Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript

World Academy of Science, Engineering and Technology International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015 IInfluennce off Interm mediaate Prinncipall Stresss on S Solutioon of Plaanar S Stabilitty Prooblemss M. Jahananddish, M. B. Zeydabadinej Z ad International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155 Abstract—In this paper, voon Mises and Drucker-Prageer yield criiteria, as typiccal ones that cconsider the effect e of interm mediate priincipal stress σ2, have been b selected and employeed for invvestigating the influence of σ22 on the solution of a typical stability s prooblem. The beearing capacity factors have bbeen calculatedd under plaane strain condition (strip foooting) and axxisymmetric coondition (ciircular footing)) using the methhod of stress characteristics toogether wiith the criteria mentioned. m Diffferent levels of σ σ2 relative to thhe other tw wo principal streesses have beenn considered. While W a higher σ2 entry in yield criterion gives a higgher bearing capacity; c its enntry in eqquilibrium equattions (axisymmetric) causes suubstantial reducttion. Keywords—IIntermediate axxisymmetric, yieeld criteria. M principal strress, plane obtaained by the envelope of ssuch circles iin - diagram m as show wn in Fig. 2. Any failure criterion sugggested for soiil, as applied to planee strain or axxisymmetric conditions shhould appear as a funcctional relationnship or in this diagram. Herre, and n arre the shear annd normal streesses t failure plaane at failure. R and are the radius of such on the circcle and the distance of its center from the orrigin, resppectively. strain, I. INTR RODUCTION ANY stabbility problem ms in geotechnnical engineeriing are in plane sstrain or axisyymmetric condditions. An exxample woould be the sstrip and circcular footingss on the surfa face of grround. At the failure f state inn these problem ms; the soil floows in planes in which the displaccements devellop. In plane strain coondition; thesee are planes iin which the strains devellop. In axxisymmetric prroblems; they are redial plannes (Fig. 1). g Mohr circle c Figg. 2 Failure enveelope obtained bby pushing the greatest A As seen; the inntermediate principal stresss 2, can takee any valuue between 3 and 1 whenn point B movves from C to A in Fig.. 2. The effectt of 2 and itss variation onn the solution have not been studiedd before. In triaxial t test for f example; 2 is eithher assumed to be equall to 3 (com mpression) orr 1 (exttension). The ambiguity aabout the valuue of 2 andd its effeects have madde the researchhers to develoop the true triaxial device [1]. The ratio b is usuaally used to reppresent its efffects. Thiss ratio is definned as: (1) Fig. 1 Planes of flow in a) Plane strain b) axxisymmetric casses The failure sstate is represented by the Mohr circle hhaving the largest diaameter, i.e., 1-3. The failure functiion is Inncrease in 2 from 3 to 1 is equivalennt to increase in b from m 0 to 1. It iis noteworthy that there is a unique relaation betw ween the Lodee angle , and the stress ratioo b, as [2]: √ R Related values of M. Jahanandishh is with the D Department of C Civil and Environmental Enngineering, Shirazz University, Shhiraz, Iran (correesponding authorr phone: +998-917-716-7268; fax: +98-713-647-3161; e-mail: [email protected] shirazuu.ac.ir). M. B. Zeydabaddinejad is with thee Department of Civil and Environmental Enngineering, S Shiraz Univeersity, Shirazz, Iran (e-mail: [email protected]). International Scholarly and Scientific Research & Innovation 9(12) 2015 1368 √ and b aree typically givven in Table I. scholar.waset.org/1999.6/10003155 (2) World Academy of Science, Engineering and Technology International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015 TA ABLE I APPROPRIATE VALUES OF B AND D -30 -60 -90 b 1 0.5 0 We begin wiith criteria thaat do not deppend on stresss level firrst. This is heelpful in realizzing the elabooration that is added duue to stress levvel dependenccy later. The simplest s form is that suuggested by Tresca [3] iin 1864; whhich is relevaant to unndrained shearring of saturated clay soils. It can be conssidered ass a special case of M-C critterion when = 0. This is usually u wrritten as: Fig. 3 Compariison between Trresca and von Mises M at uniaxiaal comprression 1,2 (3) RvonMises/RTresca International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155 II. STRESSS LEVEL INDEEPENDENT YIEELD CRITERIA whhere qu is tthe soil streength in uniiaxial comprression (uunconfined com mpressive streength). Writing this ccriterion in thee form of will resuult in: (4) whhere cu is the undrained sheear strength. A As seen; theree is no deependency on the intermeddiate principal stress 2, as is the caase with its parrent M-C criteerion. The other criterion is the oone suggested by von Misess [4] in 19913. For satuurated clay with strengthh qu in unddrained coondition it wouuld be in the fo following form m: √ (5) whhere k is the von Mises coonstant that takkes different values deepending on thhe condition of the problem.. In contrast to that of T Tresca; the vvon Mises criterion deepends on 2. Writing it inn the form off using b deefined above, we w get: . 1,15 2 √3 1.15 55 1,1 1,05 1 b 0 0,25 0,5 0,755 1 Fig. 4 The ratioo of strengths byy von Mises to tthat of Tresca as a function oof 2 and b. A As mentioned;; it is customaary in the axissymmetric casse to assuume 2=3 whhen the flow of o soil is awayy from the axxis of sym mmetry and to assume 2=1 when it is tooward it. These are knoown as von Kaarman regime. It may be ratiional to assum me 2 to be b the averagge of 3 andd 1 in planee strain conddition becaause, in this case, the floow neither can be considdered outw ward nor inwaard. But let us assume it iss a factor K of the averrage of the othher two princippal stresses soo that: (8) (6) A Accordingly w we can write: The von Misees constant k, is usually obttained from unniaxial coompression tesst. In such a ttest; and are taken eqqual to zeero. The valuee of b is zero aand the constaant k would bee equal to the unconfiined compresssive strengthh, qu. If these two crriteria are set equal e at uniaxiial compressioon as it is usuaal (Fig. 3)); envelops off both appear aas straight linees parallel to -axis onn the - diagrram. While the raddius of Mohr ccircle at failure given by Treesca is independent off 2 and b; thhe radius giveen by von Miises is T ratio of thee radii of thesee two failure ccriteria fuunction of b. The in Mohr diagram m would be a function of b so s that: (7) This functionn is drawn vs. b in Fig. 4. International Scholarly and Scientific Research & Innovation 9(12) 2015 . (9) Iff the material is linear elasstic; K=2; annd if the Poissson’s ratioo ; is 0.5; K would be 1. The value off b relevant too this condition would be b 0.5. Inn general; we can say K0 iss a reasonablee assumption ffor K in pplane strain coondition. This is consistent w with the conddition =00 relevant to undrained behhavior of satuurated clay which w givees K=1 and bb=0.5; i.e.; thhe intermediatte principal stress s wouuld be equal to the averaage of the othher two princcipal stresses. Iff we assume associated fflow rule andd take the pllastic poteential functionn g the same aas von Mises yield functionn, we can investigate thhe plane strainn condition 2=y=0, by seetting 1369 scholar.waset.org/1999.6/10003155 World Academy of Science, Engineering and Technology International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015 ⁄ equal too zero [5], [6]]. The followiing will result from this operation onn (5): International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155 0 (10) This gives bb=0.5 and = -60o; where there is maxximum deeviation of vonn Mises from Tresca (Fig. 3). Therefore;; if the sooil behavior obbeys associated flow rule annd the yield criiterion is von Mises; b should be 0.55 (Fig. 5). Suubstituting thiss value /√3. If we compaare this with tthat of off b in (6) givees Trresca, we cann conclude thaat if these tw wo criteria are to be m matched in planne strain conddition; k shoulld be equal too √3 . W We should rem member that this result has been obbtained asssuming assocciativity. But the soil behaavior is usuallly not asssociated and the value of b in this casse may be diffferent froom 0.5. It is innteresting to ssee the effect of this deviation on the solution off typical planne strain probblems. The bbearing wn in Fig. 1 onn a saturated cclay in caapacity of stripp footing show unndrained conddition can be considered aas an examplee. The am mount the valuue of b is diffferent from 0.55 can be conssidered ass the degree off nonassociativvity of soil behhavior. The soolution is obtained usinng the rigorouus method off characteristiccs [2], [77], [8]. For thee associated caase; the value of bearing caapacity factor Nc is 5.14. The soluution for nonaassociative caases is obbtained when different values of b aree put in (6) using √3 . The result is shown in Fig. 6 and this figure indicates reducttion in bearingg capacity withh increase in degree d off nonassociativvity of soil. The axisymm metric case is not as straighht forward as plane strrain. The princcipal strain 2= is not zeroo in this case tto help finnding the proper path. Thee b value can vary with diistance froom the axis off symmetry. A Another imporrtant point is thhat the intermediate principal stress enters the eqquilibrium equuations in this case. Theerefore; in conntrast to the pplane strain casse; the efffect of 2 andd b in this caase is debatabble even if thee yield crriterion dos noot include 2. Fig. 7 showss the variationn of Nc foor a circular fo footing with b as predicted by Tresca annd von M Mises criteria. The startting point oof all curvves is appproximately Nc=5.7. As (3) indicates; Trresca yield criiterion dooes not dependd on 2. Therefore; the curvve related to Tresca T in Fig. 7 indiicates that enntry of higher values off b in eqquilibrium equuations causes reduction inn bearing cappacity. Thhis is also cleaar when we ccompare the cuurves related tto von M Mises. The dashhed line curvee indicates thatt the entry of higher vaalues of b in yyield criterionn has an increeasing effect oon Nc; buut when their eeffect in equillibrium equatiions is also alllowed; their increasing effect is substtantially ceaseed (full line cuurve). Fig. 5 Comparisoon between Tresca and von Miises at plane straain conddition Nc Smoooth Strip Footinng on Saturated clay 5,14 5,1 5,06 5,02 4,98 b 4,94 0,25 0,33 0,35 0,4 0,45 00,5 0,55 0,6 0,65 0,7 0,75 Fig. 6 Effect off variation of b on Nc factor off a saturated clayy Nc Smoooth Circular Foooting on Saturatted Clay b=0 inn equil.eeq. (V) 6,6 6,4 6,2 5,8 b>0 inn equil. V) Eq. (V 5,6 Trescaa 6 5,4 5,2 5 0 0,1 0,2 0,3 0,4 0,55 b Fig. 7 Effecct of b on Nc viaa yield and equiilibrium eqs. III. STRESS LEVEL DEPPENDENT YIELD CRITERIA W We have seleected Mohr-C Coulomb (M--C) and DrucckerPragger (D-P) yield criteria for oour investigatiion in this partt [9]; but the discussionn can be extennded to Lade-D Duncan, MasuuokaNakkai and others as well. Heree, the M-C cann be considereed as an eextension of T Tresca becausee it does not ddepend on 2. D-P is sttress level deppendent form of von Mises and considerrs 2. For the sake of clarity; we lim mit our discuussion to frictiional soil. The M-C in this case can bbe written as: (11) whiich can be easiily written in International Scholarly and Scientific Research & Innovation 9(12) 2015 1370 scholar.waset.org/1999.6/10003155 form m as: World Academy of Science, Engineering and Technology International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015 The D-P criteerion in the abbsence of coheesion can be w written ass [9], [10]: (12) whhere I1 is the first invariant of stress tenssor and paarameter. It caan be easily shhown that thee D--P in this casee, is: is thhe D-P foorm of (13) International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155 √ Comparison w with similar foorm of M-C, inndicates that: √ √ the D-P parameteer , on the basis of (15). W We then substtitute this value of inn (14) and finnd an equivallent value for in term ms of b. In thiis way; the vaariation of b aaffects the matterial equivalent strenggth parameter . Values off b different from whaat is given by (16) will resuult in equivaleent friction anngles lesss than the origginal. This is cclear from Figg. 8 as we seee that the D-P circle liees inside of M M-C hexagon. Fig. 9 showss the variiation of bearinng capacity faactors Nq and N for smooth strip footting with b when =30. The value of b relevannt to assoociative behaavior in this case is 0.755 and the fiigure indiicates lower bbearing capacitty factors wheen the behavioor of soil is nonassociated; i.e.; w when b is diffferent from 0.75. 0 Theerefore; it is nnot conservativve to go with the usual beaaring capaacity factors iff the soil behaavior is really nnonassociatedd. (14) Nq Strip Footing D-P (=30) 18,9 whhich gives thee D-P parameeter , for maatching with M M-C in geeneral. Assuming thhe behavior tto be associaated; we agaain set ⁄ equal to zero [3], [[4] to get thee matching in plane strrain conditionn. This operatioon on (12) willl result in: 18,4 17,9 18,4 4069 17,4 16,9 16,4 b 15,9 (15) 15,4 0,5 0,55 0,6 0 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 Applying thee requirementt for general m matching exppressed byy (14) to this equation e resultts in: ((a) (16) Nγ whhich gives thee value of b inn terms of ffor matching oof D-P wiith M-C in pplane strain condition. c In other words; both crriteria predict the t same strenngth for the soil if the value of b is asssumed to be thhat given by (16). ( Fig. 8 shoows the matchhing of D--P and M-C inn plane strain condition. c Strip Footingg D-P ( =30) 7,9 7,65 7,4 7,,657 7,15 6,9 6,65 6,4 b 6,15 0,5 0,55 0,6 0,65 0,7 00,75 0,8 0,85 00,9 0,95 1 (b) Fig. 9 Effect of varriation of b on ((a) Nq and (b) N for strip footinng ( =330o) Fig. 8 Matchinng of Drucker-P Prager and Mohhr-Coulomb critteria The actual vaalue of b can bbe different froom what is givven by (16) due to nonnassociativity in soil behavvior. To investigate the effect of vvariation of b on the soluttion of plane strain prroblems like bbearing capacitty; we can asssume we have found International Scholarly and Scientific Research & Innovation 9(12) 2015 A As mentionedd before; thee intermediatee principal stress s partticipates in thhe equilibrium m equations in case of axial sym mmetry. In ordder to see the eeffect of its enntry in equilibrium equations on beaaring capacityy calculations;; we have chhosen the M-C criterionn because it dooes not includde 2. It is usuual to assuume 2=3 in this conditionn i.e.; b=0. Buut this is merelly an assuumption and it i is interestinng to see if 2 is really greater thann 3; how aree the values oof Nq and N influenced i byy this mattter. The resultt of investigattion for = 300o is demonstrrated by tthe curve draw wn for M-C iin Fig. 10. Thhe curve indiccates entrry of higher values of b in equilibrium eqquations resullts in 1371 scholar.waset.org/1999.6/10003155 International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155 World Academy of Science, Engineering and Technology International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015 lower values for bearing capacity factors, Nq and N. Matching of other yield criteria like D-P with M-C under axial compression relevant to bearing capacity problem of circular footings is made at the top point of Fig. 8, where b is zero. For other points; the D-P circles lies outside of the M-C hexagon. This indicates higher strength with increase in b. This effect is demonstrated by the dashed line curve in Fig. 10. When the effect of 2 being greater than 3 is considered both in equilibrium and yield equations; the full line curve of Fig. 10 is obtained. Comparison of these two curves indicates again, the reduction in Nq and N due to participation of 2 in equilibrium equations. The full line curve indicates the net effect has been increase in bearing capacity factors in case of = 30o. Nq Circular Footing (=30) 120 b=0 in equil.eq. (D-P) 105 b>0 in equil. Eq. (D-P) 90 plane strain and axisymmetric conditions. It was found that in plane strain condition; the value of 2 may be different from what is usually assumed and this may be due to nonassociativity in soil behavior. It was shown that under such a condition, a lower bearing capacity is obtained. Therefore; the usual solution to bearing capacity of strip footings would not be conservative. In the axisymmetric conditions however; 2 enters the equilibrium equations as well; and if the Mohr-Coulomb criterion is employed for bearing capacity calculation of circular footings; increase in 2 relative to other principal stresses results in decrease in bearing capacity. Therefore; if in reality, 2 is greater than 3; this effect should be considered in calculations. If other criteria that include 2 are used instead of Mohr-Coulomb; the calculated bearing capacity may be less or more than what is usually calculated, depending on the criterion used, and the amount 2 has been taken more than 3. In case of Drucker-Prager for example; the net effect has been shown here to be a little increase in the calculated bearing capacity. M-C 75 REFERENCES 60 [1] 45 30 b 15 0 0,1 0,2 0,3 0,4 (a) Circular Footing (=30) N 25 23 21 19 17 15 13 11 9 7 5 3 b=0 in equil.eq. (D-P) b>0 in equil. Eq. (D-P) M-C b 0 0,1 0,2 0,3 P. V. Lade “Assessment of test data for selection of 3-D failure criterion for sand.” Int. J. Numer. Anal. Meth. Geomech, 2006; 30:307–333. [2] R. O. Davis and A. P. S. Selvadurai, Plasticity and Geomechanics. Cambridge University Press, 2002, 287 p. [3] H. Tresca, Sur l’ecoulement des corps solids soumis `a de fortes pression, Comptes Rendus Acad. Sci. Paris, 1864, 59, 754. [4] R. von Mises, Mechanik der festen Koerper im plastisch-deformablen Zustand, Nachr. d. K. Ges. d. Wiss G¨ottingen, Math.-Phys. Kl., 1913, 582–592. [5] V. G. Solberg “Development and Implementation of Effective Stress Soil Models.” M.Sc. Thesis, Under Supervision of Dr. S. Nordal, Department of Civil and Transport Engineering, Norwegian University of Science and Technology. (2014).188 pp. [6] G. Vikash, A. Prashant “Calibration of 3-D Failure Criteria for Soils Using Plane Strain Shear Strength Data.” GeoShanghai 2010 International Conference, pp.86-91. [7] M. E. Harr, Foundation of Theoretical Soil Mechanics. McGraw-Hill, 1966, New York, 381 p. [8] C. M. Martin, “Exact bearing capacity calculations using the method of characteristics.” Turin 4, 2005, pp. 441-450. [9] H. Jiang, Y. Xie “A note on the Mohr-Coulomb and Drucker-Prager strength criteria.” Mech. Research Comunications, Vol. 38, 2011, pp. 309-314. [10] D. C. Drucker, W. Prager, ‘Soil mechanics and plastic analysis or limit design’, Quarterly of applied mathematics 10(2), 157–165. (1952). 0,4 (b) Fig. 10 Effect of variation of b on (a) Nq and (b) N for circular footing IV. CONCLUSION Tresca and Mohr-Coulomb yield criteria have long been criticized for not considering the effect of intermediate principal stress. As typical stress level- dependent and independent yield criteria that consider the effect of 2; von Mises and Drucker-Prager were selected for investigating the effect of 2 on solution of bearing capacity problem under International Scholarly and Scientific Research & Innovation 9(12) 2015 1372 scholar.waset.org/1999.6/10003155