Research Article Methods to Analyze Flexural Buckling of the Consequent Slabbed Rock Slope under Top Loading

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1 Mathematica Probems in Engineering Voume 26, Artice ID , 8 pages Research Artice Methods to Anayze Fexura Bucking of the Consequent Sabbed Rock Sope under Top Loading Hongyan Liu, Guihe Wang, and Feng Huang Coege of Engineering & Technoogy, China University of Geosciences, Beijing, Beijing 83, China Correspondence shoud be addressed to Hongyan Liu; hyan24@26.com Received 6 May 26; Accepted 26 June 26 Academic Editor: Paoo Maria Mariano Copyright 26 Hongyan Liu et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. The consequent sabbed rock sope is prone to fexura bucking faiure under its sef-weight and top oading. However, neary none of the existing studies consider the effect of the top oading on the sope fexura critica bucking height (CBH). Therefore, on the basis of Euer s Method and the fexura bucking faiure mode of the consequent sabbed rock sope, the cacuation method of the CBH of the vertica sabbed rock sope under the sef-weight is firsty proposed, and then it is extended to that of the consequent sabbed rock sope. The effect of sope dip ange, friction ange, and cohesion between the neighboring rock sabs and rock eastic moduus on the sope CBH is discussed. Secondy, the cacuation method of the CBH of the consequent sabbed rock sope under its sef-weight and top oading is proposed according to the superposition principe. Finay, on the basis of the hypothesis that the rock mechanica behavior obeys the statistica damage mode, the effect of the rock mechanica parameters n and ε on the sope CBH is studied. The resuts show that the rock strength has much effect on the sope CBH. If the rock is supposed to be a inear eastic body without faiure in Euer s Method, the resut from it is the maximum of the sope CBH.. Introduction Rock masses are often intersected by a singe set of steepy dipping discontinuities, such as reguar bedding panes or joints, forming a sab or sabbed structure. Meanwhie, the oads on the top of the sope are often encountered in the practica engineering, such as the vehice oad in the transportation engineering and the buiding oad in the civi engineering. When the sope surface is parae to the discontinuities, under the action of driving force due to the sef-weight of rock sabs and top oading, faiure by bucking may be initiated,namey,thesabsnearthetoeofsopebuckegraduay and correspondingy the sabs above the bucke side aong a weak interayer (Figure ). In the condition of the ength and width of the rock sab being far arger than its thickness, the bucking of rock sabs may be simpified as a beam stabiity probem. Because bucking of rock sabs is a common phenomenon of sope movements in sedimentary rocks, ots of studies have been conducted to understand the sip-bucking sope faiure mechanism. Harrison and Facon [] caed some naturay formed buckes, roof and wa structures, and cascades. Johnson [2] discussed bucking phenomena reated to formation of sheet structures in New Engand quarry foors. Nemčok et a. [3] and Radbruch-Ha [4] attributed some buckes to creeps. Kutter [5] briefy discussed bucking in British open pit coa mines. Cavers [6] anayzed mechanisms of singesab bucking for Euer buckes and three hinged buckes for panar or curved sabs. Aso on the basis of Euer s theory, Sun [7] obtained the bucking critica height of the rock sab with an energy equiibrium principe. Hu and Cruden [8] regarded the notion that the modes of bucking were determined by bedding thickness, and accordingy they proposed three bucking modes: Euer buckes, bock-fexure buckes, and bock buckes. Pant and Adhikary [9] and Adhikary et a. [] conducted an expicit and impicit finite eement numerica study on the mechanism of fexura bucking faiure of foiated rock sopes. Qin et a. [] proposed a catastrophe cusp mode to study the faiure mechanisms of the sip-bucking sope.pereiraandlana[2]anayzedabuckingfaiureoccurring in an open pit mine with the numerica method. Supposing that the rock eement strength obeys Weibu distribution

2 2 Mathematica Probems in Engineering mode, Zhang et a. [3] introduced the rock strength into the cacuation of the critica bucking height (CBH) of the consequent sabbed rock sope. Qi et a. [4] presented an anaytica soution on sip bucking sope faiure which fuy considers both the effect of earthquake and pore water pressure basedonenergyequiibriumtheory. However, it can be found from the existing research that athough many researchers have conducted rather profound work on the bucking faiure of the consequent sabbed rock sope,thefoowingaspectsarestitobestudiedfurther.first ofa,theoadactingonthesopeisnotperfectyconsidered in the existing studies. Athough the oads such as sef-weight of rock sabs, groundwater pressure, and the seismic force are considered in the modes proposed by many researchers [, 4], the sope is often subjected to the top oading such as the transportation and buiding oads which are not considered in the cacuation of the CBH of the consequent sabbed rock sope. Moreover, the rock is assumed to be eastic in the existing studies and cannot fai in strength which is not aways true in the practica engineering. Athough Zhang et a. [3] introduced the rock strength into the cacuation of the CBH of the consequent sabbed rock sope, they did not take into account the top oading of the sope and the cohesion between the neighboring rock sabs, and the bucking deformation defection curve equation of the rock sab they adopted is aso unreasonabe. Therefore, the present paper is aimed at presenting a comprehensive theoretica mode for the fexura bucking of the consequent sabbed rock sope under top oading. Firsty, suppose the rock to be a perfect eastic materia; the cacuation method of the CBH of the vertica sabbed rock sope is proposed, and then it is generaized to that of the consequent sabbed rock sope. Secondy, assume the rock strength obeys Weibu distribution, and then the effect of its strength propertyonthecbhoftheverticasabbedrocksopeis discussed. In a, the proposed method provides a way to consider the effect of rock strength and top oading on the CBH of the consequent sabbed rock sope. 2. Fexura Bucking Mode for the Consequent Sabbed Rock Sope under Its Sef-Weight Assuming the Rock to Be a Perfect Eastic Materia 2.. Fexura Bucking Mode for the Vertica Sabbed Rock Sope under Its Sef-Weight. Because the vertica one is a specia case of the consequent sabbed rock sope, the vertica sabbed rock sope is firsty studied, and then its resut can be generaized to the consequent sabbed rock sope. Zhang and Li [5] summarized the faiure mode of the vertica sabbed rock sopeintothefoowingtwo,namey,buckingfaiureandtopping faiure, shown in Figures 2(a) and 2(b), and the former is ony discussed here. The cassica bucking theory (Euer s Method) makes the foowing assumptions in the derivation of the formuae [6]: () The coumn is eastic and obeys Hooke slaw.(2)thesopeofthedefectioncurvecanbe approximated by a inear function. (3) The coumn is weightess. (4) The coumn is perfecty straight. y α Figure : Bucking faiure mode for the consequent sabbed rock sope. According to the assumptions above, the critica bucking oad is f G c t x p P cr = π2 EI (μ) 2, () where P cr isthecriticabuckingoad,n;e is Young s moduus of the materia, Pa; I is the moment of inertia, m 4 ; is the ength of the coumn, m; and μ is a constant describing end conditions, for pinned ends, μ=. The bucking instabiity of the vertica sabbed rock sope is triggered by the gravity of the rock sope and the cohesion and frictiona force between the neighboring rock sabs, and the oads above are inear ones aong the height of the rock sab. Therefore, it cannot be soved with the cassic bucking theory (Euer s Method), which wi otherwise ead to the engineering accident or unnecessary engineering cost. The instabiity mode for the vertica sabbed rock sope under its sef-weight shoud satisfy the foowing hypothesis: A ony the sef-weight of the rock sab and the cohesion betweentheneighboringrocksabsareincudedhere,andthe top oading of the rock sab is not considered; B the bottom of the rock sope is embedded into the ground, so it can be regarded as the fixed constraint. The bucking faiure mode of the vertica sabbed rock sope is shown as Figure 2(a). It can be seen that the upper rock sab of the sope sips down aong the bedding pane at the initia stage of the sope instabiity and its norma deformation on the bedding pane is restricted because of the neighboring rock sab. Therefore, the bucking instabiity faiure mode of the vertica sabbed rock sope can be simpified as the mechanica mode with the top siding and bottom fixation constraints, shown in Figure 2(c), with the corresponding coordinate system. Assume the depth of the rock sab perpendicuar to the pane is the unit; the critica bucking oad of the sope can be soved with the energy method. It can be seen from Figure 2(c) that the axia part of therocksabwideviatefromtheoriginaposition,namey,

3 Mathematica Probems in Engineering 3 x t dx x y (a) (b) (c) Figure 2: Faiure modes of the vertica sabbed rock sope and the pressure bar mode. (a) Bucking faiure; (b) topping faiure; (c) the cacuation mode of the pressure bar. x direction. Assume V is the defection of the rock sab; its geometrica and mechanica boundary conditions are V x= =, V x= =, (2) V =. x= According to the experiments of rock sab under uniaxia compression by Sun [7], it is reasonabe to assume the defection curve is V =δ( cos 2πx ), (3) where δ is the constant to determine. It is easiy verified that (3) satisfies the boundary condition. According to eastic theory [7], the strain energy U stored in the rock sab is EI U= 2 (V ) 2 dx = 4EIπ4 δ 2 3. (4) AsshowninFigure2(c),assumethedifferentiaength aong the defection curve is ds when x increases from x to x+dx;thedistancedλ(x) thattheoadabovex-section moves is dλ (x) =ds dx=( +(V ) 2 )dx. (5) Making Tayor extension for (5) and ignoring the highorder items, (5) is abbreviated: dλ (x) = 2 (V ) 2 dx. (6) Therefore, the dispacement λ(x) that the rock sab moves aong x-axisisasfoows: x λ (x) = 2 (V ) 2 dx = π2 δ 2 2 (x 4π 4πx sin ). (7) The corresponding potentia energy W of the rock sab is W= (q c) λ (x) dx = π2 δ 2 (q c), (8) 2 where q=γtis the gravitationa oad intensity, N/m 2 ; γ is the unit weight of the rock sab, N/m 3 ; t is the sab thickness (joint spacing), m; and c is the cohesion between the neighboring rock sabs, Pa. The overa potentia energy of the bending rock sab is Π=U W, and then according to the principe of minimum potentia energy Π/ δ =,weobtain P cr =(q c) cr = 8π2 EI cr 2, (9) where P cr is the critica force, N/m, and cr is the CBH of the vertica sabbed rock sope under its sef-weight. I = (/2)t 3 is the moment of inertia, m 4. In order to compare with the resut obtained with Euer s Method,(9)canbechangedinto (q c) cr = π 2 EI (.35 cr ) 2. () Therefore, the CBH cr of the vertica sabbed rock sope under its sef-weight is cr = 3 π2 EI.2 (q c) = 3 π2 Et 3.44 (γt c), () where the meaning of a the parameters is stated as above Fexura Bucking Mode for the Consequent Sabbed Rock Sope under Its Sef-Weight. However, the competey vertica sabbed rock sope is rare in practica engineering; amost a of the sabbed rock sops are consequent or anticonsequent.

4 4 Mathematica Probems in Engineering Therefore, in order to make the study resut more appicabe, the study on the CBH of the consequent sabbed rock sope shown in Figure is done next. Because the sef-weight, cohesion, and friction force wi a exist in the consequent sabbed rock sope, the residua driving force F x of the rock sab aong the interayer is F x =γsin α γcos α tan φ c, (2) where α is the sope dip ange (i.e., the sab dip ange), φ is the friction ange between the neighboring rock sabs, and the other parameters are as stated above. Therefore, according to the cacuation resut of the vertica sabbed rock sope, namey, substituting γt into () with γt(sin α cos α tan φ),thecbh cr of the consequent sabbed rock sope under its sef-weight is π cr = 2 Et [γt (sin α cos α tan φ) c]. (3) Sope dip ange (degree) Figure 3: Variation of sope CBH with the sope dip ange Friction ange between the neighboring rock sabs (degree) Figure 4: Variation of the sope CBH with the friction ange between the neighboring rock sabs. Itcanbeseenthattheverticasabbedrocksopeisaspecia case of the consequent one Cacuation Exampes. The cacuation exampe by Xiao and Yang [8] is discussed here. The rock sope is divided by a set of consequent joints, and the thickness of the rock sab is.63 m. The rock sope wi be prone to fexura bucking faiure under its sef-weight. The CBH cr of the sope obtained by Xiao and Yang [8] with Euer s Method is (ignoring the cohesion between the neighboring rock sabs) cr = π 2 Et 3 3 6γ (sin α cos αtgφ). (4) Itcanbeseenthattheresutobtainedfrom(3)is.6 times that obtained from (4) when the cohesion between the neighboring rock sabs is not considered. So it can be regarded that the resut obtained from (4) is too conservative, which wi ead to the unnecessary engineering cost. The reason eading to this resut is that Xiao and Yang [8] assumed the gravity of the rock sab to be a concentrated oad acting on its top, which wi reduce the CBH of the sope. Meanwhie, the cohesion between the neighboring rock sabs is aso negected in their study. Theeffectoftheparameterssuchasthesopedipangeon the CBH is discussed here. Assume E, γ, t, α, φ, c are GPa, 253 N/m 3,.63m,7,5,and5kPa,respectivey,andthe other parameters remain unchanged when one of them is studied. The variation of the sope CBH with α, c, φ, ande obtained from (3) is shown as in Figures 3 6. It can be found that the sope CBH graduay decreases with increasing the sope dip ange, but with the increase in the friction ange and cohesion between the neighboring rock sabs and the rock eastic moduus, the sope CBH increases. 3. Fexura Bucking Mode for the Consequent Sabbed Rock Sope under Sef-Weight and Top Loading Assuming the Rock to Be a Perfect Eastic Materia Athough the fexura bucking mode for the consequent sabbed rock sope under its sef-weight is ony discussed in Section 2, there are many other oads such as the transportation and buiding oads on the sope top in many conditions. Therefore, the instabiity of the consequent sabbed rock sope under the sope top oading and its sef-weight shoud be studied. When ony the top oading shown in Figure is considered, the CBH of the sope can be soved with Euer s Method, namey, (). According to boundary condition of the sope, μ shoud be equa to.7. Therefore, the CBH cr2 of the sabbed rock sope is cr2 = π2 Et p sin α, (5) where p is the top oading intensity of the sope, N/m 2,and the other parameters are as stated above. Assume the critica bucking oad of the sabbed rock sope under the foowing three oad conditions such as its sef-weight and top oading, its sef-weight, and top oading is P cr, P cr,andp cr2.then,wecanobtain P cr =P cr +P cr2. (6) If it is assumed that the critica height of the sabbed rock sope under its sef-weight and top oading is cr,and accordingtoeuer smethod,wecanobtain π 2 EI (μ cr ) 2 = π 2 EI (μ cr ) 2 + π 2 EI (μ cr2 ) 2. (7)

5 Mathematica Probems in Engineering Cohesion between the neigboring rock sabs (kpa) Figure 5: Variation of the sope CBH with the cohesion between the neighboring rock sabs Pressure acting on the sope top (MPa) Rock eastic moduus (GPa) Figure 6: Variation of the sope CBH with the rock eastic moduus. Then, we can obtain cr 2 = cr 2 + cr2 2. (8) Aso take the cacuation exampe in Section 2.3 as an exampe; the variation of the sope CBH with the top oading is studied. The parameters in Section 2.3, namey, E=GPa, γ = 253 N/m 3, t =.63 m, α=7, φ=5,andc=5kpa, areasoadopted;wecanobtain 2 cr = p 7.e (9) It can be seen from Figure 7 that the sope CBH amost ineary decreases with increasing top oading. 4. Fexura Bucking Mode for the Consequent Sabbed Rock Sope regarding the Rock to Be the Damage Materia 4.. The Statistica Damage Constitutive Mode for a Rock. WithEuer smethod,therockisassumedtobeaperfectinear eastic body and it does not fai. So ony the deformationa parametersuchastherockeasticmoduusisconsidered, whieitsstrengthoneisnot.however,therockisakindof natura damage geoogica body and it contains many randomy distributed microcracks, which wi have effect on the rock strength. The statistica damage mechanics is a powerfu too to study the occurrence, propagation, and coaescence processes of these microcracks and their effect on rock mechanica behaviors. By means of it, the distribution aw of these microcracks in rock such as norma or Weibu distributionisassumedsoastobuidupthemesoscopiceements with strength in a rock and to determine its damage state. Figure 7: Variation of the sope CBH with the oad on the top of the sope. Thus, a damage statistica constitutive mode for a rock can be set up. Ti now, much progress in the study of damage statistica constitutive modes has been made [9 22]. The estabishment of a rock damage statistica constitutive mode is mainy based on the foowing two aspects: () choose the strength criteria for the rock mesoscopic eement, for instance, the maximum principe strain criterion, Mohr-Couomb criterion, and Drucker-Prager criterion; and (2) determine the distribution aw of the rock mesoscopic eement strength, for exampe, power function distribution and Weibu distribution. The studies show that the damage constitutive mode based on Weibu distribution is better than that based on power function distribution, and its cacuation process is easier. Therefore, the damage constitutive mode based on Weibu distribution and the maximum principe strain criterion is adopted here. The strength of mesoscopic eements obeys the foowing Weibu distribution function [9]: { n ( ε n ) exp [ ( ε n ) ] ε> P (ε) = ε { ε ε { ε, (2) where ε is an eementa strength parameter or stress eve, and because the strain strength theory is adopted here, it denotes strain; n and ε are the distribution parameters, which can be obtained by fitting with the test stress-strain curve of the rock; and P(ε) isthepercentageofdamagedonesoutofthe tota number of the mesoscopic eements in the rock. Therefore, the damage D of the rock can be defined as D= E = exp [ ( ε n ) ], (2) E ε where D takes a vaue between and corresponding to damagestatesoftherockfromundamagedtofuydamaged; E and E are the eastic modui of the rock without any damage and with some damage, respectivey. Assumethemechanicabehavioroftherockmesoscopic eements obeys Hooke aw, its constitutive aw is σ=eε( D), (22) where σ and ε are the stress and strain for the rock, respectivey.

6 6 Mathematica Probems in Engineering n Figure 8: Variation of the sope CBH with n (ε =.2) ε Figure 9: Variation of the sope CBH with ε (n=2). 4.2.FexuraBuckingModefortheConsequentSabbedRock Sope under Its Sef-Weight and Top Loading Based on the Rock Statistica Damage Constitutive Mode. As stated above, the rock is a kind of natura damage geoogica body in which the damage wi increase even under itte oad. Therefore, it is unsuitabe to assume the rock to be the inear eastic body without any damage. So in order to discuss the effect of the rock damage on the CBH of the sope, the fexura bucking mode for the consequent sabbed rock sope based on Weibu statistica damage constitutive mode for the rock is set up. From (2), it can be known that the rock damage and its evoution are mainy refected by the variation of the rock eastic moduus. Therefore, substituting (2) into (8) eads to 2 cr = 3 (π 2 t 3 E exp [ (ε/ε ) n ] /.44 [γt (sin α cos α tan φ) c]) 2 + π 2 E exp [ (ε/ε ) n ]t 2 /48p sin α, (23) where p sin α+γt(sin α cos α tan φ) c ε=. (24) E Because when the bucking faiure of the sope occurs, the rock damage is rather itte, so E canberepacedwithe ; namey, ε p sin α+γt(sin α cos α tan φ) c E. (25) The effect of n and ε on the CBH of the sope is discussed. The parameters in Section 2.3, namey, E = GPa, γ = 253 N/m 3, t =.63 m, α=7, φ=5,andc=5kpa, are aso adopted; at the same time, the parameters of n and ε are assumed to be 2. and.2, respectivey, by experience. If we assume the pressure on the top of the sope is.2 MPa, the CBH of the sope is 5.3 m with (23). By comparison, the corresponding resut is 2.34 m when the rock is assumed to be a perfect eastic body. If ignoring the cacuation error, they areassumedtobeequa.theeffectsofn and ε on the CBH of thesopearediscussed.itcanbeseenfromfigure8thatthe CBH of the sope increases from 7. m to 5.3 m when n increases from to 2, which indicates that n has much effect on the cacuation resut. However, from the increase extent, the CBH of the sope increases much more when n increases from to.5, and then it increases rather ess. It shows that the effect of n on the cacuation resut is itte when it increases to a certain vaue, and the simiar resut can aso be found in Figure 9. In order to iustrate the effect of n and ε on the rock stress-strain curve, the resuts are given in Figures and. It can be seen that the rock eastic moduus and cimax strength wi both vary with them, and in some cases the variation of the rock strength is more obvious. Therefore, it can be regarded that the effect of the rock strength on the CBH of the sopeismuch.whentherockisassumedtobeaperfecteastic body, namey, its cimax strength is infinite, the CBH of the sopewireachitsmaximum,whichcanbeseenfromfigures 8 and 9. So the rock strength and deformationa behavior

7 Mathematica Probems in Engineering 7 Stress (MPa) n =.5 n= Strain n =.5 n=2 Figure : Variation of the rock stress-strain with n at ε =.2. Stress (MPa) ε =.5 ε = Strain ε =.5 ε =.2 Figure : Variation of the rock stress-strain with ε at n=2. shoud be considered at the same time in order to accuratey obtain the CBH of the sope. 5. Concusions Assuming the rock to be a perfect eastic materia, we firsty investigate the cacuation method of the CBH of the vertica sabbed rock sope under its sef-weight based on the energy principe and then extend it to that of the consequent sabbed rock sope. The effect of sope dip ange, friction ange, and cohesion between the neighboring rock sabs and rock eastic moduus on the sope CBH is discussed. It can be seen that the sope CBH graduay decreases with increasing the sope dip ange, but it wi increase with the increase in the friction ange and cohesion between the neighboring rock sabs and the rock eastic moduus. According to the superposition principe of Euer s Method, we deduce the cacuation method of the CBH of the consequent sabbed rock sope under its sef-weight and top oading. Meanwhie, the effect of top oading on the CBH of the consequent sabbed rock sope is aso discussed. It can be seen that the sope CBH amost ineary decreases with increasing top oading. Thirdy, assuming the rock mechanica behavior obeys the statistica damage mode, we estabish the corresponding cacuation method of the CBH of the consequent sabbed rock sope under its sef-weight and top oading. We aso discuss the effect of the rock strength characteristic parameters n and ε on the sope CBH, and the resuts show that their effect on the sope CBH is arge when n and ε areitte,andthenwhen they increase to a certain vaue, their effect wi become very itte.overa,theproposedmethodprovidesawaytocacuatethecbhoftheconsequentsabbedrocksopeunderits sef-weight and top oading. Finay, it is noted that the proposed method is more suitabe to the case that the top oading cannot be ignored comparing with the sef-weight of the rock sope; for exampe, there are heavy buiding and transportation oad on the top of the rock sope. Competing Interests The authors decare that they have no competing interests. Acknowedgments This study is supported by the Fundamenta Research Funds for the Centra Universities (265249/ ) of China and China Schoarship Funds (26). References [] J. V. Harrison and N. L. Facon, Gravity coapse structures and mountain ranges, as exempified in south-western Iran, Quartery the Geoogica Society of London,vo.92,no. 4, pp. 9 2, 936. [2] A. M. Johnson, Physica Processes in Geoogy, Freeman, Cooper & Co, San Francisco, Caif, USA, 97. [3] A. Nemčok, J. Pašek, and J. Rybář, Cassification of andsides and other mass movements, Rock mechanics, vo. 4, no. 2, pp. 7 78, 972. [4] D. H. Radbruch-Ha, Gravitationa creep of rock masses on sopes, in Rocksides and Avaanches,, Natura Phenomena,B. Voight,Ed.,pp ,Esevier,NewYork,NY,USA,978. [5]H.K.Kutter, Mechanismsofsopefaiureotherthanpure siding, in Rock Mechanics, Internationa Center for Mechanica Sciences, Courses and Lectures,L.Muer,Ed.,vo.65,Springer, New York, NY, USA, 974. [6] D. S. Cavers, Simpe methods to anayze bucking of rock sopes, Rock Mechanics and Rock Engineering,vo.4, no.2,pp. 87 4, 98. [7] G. Z. Sun, Rock Mass Structure Mechanics, Science Press, Beijing, China, 988, 988 (Chinese). [8] X.-Q. Hu and D. M. Cruden, Bucking deformation in the Highwood Pass, Aberta, Canada, Canadian Geotechnica Journa,vo.3,no.2,pp ,993. [9] S. R. Pant and D. P. Adhikary, Technica note: impicit and expicit modeing of fexura bucking of foiated rock sopes, Rock Mechanics and Rock Engineering,vo.32,no.2,pp.57 64, 999. [] D. P. Adhikary, H.-B. Mühhaus, and A. V. Dyskin, A numerica study of fexura bucking of foiated rock sopes, Internationa Journa for Numerica and Anaytica Methods in Geomechanics, vo.25,no.9,pp ,2. [] S. Q. Qin, J. J. Jiao, and S. J. Wang, A cusp catastrophe mode of instabiity of sip-bucking sope, Rock Mechanics and Rock Engineering, vo. 34, no. 2, pp. 9 34, 2.

8 8 Mathematica Probems in Engineering [2] L. C. Pereira and M. S. Lana, Stress-strain anaysis of bucking faiure in phyite sopes, Geotechnica and Geoogica Engineering,vo.3,no.,pp ,23. [3] L.M.Zhang,S.R.Lv,J.H.Zhang,andH.Y.Liu, Instabiity anaysis of bedding rock sope based on the statistica constitutive damage mode, Geotechnica Investigation & Survey,no.9, pp. 7 29, 24 (Chinese). [4] S.W.Qi,H.X.Lan,andJ.Y.Dong, Ananayticasoutionto sip bucking sope faiure triggered by earthquake, Engineering Geoogy, vo. 94, pp. 4, 25. [5] T.J.ZhangandY.P.Li, Linearviscoeasticitystabiityanaysis of buff rock sope, Mechanics in Engineering, vo. 25, no. 6, pp. 5 54, 23 (Chinese). [6] A. Chajes, Principes of Structura Stabiity Theory, CiviEngineering and Engineering Mechanics Series, Prentice Ha, Engewood Ciffs, NJ, USA, 974. [7] D. O. Brush and B. P. Amroth, Bucking of Bars, Pates and Shes,McGraw-Hi,NewYork,NY,USA,975. [8] S. F. Xiao and S. B. Yang, Rock Mass Mechanics, Geoogica Press, Beijing, China, 987 (Chinese). [9] W. Weibu, A statistica distribution function of wide appicabiity, Appied Mechanics, vo. 8, pp , 95. [2] D. Krajcinovic and M. A. G. Siva, Statistica aspects of the continuous damage theory, Internationa Soids and Structures,vo.8,no.7,pp ,982. [2] Z.-L. Wang, Y.-C. Li, and J. G. Wang, A damage-softening statistica constitutive mode considering rock residua strength, Computers & Geosciences,vo.33,no.,pp. 9,27. [22] H. Y. Liu and X. P. Yuan, A damage constitutive mode for rock mass with persistent joints considering joint shear strength, Canadian Geotechnica Journa, vo.52,no.8,pp.36 43, 25.

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