地质岩土英文文献翻译

1、International Journal of Rock Mechanics and Mining SciencesAnalysis of geo-structural defects in flexural topplingfailureAbbas Majdi and Mehdi AminiAbstractThe in-situ rock structural weaknesses, referred to herein as geo-structuraldefects,such as naturallyinduced micro-cracks,areextremelyresponsive

2、totensilestresses. Flexural toppling failure occurs by tensile stress caused by the momentdue totheweightof theinclinedsuperimposedcantilever-likerockcolumns.Hence,geo-structural defects that may naturally exist in rock columns are modeled by aseriesofcracksin maximum tensilestressplane.The magnitud

3、e and locationofthemaximumtensilestressin rockcolumnswithpotentialflexuraltopplingfailurearedetermined.Then, theminimum factorofsafetyforrockcolumns arecomputed by meansof principlesofsolidand fracturemechanics,independently.Next,a new equationis proposed to determine the length of critical crack in

4、 such rock columns. It hasbeen shown thatifthelengthof naturalcrack issmallerthan the lengthof criticalcrack, then the result based on solid mechanics approach is more appropriate;otherwise,theresultobtainedbased on theprinciplesof fracturemechanicsismoreacceptable.Subsequently,forstabilizationof th

5、e prescribedrockslopes,some newanalytical relationships are suggested for determination the length and diameterof the required fullygrouted rock bolts. Finally, for quickdesign ofrock slopesagainst flexural toppling failure, a graphical approach along with some designcurvesare presentedbywhich an ad

6、missible inclination of such rock slopes andorlengthofallrequiredfullygroutedrock boltsaredetermined.Inaddition,a casestudy has been used for practical verification of the proposed approaches.KeywordsGeo-structural defects, In-situ rock structural weaknesses, Criticalcrack length.1.IntroductionRock

7、masses are natural materials formed in the course of millions of years.Since during their formation and afterwards, they have been subjected to highvariable pressures both vertically and horizontally, usually, they are notcontinuous, and contain numerous cracks and fractures. The exerted pressures,s

8、ometimes, produce joint sets. Since these pressures sometimes may not besufficiently high to create separate joint sets in rock masses, they can producemicro joints and micro-cracks. However, the results cannot be considered asindependent joint sets. Although the effects of these micro-cracks are no

9、t thatpronounced compared withlargesize joint sets, yetthey may cause a drastic changeof in-situ geomechanical properties of rock masses. Also, in many instances, dueto dissolution of in-situ rock masses, minute bubble-like cavities, etc., areproduced, which cause a severe reduction of in-situ tensi

10、le strength. Therefore,one should not replace this in-situ strength by that obtained in the laboratory.On the other hand, measuring thein-siturocktensilestrengthdue totheinteractionofcomplex parameters isimpractical.Hence,an appropriateapproachforestimationof the tensile strength should be sought. I

11、n this paper, by means of principles of solid and fracture mechanics, a new approach for determination of the effect of geo-structural defects on flexural toppling failure is proposed.2. Effect of geo-structural defects on flexural toppling failure2.1. Critical section of the flexural toppling failu

12、reAs mentioned earlier,Majdi and Amini 10and Amini et al.11have proved thatthe accuratefactor of safety is equal to that calculated for a series of inclined rock columns, which, byanalogy,is equivalentto the superimposed inclinedcantileverbeamsas shown inFig.3. Accordingto the equations of limitequi

13、librium,the moment M and the shearing force Vexisting in variouscross-sectional areas in the beams can be calculated as follows:(5)( 6)Since the superimposed inclined rock columns are subjected to uniformly distributed loads.caused by theirown weight,hence, the maximum shearingforce andmoment exista

14、t the very fixedend, that is, at x=:(7)(8)Ifthe magnitudeof from Eq. (1)is substitutedof shearingforceand the maximummoment of equivalentinto Eqs. (7) and (8), then the magnitudes beam for rock slopes are computed as follows:(9)(10)where C isa dimensionlessgeometricalparameterthatisrelatedto theincl

15、inationsof therock slope, the total failure plane and the dip of the rock discontinuities that exist in rockmasses, and can be determined by means of curves shown in Fig.Mmax and Vmax will produce the normal (tensile and compressive) and the shear stresses incriticalcross-sectionalarea,respectively.

16、However, the combined effectof them willcause rockcolumns to fail. It is well understood that the rocks are very susceptible to tensile stresses,and the effect of maximum shearing force is also negligible compared with the effect of tensilestress. Thus, for the purpose of the ultimate stability, str

17、uctural defects reduce thecross-sectional area of load bearing capacity of the rock columns and, consequently, increasethe stress concentration in neighboring solid areas. Thus, the in-situ tensile strength of therock columns, the shearing effect might be neglected and only the tensile stress caused

18、 due tomaximum bending stress could be used.2.2. Analysis of geo-structural defectsDetermination of the quantitative effect of geo-structural defects in rock masses can be investigated on the basis of the following two approaches.2.2.1. Solid mechanics approachIn thismethod, which is,indeed,an old a

19、pproach,the loadsfrom the weak areasare removed and likewise will be transferred to the neighboring solid areas.Therefore,the solidareas of the rock columns,due to overloading and high stressconcentration,willeventuallyencounterwiththe prematurefailure.In thispaper,.for analysis of the geo-structura

20、l defects in flexural toppling failure, a set of cracks in critical cross-sectional area has been modeled as shown in Fig. 5. By employing Eq. (9) and assuming that the loads from weak areas are transferred to the solid areas with higher load bearing capacity (Fig. 6), the maximum stressescould be c

21、omputed by the following equation (see Appendix A for more details):（ 11）Hence, with regard to Eq. (11), for determination of the factor of safety against flexural toppling failure in open excavations and underground openings including geo-structural defects the following equation is suggested:（12）F

22、rom Eq. (12) it can be inferred that the factor of safety against flexuraltopplingfailureobtained on the basis ofprinciplesofsolidmechanics isirrelevanttothe lengthofgeo-structuraldefectsor the cracklength,directly.However, itisrelatedto thedimensionlessparameter“jointpersistence”, k, as itwas defin

23、edearlier in this paper. Fig. 2 represents the effect of parameter k on the criticalheight of the rock slope. This figure also shows the limiting equilibrium of therock mass (Fs=1) with a potential of flexural toppling failure.Fig.2.Determinationof the criticalheightof rock slopeswith a potentialof

24、flexuraltopplingfailure on the basis of principles of solid mechanics.2.2.2. Fracture mechanics approachGriffith in 1924 13, by performing comprehensive laboratory tests on theglasses, concluded that fracture of brittle materials is due to high stressconcentrations produced on the crack tips which c

25、auses the cracks to extend (Fig.3).Williamsin 1952 and 1957 and Irwinin 1957 had proposed some relationsby whichthe stress around the single ended crack tips subjected to tensile loading atinfinite is determined 14, 15 and 16. They introduced a new factor in theirequations calledthe “stress intensit

26、y factor” which indicates the stresscondition at the crack tips. Therefore if this factor could be determinedquantitatively in laboratorial, then, the factor of safety corresponding to thefailure criterion based on principles of fracture mechanics might be computed.Fig. 3.Stress concentration at the

27、 tip of a single ended crack under tensile loadingSimilarly, the geo-structural defects exist in rock columns with a potentialof flexuraltopplingfailure could be modeled. As it wasmentioned earlier in thispaper, cracks could be modeled in a conservative approach such that the locationof maximum tens

28、ile stress at presumed failureplane to be considered asthe crackslocations (Fig. 3). If the existing geo-structural defects in a rock mass, aremodeled witha seriescracksin the totalfailureplane,then by means of principlesof fracturemechanics,an equationfordeterminationof the factorof safetyagainstfl

29、exural toppling failure could be proposed as follows:（ 13）where KIC is the criticalstressintensityfactor.Eq. (13)clarifiesthatthefactor of safety against flexural toppling failure derived based on the method offracture mechanics is directly related to both the“joint persistence” and the“lengthof cra

30、cks ”.As such the lengthof cracks existingin the rock colum ns playsimportantrolesin stressanalysis.Fig.10 shows the influenceofthe cracklengthon the critical height of rock slopes. This figure represents the limitingequilibrium of the rock mass with the potential of flexural toppling failure. Asit

31、can be seen, an increase of the crack length causes a decrease in the critical.height of the rock slopes. In contrast to the principles of solid mechanics, Eq.(13) or Fig. 4 indicates either the onset of failure of the rock columns or the inception of fracture development.Fig.4.Determinationof the c

32、riticalheightof rock slopeswith a potentialof flexuraltopplingfailure on the basis of principle of fracture mechanics.3. Comparison of the results of the two approachesThe curves shown in Fig.representEqs. (12) and (13),respectively.The figuresreflectthe quantitativeeffectof the geo-structuraldefect

33、son flexuraltopplingfailure on the basis of principles of solid mechanics and fracture mechanicsaccordingly. For the sake of comparison, these equations are applied to one kindof rock mass (limestone)with the followingphysicaland mechanicalproperties16:,=20kN/m3, k=0.75.Inany case studies,a safeand

34、stableslopeheightcan be determinedby usingEqs. (12) and (13), independently. The two equations yield two different slopeheights out of which the minimum height must be taken as the most acceptable one.By equating Eqs. (12) and (13), the following relation has been derived by whicha crack length, in

35、this paper called critical length of crack, can be computed:（ 14a）where ac is the half of the average critical length of the cracks. Since acappears on both sidesof Eq. (14a),the criticallengthof the crack could be computedby trial and error method. If the length of the crack is too small with respe

36、ct torock column thickness, then the ratio t/(t-2ac) is slightly greater than one.Therefore one may ignore the length of crack in denominator, and then this ratio becomes 1. In this case Eq. (14a) reduces to the following equation, by which the critical length of the crack can be computed directly:.

37、(14b)Itmust be borninmind thatEq. (14b)leads to underestimatethecriticallengthof the crack compared with Eq. (14a). Therefore, for an appropriate determinationof the quantitativeeffectof geo-structuraldefectsinrockmass againstflexuraltopplingfailure,thefollowing3 conditionsmust be considered:(1)a=0;

38、(2)aac.In case 1, there are no geo-structural defects in rock columns and so Eq. (3)willbe used forflexuraltopplinganalysis.In case 2,the lengthsofgeo-structuraldefects are smaller than the critical length of the crack. In this case failure ofrockcolumn occursdue totensilestressesforwhich Eq. (12),b

39、ased on theprinciplesof solidmechanics,shouldbe used. Incase 3,the lengthsof existinggeo-structuraldefects are greater than the critical length. In this case failure will occur dueto growing cracksforwhichEq. (13), based on the principlesoffracturemechanics,should be used for the analysis.The result

40、sof Eqs. (12) and (13) forthe limiting equilibrium both areshownin Fig. 11. For the sake of more accurate comparative studies the results of Eq.(3),which representstherock columnswith no geo-structuraldefectsarealsoshownin the same figure. As it was mentioned earlier in this paper, an increase of th

41、ecracklengthhas no directeffecton Eq. (12),whichwas derivedbased on principlesof solid mechanics, whereas according to the principles of fracture mechanics, itcauses to reduce the value of factor of safety. Therefore, for more in-depthcomparison, theresultsof Eq. (13),for different values ofthe crac

42、klength, arealso shown in Fig. As can be seen from the figure, if the length of crack is lessthan the critical length (dotted curve shown inFig. 11), failure is considered tofollow the principles of solid mechanics which results the least slope height.However,ifthelengthofcrack increases beyond thec

43、riticallength,therockcolumnfailsdue tohighstressconcentrationatthecracktipsaccordingtotheprinciplesof fracture mechanics, which provides the least slope height. Hence, calculation of critical length of crack is of paramount importance.4. Estimationof stablerock slopes with a potentialof flexuraltopp

44、lingfailureIn rock slopes and trenches, except for the soil and rock fills, the heightsaredictatedbythenaturaltopography.Hence, thedesiredslopes must be designedsafely.Inrockmasses withthepotentialofflexuraltopplingfailure,withregardto the length of the cracks extant in rock columns the slopes can b

45、e computed byEqs. (3), (12), and (13) proposed in this paper. These equations can easily beconvertedintoa seriesofdesigncurvesforselectionof the slopestoreplace thelengthy manual computations aswell. Fig. 12, Fig. 13, Fig.14 and Fig.15show several such design curves with the potential of flexural to

46、pping failures.Ifthelengthsofexistingcracksin therockcolumnsare smallerthanthecriticallength of the crack, one can use the design curves, obtained on the basis ofprinciplesofsolidmechanics,shown inFig.12and Fig.13,fortherockslopedesign purpose. If the lengths of the cracks existing in rock columns a

47、re greaterthan the critical length of the crack, then the design curves derived based onprinciples of fracture mechanics and shown in Fig. 14 and Fig. 15 must be usedforthe slopedesignintention.In all,thesedesign curves,withknowingtheheightof therock slopesand the thicknessof the rock columns, param

48、eter(H2/t)iscomputed,and then from the design curves the stable slope is calculated. It must be born inmind that all the aforementioned design curves are valid for the equilibriumcondition only, that is, when FS=1. Hence, the calculated slopes from the abovedesigncurves,forthefinalsafedesignpurpose

49、must be reducedbased on thedesiredfactorofsafety.Forexample,iftheinformationregardingtoone particularrockslopearegiven17:k=0. 25, =10,t=10MPa, =20kN/m3,=45,H=100 m,t=1m, aca=0.1m, and then according to Fig. 12which representsthe conditionof equilibriumonly.can be taken any valueslessthan the above m

50、entionedon the desired factor of safety.the design slope will be 63,Hence, the finaland safeslopeone, which issolelydependent.k=0.25.Fig. 5.Selection of critical slopes for rock columns with the potential of flexural topplingfailure on the basis of principles of solid mechanics whenFig. 6.Selection of critical slopes for rock columns with the potential of flexural topplingfailure based on principle