943. Theoretical soil mechanics. New York: Wiley.
Tranter, C. J. 1951. Integral transforms in mathematical physics. 40–45. New York: John Wiley & Sons, Inc.
Vesic, A. S. 1975. “Bearing capacity of shallow foundations.” In Foundation engineering handbook. 1st ed., edited by H. F. Winterkorn and H. Y. Fang, 121–147. New York: Van Nostrand Reinhold Company.
Xie, H., Q. Wang, J. Wu, and Y. Chen. 2019. “Analytical model for methane migration through fractured unsaturated landfill cover soil.” Eng. Geol. 255: 69–79. https://doi.org/10.1016/j.enggeo.2019.04.018.
Zhou, H., G. Zheng, X. Yin, R. Jia, and X. Yang. 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.
Information & Authors
Information
Published In
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Mar 18, 2020
Accepted: Sep 4, 2020
Published online: Dec 7, 2020
Published in print: Feb 1, 2021
Discussion open until: May 7, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.