LIMIT ANALYSIS AND LIMIT EQUI LIBRIUM SOLUTIONS IN SOIL .
------------ . . .Soil Mechanics. . - .-- .and Theories of PlasticityLIMIT ANALYSISAND LIMIT EQUI LIBRIUMSOLUTIONS IN SOIL MECHANICSbyWai F. ChenCharles R. ScawthornFritz Engineering Lab9ratory Report No. 355.3. . .-
Soil Mechanics and Theories of PlasticityLIMIT ANALYSIS AND LIMIT EQUILIBRIUMSOLUTIONS IN SOIL MECHANICSbyWai F. :.ChenCharles R. :.Scawthornr.,Fritz Engineering LaboratoryDepartment of Civil EngineeringLehigh UniversityBethlehem, PennsylvaniaJune 1968Fritz Engineering Laboratory Report No. 355.3
TABLE OF CONTENTSPageABSTRACT11.INTRODUCTION22.FUNDAMENTALS OF LIMIT ANALYSIS62.1Basic Concepts6-2.2Limit Theo.rems92.3The Dissipation Functions10THE STABILITY OF VERTICAL CUTS173.1Limit Equilibrium Analysis173.2Plastic Limit Analysis193.3Soil Unable to Take Tension213.4.5.LATERAL EARTH PRESSURES254.125Introduction4.2 Plastic Limit Analysis27THE BEARIN; CAPACITY OF SOILS335.1Limit Equilibrium Analysis335.2Plastic Limit Analysis--Upper Bounds35-5.3Plastic Limit Analysis--Lower Bounds415.4Limit Analysis of Three Dimensional Problems436.CONCLUSIONS'467.ACKNOWLEDGMENTS478 TABLE AND FIGURE S489.APPENDIX7010.NOMENCLATURE7311 .REFERENCES75
ABSTRACTIdealizations in soil mechanics are usually necessary inorder to obtain solutions and to have these solutions in a readilyapplicable form.Limit qui1ibriumhas been a method of solvingvarious soil stability problems.One weakness of the limit equilibrium method has been theneglect of the stress-strain relationship of the soil.Accordingto the mechanics of. solids, this condition must be satisfied for acomplete solution.criterion and itsrelationship.'Limit analysis, through the concept of a yield ssociatedflow rule, considers the stress-strainHowever, a soil with cohesion and internal frictionis not modeled accurately by a theory of perfect plasticity.Never-theless, indications are that the stability problems in soil mechanicswill, in time, be computed on the basis of the limit theorems of plasticity.A discussion is given, therefore, of the significance ofthe limit analysis inidealizations.ter of the real behavior of soils and theirWith this background, the meaning of existing limitequilibrium solutions is discussed, and the power and simplicity ofapplication of the limit analysis method is demonstrated.
1.INTRODUCTIONSoil mechanics, a science of relatively recent origin, hasbeen well developed since Karl Terzaghi'sl pioneering efforts in theearly twentieth century.2As pointed out by Drucker , one peculiarfeature of soil mechanics has. been the lack of interrelation between methods used to treat similar problems with different purposesin mind.In foundation problems, for instance, the stressdis ributionunder a footing is determined from Boussinesq's solution for the stressdistribution under a vertical load on a semi-infinite plane--a solutionfrom linear elasticity theory.' On the other hand, the bearing capacityof a footing is determined using limit equilibrium (or the sliplinelution) of plasticity theory.80-The calculation of the settlement of afooting actually utilizes visco-elastic theory to describe the materialbehavior with time.The reasons for treating the problems differently are evident:The key to obtaining the complete solution, sparting with a considerationof elastic action and proceeding to contained plastic flow, and final1yunrestricted plastic flow, requires the basic knowledge of the stressstrain relations for soils in the elastic as well as the inelastic range.No such general relations have been determined as yet, for an inelasticsoil.Even with an appropriate idealized stress-strain relationship', thedetails of the distribution of stress and strain in the complete solutionare. far too complicate'd and probably not possible.-2-·The solutions, if
possible to obtain, would be too involved and impractical for application.Necessary idealizations and simplifications are, there-fore, used by engineers to obtain solutions for practical problems.These simplified analyses must, ultimately, be justified by com.parison with available rigorous solutions, or, in the event theseare lacking, experimental results.As an illustration of this point, let us examine a problemand the simplifications involved.In order to solve stability prob-lems, which are the primary concern of this paper, such as lateralearth pressures, bearing capacity, or stability of vertical cuts,use is made of only limit equilibrium conditions3(that is, the stressconditions in ideal soils immediately preceeding ultimate failure).No thought is given to the corresponding state of strain.This methodof. attack raises doubts, if one recalls from mechanics of solids, thata valid solution requires satisfying the boundary conditions, equationsof equilibrium (or of motion), equations of compatibility, and thestress-strain relationship.It is this stress-strain relationshipwhich connects equilibrium to compatibility, and which distinguisheselasticity from plasticity or visco-elasticity theories.Withoutcon sidering the stress-strain relationship, a so-called solution is merelya guess.On the other hand, the limit equilibrium approach simplifiesthe theoretical analysis drastically, and provides good predictionsof the ultimate load of soil stability problems which are otherwise unavailable for practical applications.-3-
A more rigorous approach is to consider the stress-strain relationship in an idealized manner.This idealization, termed normality(or the flow rUle)4, establishes the limit theoremsnalysis is based.5on which limit a-Within the framework of this assumption, the approachis rigorous and the techniques are competitive with those of limit equilibrium, in some instances being much simpler.rems of Drucker, Prager, and Greenberg5The plastic limit theo-may conveniently be employed toobtain upper and lower bounds of the ultimate load for stability problems, such as the critical heights of unsupported vertical cuts, or thebearing capacity of nonhomogeneous soils.It is the objective of this paper, through a review of thestandard and widely known techniques used in the solutions of soil stability problems, to accomplish two purposes.The first is to discussthe meaning and nature of existing, "classical", soil mechanics solutionsfrom the limit analysis point of view.Many of the techniques will beshown to implicitly contain the basic philosophy of one or both of theplastic limit theorems.The second purpose is to demonstrate the usefulness and powerof the plastic limit theorems in developing a limit analysis technique.Useful information, although sometimes crude, will be quickly obtained.It will be seen, by comparing numerical results of the classical andlimit analysis solutions, that good agreement is usually obtained.Thelimit analysis technique will provide new solutions, or an alternativemethod which is more rational than existing techniques.-4-
(1)The critical height of a vertically unsupported cut,(2)Active and passive lateral earth pressure, and(3)The bearing capacity of soils.-5-
2.2.1FUNDAMENTALS OF LIMIT ANALYSISBasic ConceptsThe mechanical behavior of soils is usually described ashaving both cohesion, c, and internalfriction, .The resistance ofsoils to deformation is furnished by cohesion and friction across thepossible slip planes in a mass of material.The generally acceptedllaw of failure in soil mechanics is Coulomb's criterion , which statesthat slip occurs when, on any plane at any point in a mass of soil,the shearstress,' ( 0) reaches an amount that depends linearly uponthe cohesion and the normal stress, cr (here taken to be positive incompression), i.e. (Fig. 1)1" C crtan (1)For illustrative purposes, a simple physical model, shown inF g.2, may be helpful.In the figure, a layer of dense granularterial is subjected to the action of two forces.The force Pnma acts atright angles to the plane 1-1, whereas the other, the force Pt,"actstangentially to that plane.Let us further assume that Pstant during the experiment, whereas, Ptto the value which will produce sliding.value Ptnremains con-gradually increases from zeroAt the instant of sliding, themust not only overcome cohesion, but also must exceed the re-sistance furnished by two types of friction.The first of these ariseson the contact surfaces of adjoining particles and is termed ,surfacefriction.The second, offered by the interference-6-o the particles
themselves to changes of their relative position, is termed interlocking friction.It is this interlocking friction thatrequ resthedisplacement upward, as well as the usual displacement to the side.The displacement vector must, therefore, make an angleeto the slipplane.If soil were idealized as perfectly plastic with Coulomb'slaw of yielding, then Eq. (1) defines the yield curve in the stressspacecr, .If now a stress state, represented by a vector from theorigin, is increased from zero, yield will be incipient when the vec'tor reaches the curve (two straight lines).For a perfectly plasticmaterial, the vector representing the stress- state at any given pointcan never protrude beyond the curve, since it is an unattainable stressstate in granular media.Let us further assume that the stress-strain relation issuch that the plastic strain rate vector is always normal to the yieldcurve when their corresponding axes are superimposed (Fig. 1).It canbe seen from the figure that this is equivalent to assuming e inFig. 2.The perfectly-plastic idealization with associated flow rule(normality) is illustrated by a block shearing on a horizontal plane,Fig. 3a.Volume expansion is seen to be a necessary accompaniment toshearing deformation according to the idealizations.proposed by Drucker and Prager6This theory was7and generalized later by Drucker , and8Shield .-7-
In contrast to the above mentioned effort, one may idealizesoil as a frictional material for which the interlocking friction isignored.Deformation occurs by the smooth sliding of adjacent sur-faces of material points (see Fig. 3b).Iftan in expression (1)denotes the coefficient of friction between adjacent surfaces ofm terial points along the plane, expression (1) becomes the well-knownCoulomb friction limit condition for the shear strength of soil.Theimportant difference between CoulonID friction and perfectly-plasticCoulomb action is seen in Fig. 3, where frictional sliding is horizontal while perfectly-plastic shearing involves large upward vertical motion.If the plastic strain rate vector is superimposed to. theCoulomb limit curves (assumed as yield curve), the normality rule doesnot hold (See Fig. 1).The extent of this endeavor is described in a9recent work by Dais .Real soils are quite complex and are still imperfectly understood.plastic.They are neither truly frictional in behavior, nor are theyHence, any such idealized treatment, as discussed above,will either result in some differences between predictions and experimental facts, or will entertain certain mathematical difficulties.For example, the dilatation which is predicted by perfect plastictheory to accompany the shearing action will usually be larger than"that found in practicelO The inadequacy of a perfectly plasticidealiz-ation has been discussed by Drucker 2 ,11,12 and DeJong 13 amongothers.The lack of uniqueness for solutions to problems using fric-9tion theory has been exhibited and explored by Dais .-8-
In order to improve upon the perfectly plastic theory,Drucker, Gibson, and Henkel introduced the strain-hardening theories"14 ,o f 801'1 p 1 ast1c1tyWh'1C hwere 1 ater exten d e d b y Jen1'ke nd Sh'1e·1d 15The work-hardening plastic action may involve upward or downward vertical motion, or neither, of the sliding block as illustrated in Fig.3, which qualitatively agrees with experimental data.Recently, moresophisticated theories have been proposed by Weidler and PaslaySpencer1718 19,and Sobotka'16,on non-homogeneous soils in an attempt toovercome some of the known deficiencies in previous theories,C1earl the development of a more sophisticated theory will almost always bringa more elaborate stress-strain relation.Solutions to practically im-portant problems, on the contrary, become exceedingly difficult toobtain, if the stress-strain relation is too involved.A compromisemust, therefore, be made between convenience and physical reality,In certain circumstances, such as in the stability problemsof soil mechanics, there appears to be reasonable justification forthe adoption of a limit analysis approach based upon Coulomb's yieldcriterion and its associated flow rule in soils, as discussed in Section 1.2,2Limit TheoremsThe foundations of limit analysis are the two limit theo-5rerns,For any body or assemblage of bodies of elastic-perfectlyplastic material they may be stated, in terminology appropriate to-9-
soil mechanics, as:Theorem 1 (lower bound)--If an equilibrium distribution of stress canbe found which balances the applied load andnowhere violates the yield criterion, whichincludes c, the cohesion, and ,the angleof internal friction, the soil mass will notfail, or will be just at the point of failure.Theorem 2 (upper bound)--The soil mass will collapse if there is anycompatible pattern of plastic deformationfor which the rate of work of the external·loads exceeds the part of internal dissipation.According to the statement of the theorems, in order to properly bound the "true" solution, it is necessary to find a compatiblefailure mechanism (velocity field or flow pattern) in order to obtainan upper bound solution.A stress field satisfying allTheorem 1 will be required for a lower hound solution.cond tionsofIf the upperand lower bounds provided by the velocity field and stress field coincide, the exact value of the collapse, or limit, load is determined.2.3The Dissipation FunctionsAs stated in the upper bound theorem, it is necessary to com-pare the rate of internal dissipation of energy with the rate of workof external forces.The dissipation of energy, D, per unit volume dueto a plastic strain ratei ,therefore, of primary importance.-10-It can
be shown in general that the dissipation function has the simplerform20D(2)where e t denotes a positive principal component of the plastic strainrate tensor.For the particular case of plane strain, the expression (2)reduces toDwhere'\11max-[( )2 y'xyxyc cos0'\1I(3)max2 J1 / 2 18·. the maX1mum"ra t e a feng1neer1ngshear strain.An alternative derivation in terms more familiar to the engineer will be discussed in what follows.stricted to the plane strain case.The discussion will be re-A number of familiar shear de-formation zones, which are especially useful for soil mechanics, aretreated as illustrative examples.Some of the results have been dis-nection with later application.cussed by ChenThe results are adequate in con-21 .Homogeneous Shearing Zone--The energy dissipated in the homogeneous shearing zone, Fig. 4, is-11-
(4).in whichY is the shear strain rate and t is the thickness of the zone.The dimension perpendicular to the plane of the paper in Fig. 4 istaken as unity and the width of the zone is denoted by b.Then therate of energy dissipated per unit volume, D, is the total dissipationin Eq. (4) divided by the volume, bt:or(5).Dy (1" - cr tan0)Since the Coulomb yield criterion must be satisfied in the plastic zoneit follows from Eq. (1) that( 6)D c 'VIt should be noted that the shear strain rate,zone is not the maximum shear strain rate yy ycos0.maxmaxY, in the,but is related byThis is the consequence of volume expansion accompaniedby plastic shearing.The Mohr circle shown in the figure indicates.clearly the relationship between y and ymax.Figure 5 shows a number of examples of differently shapedhomogeneous shearing zones.Fig. 5a is a part of Fig-. 4 as shown by-12-
the dotted lines in the figure.Fig. 5b is the half field of homo-geneous deformation of Fig. Sa, while a proper rigid body rotationof Fig. Sb results in the interesting field of Fig. Sc.Narrow transition layer--If the thickness t in Fig. 4 isvery thin, the homogeneous shearing zone may beimagined as in Fig.6, to be a simple discontinuity with a discontinuous tangential velocityauOu tan0. tyand a discontinuous normal separation Ov tytan0 The rate of dissipation of work per unit of discontinuitysurface isPDA bPOu - b n Ovtor(7)D Aeu( -cr tan0) couFigure 6 shows clearly that a simple slip Ou must always beaccompanied by a separation ov for0 O.The familiar circular surfaceof discontinuity is, therefore, not a permissible surface for rigidbody sliding because of the separation requirement forc- plane surface and the logarithmic spiral surface of anglesoils. Theare theonly two surfaces of discontinuity which permit rigid body.motions relative to a fixed surface.Zone of radial shear when c- Anapproximation to a zone ofradial shear is given in Fig. 7a where six rigid triangles at an equal-13-
central angle be are shown.Energy dissipation takes place along theradial lines O-A, O-B, O-C, etc. due to the discontinuity in velocitybetween the triangles.Energy is also dissipated on the discontinuoussurface D-A-B-C-E-F-G since the material below this surface is considered at rest.Since the material must remain in contact with thesurface D-A-B-C-E-F-G, the triangles must move parallel to the arcsurfaces.other.The rigid triangles must also remain in contact with eachHence, the compatible velocity diagram of Fig. 7b shows thateach triangle of the mechanism must have the same speed.With expression (7), the rate of dissipation of energy caneasily be calculated.O B,The energy dissipation along the radial linefor example, is the cohesion c multiplied by the relative ve-locity, au, and the length of the line of discontinuity, r:· 2 e)c r (2V S1n( 8)where the relative velocity 6u, appears as (2V sin e).Similarly,the energy dissipation along the discontinuous surface A-B isc (2r sin e)Vwhere the length of A-B is (2r sin e)and 6u(9) y.Since the energydissipation along the radial line O-B is the same as along the arcsurface A-B, it is natural to expect that the total energy dissipation-14-
in the zone of radial shear, D-O-G, with a central angle@ will beidentical with the energy dissipated along the arc D-G.This isevident since Fig. 7a becomes closer and closer to the zone of radialshear as the number n increases.At the limit, when n approachesinfinity, the zone of radial shear is recovered.The total energydissipated in the zone of radial shear is the sum of the energy dissipated along each radial line when the number n approaches infinity: D lim (2 c r V sin 2n) nI'}'-tJll'COor D 2c·r V lim n sin2n11'-""'"00or(10)D c V (r )where@be nLog spiral zone of·c-0 soils--The extension of the abovediscussion to the more general case of a log spiral zone ofsoils is evident.c- A picture of six rigid triangles, at an equal angleA8 to each 9ther, is shown in Fig. 8a.It is found that the energydissipation in a log spiral zone of c-0 soil is equal to the energydissipated along the spiral discontinuity surface, which is:. 15-
D cSrVde 6cS0(r -expetan ) (V0 0expetan0) deorD 21C Var o cot (exp2@Dtan0 - 1)where V , r , and8 are shown in Fig. 8a.o0A detailed discussion ofthe log spiral zone is given in the Appendix.-16-(11)
3.3.1THE STABILITY OF VERTICAL CUTSLimit Equilibrium AnalysisThe comparison between limit equilibrium and plastic limitanalysis can be illustrated by evaluatirig the stability of soil in avertical bank.The height at which an unsupported vertical cut, asillustrated in Fig. 9, will collapse due to the weight of the soilwill be defined as the critical height, H .crThe conventional a-nalysis (limit equilibrium) will be examined first and then comparedto the method of limit analysis.It is common practice to evaluate this problem by the equilibrium method.at an angleeThe failure surface is assumed to be a plane inclinedto the horizontal (See Fig. 9) and Coulomb's law offailure is applied.From
neglect of the stress-strainrelationship of the soil. According to the mechanics of. solids, this condition must be satisfied for a ' complete solution. Limit analysis, through the concept of a yield criterion and its ssociatedflow rule, considers the stress-strain relationship. However, a soil with cohesion and internal friction
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Does anyone know what kind of triangle this is? 'Equilateral' or 'Equiangular' Both are correct. What do "equi" and "lateral" mean and does anyone know where the words come from? 'Equal lengths’ or ‘Equal sides’ The prefix ‘equi-‘ means ‘equal’ and ‘lateral’ means ‘sides’, both from Latin. Also equiangular means having
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Test Day Written Exam . Atterberg Limits 31 - Liquid Limit - - Plastic Limit - Plasticity Index Plastic Limit of Soils AASHTO T 90 Plastic Limit 32. 2022 15 Introduction Plastic limit is the lowest moisture content at which the soil remains plastic A soil is at its plastic limit when
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