Effect of undercut on the lower bound stability of vertical rock escarpment using finite element and power cone programming

doi:10.1007/s11709-022-0841-1

Front. Struct. Civ. Eng.

2022, Vol. 16

Issue (8) : 1040-1055 https://doi.org/10.1007/s11709-022-0841-1

RESEARCH ARTICLE

Effect of undercut on the lower bound stability of vertical rock escarpment using finite element and power cone programming

Shuvankar DAS, Debarghya CHAKRABORTY(

)

Department of Civil Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India

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Abstract

In the present study, the stability of a vertical rock escarpment is determined by considering the influence of undercut. Lower bound finite element limit analysis in association with Power Cone Programming (PCP) is applied to incorporate the failure of rock mass with the help of the Generalized Hoek-Brown yield criterion. The change in stability due to the presence of undercut is expressed in terms of a non-dimensional stability number (σ_ci/γH). The variations of the magnitude of σ_ci/γH are presented as design charts by considering the different magnitudes of undercut offset (H/v_u and w_u/v_u) from the vertical edge and different magnitudes of Hoek-Brown rock mass strength parameters (Geological Strength Index (GSI), rock parameter (m_i,), Disturbance factor (D)). The obtained results indicate that undercut can cause a severe stability problem in rock mass having poor strength. With the help of regression analysis of the computed results, a simplified design equation is proposed for obtaining σ_ci/γH. By performing sensitivity analysis for an undisturbed vertical rock escarpment, we have found that the undercut height ratio (H/v_u) is the most sensitive parameter followed by GSI, undercut shape ratio (w_u/v_u), and m_i. The developed design equation as well as design charts can be useful for practicing engineers to determine the stability of the vertical rock escarpment in the presence of undercut. Failure patterns are also presented to understand type of failure and extent of plastic state during collapse.

Keywords undercut vertical escarpment stability Hoek-Brown yield criterion PCP

Corresponding Author(s): Debarghya CHAKRABORTY

Just Accepted Date: 23 August 2022 Online First Date: 01 November 2022 Issue Date: 02 December 2022

Cite this article:

Shuvankar DAS,Debarghya CHAKRABORTY. Effect of undercut on the lower bound stability of vertical rock escarpment using finite element and power cone programming[J]. Front. Struct. Civ. Eng., 2022, 16(8): 1040-1055.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-022-0841-1
https://academic.hep.com.cn/fsce/EN/Y2022/V16/I8/1040

Fig.1 (a) Formation of undercut due to erosion; (b) chosen domain and associated stress boundary conditions; (c) a typical finite element mesh used in the analysis for GSI = 90, m_i = 15, D = 0, H/v_u = 3, w_u/v_u = 2.

Tab.1 Mesh convergence study for a vertical rock escarpment having H/v_u = 6, w_u/v_u = 3, GSI = 60, m_i = 15, and D = 0

Fig.2 Variation of stability number (σ_ci/γH) with w_u/v_u and GSI for H/v_u = 3, (a) m_i = 5, D = 0; (b) m_i = 15, D = 0; (c) m_i = 25, D = 0; (d) m_i = 35, D = 0; (e) m_i = 5, D = 0.5; (f) m_i = 15, D = 0.5; (g) m_i = 25, D = 0.5; (h) m_i = 35, D = 0.5; (i) m_i = 5, D = 1; (j) m_i = 15, D = 1; (k) m_i = 25, D = 1; (l) m_i = 35, D = 1.

Fig.3 Variation of stability number (σ_ci/γH) with w_u/v_u and GSI for H/v_u = 5, (a) m_i = 5, D = 0; (b) m_i = 15, D = 0; (c) m_i = 25, D = 0; (d) m_i = 35, D = 0; (e) m_i = 5, D = 0.5; (f) m_i = 15, D = 0.5; (g) m_i = 25, D = 0.5; (h) m_i = 35, D = 0.5; (i) m_i = 5, D = 1; (j) m_i = 15, D = 1; (k) m_i = 25, D = 1; (l) m_i = 35, D = 1.

Fig.4 Variation of stability number (σ_ci/γH) with w_u/v_u and GSI for H/v_u = 7, (a) m_i = 5, D = 0; (b) m_i = 15, D = 0; (c) m_i = 25, D = 0; (d) m_i = 35, D = 0; (e) m_i = 5, D = 0.5; (f) m_i = 15, D = 0.5; (g) m_i = 25, D = 0.5; (h) m_i = 35, D = 0.5; (i) m_i = 5, D = 1; (j) m_i = 15, D = 1; (k) m_i = 25, D = 1; (l) m_i = 35, D = 1.

Fig.5 Failure patterns obtained for undercut having m_i = 5, D = 0, H/v_u = 5, (a) w_u/v_u = 1, GSI = 50; (b) w_u/v_u = 1, GSI = 90; (c) w_u/v_u = 3, GSI = 50; (d) w_u/v_u = 3, GSI = 90; (e) w_u/v_u = 5, GSI = 50; (f) w_u/v_u = 5, GSI = 90; failure patterns obtained for w_u/v_u = 5, m_i = 5, H/v_u = 5, (g) GSI = 50, D = 1; (h) GSI = 90, D = 1.

Fig.6 Variation of stability number (σ_ci/γH) with w_u/v_u and (a) H/v_u having GSI = 10, 50, and 90, m_i = 5, D = 0; (b) GSI having m_i = 5, D = 0, H/v_u = 3, 5, and 8; (c) m_i having GSI = 50, D = 0, H/v_u = 3, 5, and 8; (d) D having GSI = 50, m_i = 5, H/v_u = 3, 5, and 8.

Fig.7 Variation of stability number (σ_ci/γH) with GSI having m_i = 10, D = 0 for vertical escarpment without any undercut.

coefficient	value
$r 1$	3.843
$r 2$	0.687
$r 3$	?0.344
$r 4$	0.013
$r 5$	?0.262
$r 6$	?0.169
$r 7$	?0.081
$r 8$	0.868
$r 9$	0.098
$r 10$	3.263
$r 11$	0.208
$r 12$	0.151
$r 13$	?0.067

Tab.2 The obtained values of the constant coefficients

Fig.8 Comparison of predicted σ_ci/γΗ from proposed equation with computed σ_ci/γΗ.

Tab.3 Input parameters to perform the sensitivity analysis for undisturbed vertical rock escarpment

Fig.9 Tornedo Chart obtained from the sensitivity analysis.

1	T Sunamura. Geomorphology of rocky coasts. Coastal Morphology and Research, 1994, 3: 174–175
2	P Budetta, G Galietta, A Santo. A methodology for the study of the relation between coastal cliff erosion and the mechanical strength of soils and rock masses. Engineering Geology, 2000, 56(3−4): 243–256 https://doi.org/10.1016/S0013-7952(99)00089-7
3	N Abderahman. Evaluating the influence of rate of undercutting on the stability of slopes. Bulletin of Engineering Geology and the Environment, 2007, 66(3): 303–309 https://doi.org/10.1007/s10064-006-0078-6
4	M L Chu-Agor, G A Fox, R M Cancienne, G V Wilson. Seepage caused tension failures and erosion undercutting of hillslopes. Journal of Hydrology (Amsterdam), 2008, 359(3−4): 247–259 https://doi.org/10.1016/j.jhydrol.2008.07.005
5	V E Mirenkov. On probable failure of an undercut rock mass. Journal of Mining Science, 2009, 45(2): 105–111 https://doi.org/10.1007/s10913-009-0014-9
6	A P Young, R T Guza, W C O’Reilly, J E Hansen, P L Barnard. Short-term retreat statistics of a slowly eroding coastal cliff. Natural Hazards and Earth System Sciences, 2011, 11(1): 205–217 https://doi.org/10.5194/nhess-11-205-2011
7	O Katz, A Mushkin. Characteristics of sea-cliff erosion induced by a strong winter storm in the eastern Mediterranean. Quaternary Research, 2013, 80(1): 20–32 https://doi.org/10.1016/j.yqres.2013.04.004
8	P Budetta, C D Luca. Wedge failure hazard assessment by means of a probabilistic approach for an unstable sea-cliff. Natural Hazards, 2015, 76(2): 1219–1239 https://doi.org/10.1007/s11069-014-1546-0
9	B Ukritchon, R Ouch, T Pipatpongsa, M H Khosravi. Investigation of stability and failure mechanism of undercut slopes by three-dimensional finite element analysis. KSCE Journal of Civil Engineering, 2018, 22(5): 1730–1741 https://doi.org/10.1007/s12205-017-2011-x
10	P C Augustinus. Rock mass strength and the stability of some glacial valley slopes. Zeitschrift für Geomorphologie, 1995, 39(1): 55–68 https://doi.org/10.1127/zfg/39/1995/55
11	M TsesarskyY H HatzorI LeviathanU SaltzmanM Sokolowksy. Structural control on the stability of overhanging, discontinuous rock slopes. In: Alaska Rocks, 40th US Symposium on Rock Mechanics (USRMS). Anchorage, AK: American Rock Mechanics Association, 2005
12	J L Briaud. Case histories in soil and rock erosion: Woodrow Wilson bridge, Brazos River Meander, Normandy Cliffs, and New Orleans Levees. Journal of Geotechnical and Geoenvironmental Engineering, 2008, 134(10): 1425–1447 https://doi.org/10.1061/(ASCE)1090-0241(2008)134:10(1425
13	M Tsesarsky, Y H Hatzor. Kinematics of overhanging slopes in discontinuous rock. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 135(8): 1122–1129 https://doi.org/10.1061/(ASCE)GT.1943-5606.0000049
14	Y S Hayakawa, Y Matsukura. Stability analysis of waterfall cliff face at Niagara Falls: An implication to erosional mechanism of waterfall. Engineering Geology, 2010, 116(1−2): 178–183 https://doi.org/10.1016/j.enggeo.2010.08.004
15	P Budetta. Stability of an undercut sea-cliff along a cilento coastal stretch (Campania, Southern Italy). Natural Hazards, 2011, 56(1): 233–250 https://doi.org/10.1007/s11069-010-9565-y
16	S K Banerjee, D Chakraborty. Influence of undercut and surface crack on the stability of a vertical escarpment. Geomechanics and Engineering, 2017, 12(6): 965–981
17	E Hoek, C Carranza-Torres, B Corkum. Hoek-Brown failure criterion—2002 edition. Proceedings of NARMS-Tac, 2002, 1(1): 267–273
18	E Hoek, E T Brown. The Hoek–Brown failure criterion and GSI–2018 edition. Journal of Rock Mechanics and Geotechnical Engineering, 2019, 11(3): 445–463 https://doi.org/10.1016/j.jrmge.2018.08.001
19	R S Merifield, A V Lyamin, S W Sloan. Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion. International Journal of Rock Mechanics and Mining Sciences, 2006, 43(6): 920–937 https://doi.org/10.1016/j.ijrmms.2006.02.001
20	A J Li, R S Merifield, A V Lyamin. Effect of rock mass disturbance on the stability of rock slopes using the Hoek–Brown failure criterion. Computers and Geotechnics, 2011, 38(4): 546–558 https://doi.org/10.1016/j.compgeo.2011.03.003
21	J Kumar, D Mohapatra. Lower-bound finite elements limit analysis for Hoek-Brown materials using semidefinite programming. Journal of Engineering Mechanics, 2017, 143(9): 04017077 https://doi.org/10.1061/(ASCE)EM.1943-7889.0001296
22	J Kumar, O Rahaman. Lower bound limit analysis of unsupported vertical circular excavations in rocks using Hoek−Brown failure criterion. International Journal for Numerical and Analytical Methods in Geomechanics, 2020, 44(7): 1093–1106 https://doi.org/10.1002/nag.3051
23	S Das, D Chakraborty. Effect of interface adhesion factor on the bearing capacity of strip footing placed on cohesive soil overlying rock mass. Frontiers of Structural and Civil Engineering, 2021, 15(6): 1494–1503 https://doi.org/10.1007/s11709-021-0768-y
24	S Das, D Chakraborty. Effect of soil and rock interface friction on the bearing capacity of strip footing placed on soil overlying Hoek–Brown rock mass. International Journal of Geomechanics, 2022, 22(1): 04021257 https://doi.org/10.1061/(ASCE)GM.1943-5622.0002225
25	S Das, K Halder, D Chakraborty. Bearing capacity of interfering strip footings on rock mass. Geomechanics and Geoengineering, 2022, 17(3): 883–895 https://doi.org/10.1080/17486025.2021.1903091
26	J Lysmer. Limit analysis of plane problems in soil mechanics. Journal of the Soil Mechanics and Foundations Division, 1970, 96(4): 1311–1334 https://doi.org/10.1061/JSFEAQ.0001441
27	S W Sloan. Lower bound limit analysis using finite elements and linear programming. International Journal for Numerical and Analytical Methods in Geomechanics, 1988, 12(1): 61–77 https://doi.org/10.1002/nag.1610120105
28	S W Sloan. Geotechnical stability analysis. Geotechnique, 2013, 63(7): 531–571 https://doi.org/10.1680/geot.12.RL.001
29	A Makrodimopoulos, C Martin. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. International Journal for Numerical Methods in Engineering, 2006, 66(4): 604–634 https://doi.org/10.1002/nme.1567
30	K Krabbenhøft, A V Lyamin, S W Sloan. Three-dimensional Mohr−Coulomb limit analysis using semidefinite programming. Communications in Numerical Methods in Engineering, 2008, 24(11): 1107–1119 https://doi.org/10.1002/cnm.1018
31	J Kumar, O Rahaman. Lower bound limit analysis using power cone programming for solving stability problems in rock mechanics for generalized Hoek–Brown criterion. Rock Mechanics and Rock Engineering, 2020, 53(7): 3237–3252 https://doi.org/10.1007/s00603-020-02099-y
32	D Chakraborty, J Kumar. Stability of a long unsupported circular tunnel in soils with seismic forces. Natural Hazards, 2013, 68(2): 419–431 https://doi.org/10.1007/s11069-013-0633-y
33	MATLAB. Version 8.5. Natick, MA: MathWorks, 2015
34	ApS MOSEK. Version 9.0. Copenhagen: MOSEK, 2020
35	R L Michalowski, D Park. Stability assessment of slopes in rock governed by the Hoek−Brown strength criterion. International Journal of Rock Mechanics and Mining Sciences, 2020, 127: 104217 https://doi.org/10.1016/j.ijrmms.2020.104217
36	T RabczukG ZiS BordasH Nguyen-Xuan. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2437−2455
37	T Rabczuk, T Belytschko. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
38	T Rabczuk, T Belytschko. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29−30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
39	P Areias, J Reinoso, P P Camanho, Sá J C de, T Rabczuk. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360 https://doi.org/10.1016/j.engfracmech.2017.11.017
40	P Areias, T Rabczuk. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41 https://doi.org/10.1016/j.finel.2017.05.001
41	P Areias, M A Msekh, T Rabczuk. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143 https://doi.org/10.1016/j.engfracmech.2015.10.042
42	H Ren, X Zhuang, T Rabczuk. A higher order nonlocal operator method for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113132 https://doi.org/10.1016/j.cma.2020.113132
43	X Zhuang, H Ren, T Rabczuk. Nonlocal operator method for dynamic brittle fracture based on an explicit phase field model. European Journal of Mechanics. A, Solids, 2021, 90: 104380 https://doi.org/10.1016/j.euromechsol.2021.104380

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