Abstract
This study communicates a novel seismic stability assessment of rock slopes based on the concept of the limiting slope face (LSF) combined with the method of stress characteristics (MSC) and the modified pseudo-dynamic (MPD) approach. The slope geometry with a target factor of safety (FS) of 1.0 is derived, precluding the need for any preordained slip surfaces in the analysis. Subsequently, the derived LSF in cognition with the morphological aspects indeed acts as a self-guided stability index for rock slopes. Besides, a realistic characterization of the dynamic properties of input earthquake motions satisfying the zero stress boundary conditions is apprehended through the coherent utilization of the MPD approach. The generalized Hoek–Brown (GHB) strength criterion is engaged to capture the factual non-linearity present in the rock strength. Compared to the reported investigations, the present results indicate that the developed curvilinear LSFs are steeper than the traditional linear slopes commonly encountered in the conventional practice. A parametric study accounting for the effect of different influential parameters on the behavior of LSFs is performed in view of various prospective design challenges in rock engineering. With a rise in the horizontal seismic acceleration coefficient (kh) from 0.1 to 0.3, a nearly threefold increase in the magnitude of the major principal stress orientation (ψ) at the slope crest but along the slope can be observed. Such enhancement in ψ indicates significantly flat LSF. A sudden rise in the magnitude of ψ can also be observed at the fundamental frequencies of seismic waves due to resonance. However, at kh = 0.1, such aptness declines by 54% at the first fundamental frequency of the shear wave as the rock mass damping increases from 5 to 15%. Thus, the present approach attributes to a rational way for seismic design and stability assessment of rock slopes. Several real-life case studies adopting the current LSF concept further exhibit the accuracy, rationality, and robustness of the proposed methodology.
Highlights
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Concept of limiting slope face is introduced for the seismic performance of rock slopes.
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Analysis is performed using an integrated framework of generalized Hoek-Brown criterion, method of stress characteristics and modified pseudo-dynamic approach.
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An adaptive collapse mechanism is investigated in response to varying seismic wave characteristics and rock mass parameters.
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A comprehensive review of other prevailing analytical seismic approaches is provided.
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Rationality of the results is ensured through validation with different published case studies.
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Data Availability
Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Abbreviations
- α :
-
Horizontal inclination of a linear slope
- α h :
-
Non-dimensional horizontal seismic acceleration normalized with respect to g
- α v :
-
Non-dimensional vertical seismic acceleration normalized with respect to g
- β a :
-
A parameter as defined in Eq. (9)
- γ :
-
Unit weight of the rock mass
- δ :
-
A parameter as defined in Eq. (23)
- ζ a :
-
A parameter as defined in Eq. (9)
- θ :
-
Angle made by the major principal stress with the positive z-axis
- θ g :
-
Magnitude of θ along the top surface of the slope (OG)
- θ s :
-
Magnitude of θ along the limiting slope face (OA)
- μ :
-
A parameter as defined in Eq. (15)
- ξ :
-
Damping ratio of the rock mass
- ρ :
-
Instantaneous friction angle as defined in Eq. (17)
- ρ g :
-
Magnitude of ρ along the top surface of the slope (OG)
- ρ s :
-
Magnitude of ρ along the limiting slope face (OA)
- σ +, σ − :
-
Axes in the two-dimensional curvilinear coordinate system representing the positive and negative characteristics, respectively
- σ 1 :
-
Major principal stress
- σ 3 :
-
Minor principal stress
- σ ci :
-
Uniaxial compressive strength of the intact rock
- σ ng :
-
Normal stress on the top surface of the slope (OG)
- σ x :
-
Normal stress on the x plane
- σ z :
-
Normal stress on the z plane
- τ g :
-
Shear stress on the top surface of the slope (OG)
- τ xz :
-
Shear stress in the xz plane
- χ :
-
Horizontal inclination of the top surface of the slope (OG), as shown in Fig. 15a
- ψ :
-
Magnitude of θ along the limiting slope face (OA), but at the slope crest (O)
- ω :
-
Angular frequency of seismic waves = 2π/T
- a :
-
Dimensionless Hoek–Brown material parameter, as defined in Eq. (7a), representing the characteristics of rock mass
- a h(z, t):
-
Horizontal seismic acceleration in the rock mass at depth z and time t
- a v(z, t):
-
Vertical seismic acceleration in the rock mass at depth z and time t
- A a :
-
A parameter as defined in Eq. (9)
- C p, C pz :
-
Dimensionless parameters, as defined in Eqs. (4b) and (4a), respectively
- C s, C sz :
-
Dimensionless parameters, as defined in Eqs. (3b) and (3a), respectively
- d :
-
Depth of rigid bed from the base of the slope
- D :
-
Disturbance factor of the rock mass
- f :
-
Macroscopic yield condition
- f a :
-
Amplification factor for seismic waves
- F :
-
Function defining the yield criterion in Eq. (12)
- g :
-
Acceleration due to gravity
- GSI :
-
Geological strength index
- h :
-
Height of the slope having a horizontal top surface (OG), as shown in Fig. 1a
- h′ :
-
Height of the slope having an inclined top surface (OG), as shown in Fig. 15a
- H :
-
Depth of the rigid bed from the top surface (OG) of the slope
- k :
-
A parameter as defined in Eq. (9)
- k h :
-
Horizontal seismic acceleration coefficient
- k v :
-
Vertical seismic acceleration coefficient
- m :
-
A parameter as defined in Eq. (15)
- m b :
-
Dimensionless Hoek–Brown material parameter, as defined in Eq. (7b)
- m i :
-
Hoek–Brown constant of intact rock representing the hardness of the rock
- N :
-
Stability number = σci/(γh.FS)
- N cr :
-
Critical stability number corresponding to FS = 1.0
- p :
-
Average stress
- p g :
-
Magnitude of p along the top surface of the slope (OG)
- p s :
-
Magnitude of p along the limiting slope face (OA)
- q :
-
Uniformly distributed surcharge
- R :
-
Radius of the Mohr circle
- R g :
-
Magnitude of R along the top surface of the slope (OG)
- R s :
-
Magnitude of R along the limiting slope face (OA)
- s :
-
Dimensionless Hoek–Brown material parameter as defined in Eq. (7c) representing the degree of fragmentation of rock
- S p, S pz :
-
Dimensionless parameters, as defined in Eqs. (4b) and (4a), respectively
- S s, S sz :
-
Dimensionless parameters, as defined in Eqs. (3b) and (3a), respectively
- t :
-
Time
- T :
-
Period of lateral shaking
- u h(z, t):
-
Horizontal displacement of the rock mass at depth z and time t
- u v(z, t):
-
Vertical displacement of the rock mass at depth z and time t
- u h 0 , u v 0 :
-
Amplitudes of the finite harmonic displacement at the rigid base along the horizontal and the vertical directions, respectively
- u hb , u vb :
-
Finite harmonic displacements at the rigid base along the horizontal and the vertical directions, respectively
- V p :
-
Velocity of primary wave
- V s :
-
Velocity of shear wave
- x, z :
-
Axes in the regular two-dimensional Cartesian coordinate system
- x′, z′ :
-
Axes in the transformed two-dimensional Cartesian coordinate system, as shown in Fig. 15a
- y s 1, y s 2 :
-
Dimensionless parameters, as defined in Eq. (3c)
- y p 1, y p 2 :
-
Dimensionless parameters as defined in Eq. (4c)
- X :
-
Body force per unit volume in the x-direction
- Z :
-
Body force per unit volume in the z-direction
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Acknowledgements
The first author acknowledges the Ministry of Education, Government of India, for Prime Minister’s Research Fellowship grant.
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Conceptualization: PG; methodology: SN and PG; formal analysis and investigation: SN; writing–original draft preparation: SN; writing—review and editing: PG; resources: SN and PG; supervision: PG.
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Nandi, S., Ghosh, P. Seismic Stability Assessment of Rock Slopes Using Limiting Slope Face Concept. Rock Mech Rock Eng 56, 5077–5102 (2023). https://doi.org/10.1007/s00603-023-03308-0
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DOI: https://doi.org/10.1007/s00603-023-03308-0