Evaluating the Nγ Coefficient for Shallow Strip Footings on Sloping Ground by the Method of Stress Characteristics
Publication: International Journal of Geomechanics
Volume 24, Issue 5
Abstract
This study presents a new solution algorithm to calculate the bearing capacity coefficient Nγ for a fully rough strip footing located on an infinite slope considering different soil friction angles φ and ground slopes β° using the stress characteristics method. The results showed that Nγ coefficients decreased with an increase in slope inclination, and roughness contribution prevented further reduction, particularly for higher φ values. The higher contribution of the upside-sloping ground for footing laid on the infinite slope led to a lower reduction of the Nγ coefficients compared with the footing placed at the slope vicinity in terms of a larger area affected by the stress field, and a larger plastic region of the downface slope. The reduction value was more eminent for steeper slopes β ≥ 20° due to the tendency for nonsymmetrical failure pattern, especially for higher φ values. The findings of the current research and those described in the literature were in good agreement, so much so that the lower-bound limit analysis and the current solution technique virtually reflect the same trend. The load inclination values for footings located adjacent to the slope with β ≤ 30° were not meaningful, whereas for footing resting on an infinite slope with β ≥ 25° it exceeds 15°, which reflects the fact that its effect should not be neglected, especially for ground slopes greater than 20°. By increasing β°, the maximum values of plastic length and plastic depth on the left and right sides of the footing decrease and increase, respectively. The maximum depth of the plastic region for β ≥ 20° from around the right side of the footing shifts toward the downward-sloping face. The maximum width of the elastic wedge gradually reduces, and the intersection point of the left and right plastic regions slowly shifts to the left and upside of the footing.
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Data Availability Statement
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Notation
The following symbols are used in this paper:
- B
- width of the footing;
- Fml
- parts of the footing base where the shear stress is fully mobilized along the left side of the footing;
- FmR
- parts of the footing base where the shear stress is fully mobilized along the right side of the footing;
- i
- inclination angle of the total resultant force;
- Nγ
- bearing capacity factor due to unit weight;
- q
- overburden pressure;
- qh
- total horizontal component of the resultant force;
- qv
- total vertical component of the resultant force;
- qu
- mean contact pressure beneath the rough footing;
- R
- radius of Mohr’s circle;
- wnpw
- maximum width of the nonplastic curved wedge;
- x
- x-coordinate;
- xdeR
- maximum depth value that the failure region reaches along the downface of the slope;
- xdl
- maximum depth of the plastic region beneath the footing on the left side of the footing;
- xdR
- maximum depth of the plastic region beneath the footing on the right side of the footing;
- xs
- maximum depth of the nonplastic curved wedge;
- y
- y-coordinate;
- ycl
- maximum length of the plastic region along the ground surface on the left side of the footing;
- ycR
- maximum length of the plastic region along the ground surface on the right side of the footing;
- α
- alpha stress characteristic lines;
- β
- beta stress characteristic lines;
- β°
- slope inclination of the ground;
- γ
- unit weight of the soil;
- δ
- roughness of the footing;
- ɛ
- angle of body forces with the vertical direction;
- θgl
- orientation of σ1 with x-axis along the sloping ground surface on left side of the footing;
- θgR
- orientation of σ1 with x-axis along the sloping ground surface on right side of the footing;
- θf
- orientation of σ1 with x-axis along the footing base;
- θ
- orientation of σ1 with x-axis;
- σ
- distance between Mohr’s circle center and the point where the failure envelope joins with σ-axis;
- σn
- normal stress acting along the ground slope;
- σ1
- major principal stress;
- σxx
- normal stress on a plane perpendicular to x-axis;
- σyy
- normal stress on a plane perpendicular to y-axis;
- τnt
- shear stress acting along the ground slope;
- τxy
- shear stress acting on a plane perpendicular to x-axis and in the direction of y-axis;
- μ
- angle between σ1 and α and β characteristic lines; and
- φ
- internal friction angle of the soil.
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Published In
International Journal of Geomechanics
Volume 24 • Issue 5 • May 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Jun 13, 2023
Accepted: Nov 20, 2023
Published online: Mar 6, 2024
Published in print: May 1, 2024
Discussion open until: Aug 6, 2024
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