Limit Analysis And Limit Equilibrium analysis in Soil mechanics

7/28/2019 Limit Analysis And Limit Equilibrium analysis in Soil mechanics

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Soil Mechanics and Theories of Plasticity

LIMIT ANALYSISAND LIMIT EQUI LIBRIUMSOLUTIONS IN SOIL MECHANICS

byWai F. ChenCharles R. Scawthorn

Fritz Engineering Lab9ratory Report No. 355.3


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Soil Mechanics and Theories of Plas t ic i ty

LIMIT ANALYSIS AND LIMIT EQUILIBRIUMSOLUTIONS IN SOIL MECHANICS

byWai F. :.ChenCharles R. : .Scawthorn

r.,

Fri tz Engineering LaboratoryDepartment of Civi l EngineeringLehigh UniversityBethlehem, Pennsylvania

June 1968

Fri tz Engineering Laboratory Report No. 355.3


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TABLE OF CONTENTS

ABSTRACT1. INTRODUCTION2. FUNDAMENTALS OF LIMIT ANALYSIS

2.1 Basic Concepts2.2 Limit Theo.rems2.3 The Dissipation Functions

3. THE STABILITY OF VERTICAL CUTS

3.1 Limit Equilibrium Analysis3.2 Plast ic Limit Ana lysi s3.3 Soil Unable to Take Tension

4. LATERAL EARTH PRESSURES4.1 Introduction4.2 Plast ic Limit Analysis

5. THE BEARIN; CAPACITY OF SOILS5.1 Limit Equilibrium Analysis5.2 Plast ic Limit Analysis--Upper Bounds5.3 Plast ic Limit Analysis--Lower Bounds5.4 Limit Analysis of Three Dimensional Problems

6. CONCLUSIONS7. ACKNOWLEDGMENTS

8 TABLE AND FIGURES9. APPENDIX

10 . NOMENCLATURE11 . REFERENCES

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ABSTRACT

Ideal izat ions in soi l mechanics are usually necessary inorder to ob ta in solut ions and to have these solutions in a readilyapplicable form. Limit ~ q u i 1 i b r i u m has been a method of solvingvarious so i l s tabi l i ty problems.

One weakness of the l imi t equilibrium method ha s been theneglect of the stress-strain relationship of the soi l . Accordingto th e mechanics of. solids , this condition must be sat isf ied for a 'complete solution. Limit analysis, through the concept of a yieldcri ter ion and i t s ~ s s o c i a t e d flow rule , considers the s t ress-s t ra inrelationship. However, a so i l with cohesion and internal fr ict ioni s not modeled accurately by a theory of perfect plas t ic i ty . Never-theless, indications are that the s tabi l i ty problems in soi l mechanicsw ill, in time, be computed on the basis of the l imi t theorems of plas-t ic i ty . A discussion is given, therefore, of th e signif icance ofthe l imit analysis in t e r ~ of the real behavior of soi ls and theiridealiza t ions . With th is background, the meaning of existing l imitequil ibr ium solutions is discussed, and th e power and simplicity ofapplication of the l imit analysis method is demonstrated.


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1 . INTRODUCTION

Soil mechanics, a science of re la t ively recent origin, hasbeen well developed since Karl Terzaghi ' s l pioneering ef for ts in theearly twentieth century. 2As pointed out by Drucker , one peculiarfeature of so i l mechanics has. been the lack of interre la t ion be-tween methods used to t r ea t similar problems with different purposesin mind.

In foundation problems, for instance, the s tress d i s ~ r i b u t i o nunder a footing is determined from Boussinesq's solution for the s tressdis tr ibution under a ver t ica l load on a s emi- in f in i te p lane -- a so lu ti onfrom l inear elas t ic i ty theory. ' On the other hand, th e bearing capacityof a footing i s determined using l imi t equ il ib rium (or the s l ipl ine 80 -lut ion) of plas t ic i ty theory. The calculat ion of the settlement of afooting actual ly ut i l izes visco-elas t ic theory to describe the materialbehavior with t ime.

The reasons for treating th e problems differently are evident:The key to obt ain ing th e complete solution, spart ing with a considerationof elas t ic action and proceeding to contained plas t ic flow, and final1yunrestricted plast ic flow, requires the basic knowledge of th e s tress-strain re la t ions for so i ls in the elas t ic as well as the ine las t ic range.No such general re la t ions have been determ ined as yet , for an ine las t icso i l . Even with an appropriate idea l ized stress-strain relationship', th edeta i ls of the dis tr ibution of s tress and strain in the complete solutionare. far to o complicate'd and probably not possible. The solutions, i f

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possible to obtain, would be t oo invo lved and impractical for applicat ion. Necessary idealizations and s impl if ica ti ons a re , therefore, used by engineers to o bta in s olu tio ns for practica l problems.These simplified analyses must, ultimately, be just i f ied by com-.parison with available rigorous s olu tio ns , o r, in the event theseare lacking, experimental results .

As an i l lus t ra t ion of th is point, le t us examine a problemand the simplif icat ions involved. In order to solve s tabi l i ty problems, which are th e p rimary concern of this paper, such as l a tera learth pressures, bearing capacity, or s tabi l i ty of ver t ica l cuts,use is made of only l imi t equilibrium conditions3 ( tha t is , the s tresscondit ions in ideal soils immediately preceeding ultimate fa i lure) .No thought is given to the corresponding s ta te of s t ra in . This methodof. attack raises doubts, i f one recalls from mechanics of sol ids , thata val id solution requires satisfying the boundary conditions, equationso f equ il ib rium (or of motion), equations o f compa tibi li ty , and thes t ress-s t ra in relationship. I t is this s t ress-s t ra in relat ionshipwhich connects equ il ib rium to compatibili ty, and which distinguisheselas t ic i ty from plas t ic i ty or visco-elas t ic i ty theories. Without c o n ~sidering the s t ress-s t ra in relationship, a so -ca ll ed solut ion is merelya guess. On th e other hand, the l imit equilibrium approach simplif iesthe theoret ical analysis drastically , and provides good predictionsof th e ultimate load of soi l s tabi l i ty problems which a re o therwis e unavailable for pract ical applicat ions.

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A more r igorous approach is to consider the s t ress-s t ra in re-la tio nsh ip in an idealized manner. This ideal izat ion, termed normality(or th e flow rUle)4, establishes the l imit theorems5 on which l imi t a-nalysis is based. Within the framework of th is assumption, the approachis rigorous and the tech niq ues are competitiv e with those of l imit equi-librium, in some ins tances being much s imple r . The p la st i c l im it theo-

5rems of Drucker, Prager, and Greenberg may conveniently be employed toobtain upper and lower bounds of the ultimate load for s tabi l i ty prob-lems, such as the cr i t ica l heights of unsupported ver t ica l cuts , or thebearing capacity of nonhomogeneous soi ls .

I t is the obje ctiv e o f this paper, through a review of th estandard and widely known techniques used in the solutions of so i l s ta-bi l i ty problems, to accomplish two purposes. The f i r s t is to discussth e meaning and nature of exis t ing, "class ical" , so i l mechanics solutionsfrom the l imit analysis point of view. Many of th e techniques will beshown to implicit ly contain th e basic philosophy of one or both of th eplas t ic l imit theorems.

The second purpose is to demonstrate th e usefulness and powerof th e plast ic l imit theorems in developing a l imit analysis technique.Useful informat ion, al though sometimes crude, wil l be quick ly ob ta ined.I t will be seen, by comparing numerical results of the class ical andl imi t analysis solutions, that good agreement is usually obtained. Thel imi t analysis technique will provide new solutions, or an al ternat ivemethod which is more ra t ional than existing techniques.

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(1 ) The cri t ical height of a vert ical ly unsupported cut,(2 ) Active and passive lateral earth pressure, and(3 ) The b earin g c ap acity o f soi ls .

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2. FUNDAMENTALS OF LIMIT ANALYSIS

2.1 Basic ConceptsThe mechanical behavior of soi ls i s usually described as

having both cohesion, c, and internal f r i c t i o n , ~ . The resistance ofsoi ls to deformation is furnished by cohesion and fr ict ion across th epossible slip planes in a mass of material . The general ly acceptedlaw of failure in so i l mechanics is Coulomb's cri ter ion l , which sta testhat slip occurs when, on any plane a t any point in a mass of so i l ,the shear s t r e s s , ' ~ (>0) reaches an amount tha t depends l inearly uponth e cohesion and th e normal s t ress , cr (he re ta ken to be posit ive incompression), i . e . (Fig. 1)

1" = C + cr t a n ~ (1)

For i l lus t ra t ive purposes, a simple physical model, shown inF ~ g . 2, may be helpful. In the f igure, a layer of dense granular

stant during t he exper iment, whereas, Pt gradual ly increases from zeroto the value which wil l produce sl id ing. At the instant of sliding, the

The force P acts atne r i a l is subjected to th e action of two forces.r ight angles to the plane 1-1, whereas the other, the force Pt ,"actstangentially to that plane. Let us further assume tha t P remains conn

value P t must not only overcome cohesion, but also must exceed the resistance furnished by two types of fr ict ion. The f i r s t of these ariseson th e con tact sur fa ce s of adjoining part icles and is termed ,surfacefr ict ion. The second, offered by the interference th e part icles

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themselves to changes of the i r re la t ive posi t ion, is termed inter-locking f r ic t ion. I t is th is interlocking f r ic t ion that r e q u ~ r e s thedisplacement upward, as well as the usual displacement to th e side.The displacement vector must, therefore, make an angle e to the s l ipplane.

I f so i l were idealized as perfect ly plast ic with Coulomb'slaw of yielding, then Eq. (1 ) defines th e yield curve in the stressspace c r , ~ . I f now a s tre ss s ta te , represented by a vector from th eorigin, is increased from zero, yield wil l be incipient when the vec-

' t o r r ea ches th e curve (two s tra ight l ines) . For a perfectly plast icmaterial , the v ec to r r ep re se nt in g the stress- s ta te a t any given pointcan never protrude beyond th e curv e, since it i s an unattainable stresss ta te in granular media.

Let us further assume that the s t ress-s t ra in re la t ion issuch that the p la sti c s tr ai n rate vector is always normal to th e yieldcurve when their corresponding axes are superimposed (Fig. 1). I t canbe seen from th e figure that th is is equivalent to assuming e= inFig. 2. The perfectly-plas t ic ideal izat ion with associated flow rule(normality) i s i llu st ra te d by a block shearing on a horizontal plane,Fig. 3a. Volume expansion is seen to be a necessary accompaniment toshearing deformation according to the ideal izat ions. This theory was

6 7proposed by Drucker and Prager and generalized l a ter by Drucker , andShield8 .

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In contrast to th e above mentioned effor t , one may idealizesoil as a fr ic t ional material for which the interlocking f r ic t ion isignored. Deformation occurs by the smooth sliding of adjacent surfaces of material points (see Fig. 3b). I f t a n ~ in expression (1)denotes the coeff ic ient of f r ic t ion between adjacent surfaces ofte r ia l points along the plane, expression (1 ) becomes the well-knownCoulomb f r ic t ion l imit condition for th e shear strength of so i l . Theimportant difference between CoulonID f r ic t ion and perfectly-plas t icCoulomb action i s seen in Fig. 3, where fr ic t ional s l id ing i s horizontal while perfectly-plas t ic shearing involves large upward ver t i -cal motion. I f the plas t ic strain rate vector is superimposed to. theCoulomb l imit curves (assumed as yield curve), the normality rule doesnot hold (See Fig. 1). The extent of th is endeavor i s described in arecent work by Dais9 .

Real soils are quite complex and are s t i l l imperfectly understood. They are nei ther t ruly fr ic t ional in behavior, nor are theyplas t ic . Hence, any such idealized treatment, as discussed above,wil l either result in some differences between predict ions and experimental facts , or wil l enterta in cer ta in mathematical di f f icu l t ies .Fo r example, th e di la ta t ion which is predicted by perfect plas t ictheory to accompany th e shearing action wil l usually be larger than"that found in practice lO The inadequacy of a perfect ly plast icidealiz-ation has been discussed by Drucker2 ,11,12 and DeJong13 amongothers. The lack of uniqueness for solutions to problems using fr ic-t ion theory has been exhibited and explored by Dais9 .

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In order to improve upon the perfect ly plast ic theory,Drucker, Gibson, and Henkel i nt roduced the strain-hardening theories

f '1 1 " 14 h ' h 1 d d b 'k d Sh' 1d15o 801 p ast1c1ty , W 1C were a te r exten e y Jen1 e 1eThe work-hardening plastic action may involve upward or downward ver-t i ca l motion, or nei ther , of the sl iding block as i l lus t ra ted in Fig.3, which qual i ta t ively agrees with experimental data. Recently, more

16sophi s ti cat ed theor ies have been proposed by Weidler and Paslay ,17 18 19Spencer , a n d Sobo tka ' on non-homogeneous so i l s in an attempt to

overcome some of the known deficiencies in previous theor ies , C 1 e a r l ~the development of a more sophist icated theory wil l almost always bringa more elaborate s t ress - s t ra in re la t ion. S olu tio ns to pract ica l ly im-portant problems, on the contrary, become exceedingly di f f icu l t toobtain, i f the s t ress - s t ra in relat ion is too involved. A compromisemust, therefore, be made between convenience and physical rea l i ty ,

In cer ta in circumstances, such as in the s tab i l i ty problemsof so i l mechanics, there appears to be reasonable jus t i f ica t ion fort he ado pti on of a l imit analysis approach based upon Coulomb's yieldcr i ter ion and i t s associated flow rule in so i ls , as discussed in Sec-t ion 1.

2,2 Limit TheoremsThe foundations of l imit analysis are the two l imi t theo-

5rerns, For any body or assemblage of bodies of elast ic-perfect lyplast ic material they may be sta ted , in terminology appropriate to

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so i l mechanics, as:Theorem 1 (lower bound)--If an equilibrium distr ibut ion of stress can

be found which balances th e app li ed load andnowhere violates the yield cr i ter ion, whichincludes c , t he cohes ion, and th e angleof internal fr ic t ion, the so i l mass wil l notfa i l , or wil l be jus t a t th e point of fa i lure .

Theorem 2 (upper bound)--The so i l mass will collapse i f there is anycompatible pattern of plast ic deformationfor which the rate of work of th e externalloads exceeds th e par t of internal dissi pation.

According to th e s ta temen t of the theorems, in order to proper ly bound the " tr ue " s ol ut ion , i t is necessary to find a compatiblefai lure mechanism (velocity f ie ld or flow pattern) in order to ob tainan upper bound solution. A s tress f ie ld sat is fying a l l c o n d ~ t i o n s ofTheorem 1 wil l be re qu ire d f or a lower hound solut ion. I f the upperand lower bounds provided by the veloci ty f ie ld and stress f ie ld coincide, th e exact value of the collapse, or l imit , load i s determined.

2.3 The Dissipation FunctionsAs stated in th e upper bound theorem, i t is necessary to com-

pare th e rate of internal diss ipat ion of energy with the ra te of workof external forces. The diss ipat ion of energy, D, per uni t volume dueto a plast ic s tra in rate therefore, of primary importance. I t can

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be shown in general that the diss ipat ion function has the simplerform20

D (2)

where e t denotes a posi ti ve p ri nc ip al component of the p la stic s tr ainrate tensor.

For the part icular case of plane s t ra in , th e expression (2)reduces to

D c cos0 '\1I max (3)

where '\1 _- [ ( ~ )2 + y' 2J1 / 2 . th t f"1 18 e maX1mum ra e a eng1neer1ngmax x y xyshear s t ra in .

An al ternat ive derivation in terms more familiar to the engi-neer wil l be discussed in what follows. The discussion wil l be re -s t r ic ted to the plane s t ra in case. A number of familiar shear de-formation zones, which are especially useful for so i l mechanics, aret reated as i l lus t ra t ive examples. The resu l t s are adequate in con-nection with la te r applicat ion. Some of the resu l t s have been dis-

21cussed by Chen .

Homogeneous Shearing Zone--The energy dissipated in the homo-geneous shearing zone, Fig. 4, i s

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the d otte d l ines in the figure. Fig. 5b is the hal f f ie ld of homo-geneous deformation of Fig. Sa, while a proper r igid body rotationof Fig. Sb results in the i nt er es ti ng f ie ld of Fig. Sc .

Narrow t rans i t ion layer-- I f the thickness t in Fig. 4 isvery th in , the homogeneous shearing zone may be i m a g i n e d ~ as in Fig.6, to be a simple discontinuity with a discontinuous tangent ia l ve-loci ty au = ty and a discontinuous normal separation Ov = ty tan 0 =Ou tan0. The ra te of dissipat ion of work per unit of discontinuitysurface is

Pt PDA = b Ou - bn Ovor (7)

DA = eu - cr tan0) = cou

Figure 6 shows c le ar ly t ha t a simple s l ip Ou must always beaccompanied by a separation ov for 0 ~ O . The familiar circular surfaceof discontinuity is , therefore, not a p erm is sib le s ur fa ce fo r r igidbody sliding because of the separation requirement for so i ls . Theplane surface and the logarithmic spiral surface of angle are theonly two s urfa ce s o f discontinuity which permit r igid body.motions rela-t ive to a fixed surface.

Zone of radial shear when ~ = c - ~ A n approximation to a zone ofradial shear is given in Fig. 7a where six r igid tr iangles a t an equal

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central angle be are shown. Energy dissipation ta kes p la ce along theradia l l ines O-A, O-B, O-C, etc . due to th e d is con ti nu it y i n veloci tybetween th e triangles. Energy is a lso d is sip a ted on the discontinuoussurface D-A-B-C-E-F-G since th e material below th is surface is con-sidered a t res t . Since the material must remain in contact with thesurface D-A-B-C-E-F-G, the t r iangles must move parallel to th e arcsurfaces. The r igid t r iangles must also remain in contact with eachother. Hence, th e compatible veloci ty diagram of Fig. 7b shows tha teach triangle of the mechanism must have the same speed.

With expression (7), the rate of diss ipat ion of energy caneasi ly be calculated. The energy dissipation along the radia l l ine

O ~ B , fo r example, is the cohesion c multipl ied by the re la t ive ve-loc i ty , au , and the le ng th o f the l ine of d iscont inu it y , r :

(2 c r V S1n 2

where the re la t ive veloci ty 6u, appears as (2V s in ~ e ) .

( 8)

Similarly,th e en ergy dissipation along t he d iscont inuous surface A-B i s

c (2r s in V (9)

where the length of A-B is (2r s in and 6u =y. Since the energydissipation along the radia l line O-B is the same as along the arcsurface A-B, i t is natural to expect tha t the to ta l energy dissipation

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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 sin ce F ig . 7a becomes closer and clo ser to the zone of radialshear as the number n increases. At th e l imit , when n approachesinf ini ty , the zone of radial shear is recovered. The to ta l energydissipated in the zone of radial shear i s the sum of the energy dis-sipated along each radial l ine when th e number n approaches inf ini ty:

lim (2 D = c r V s in 2n) nI'}'-tJll'CO

or2cr V D = lim n sin-2n

11'-""'"00

or (10)D = c V ( r )

where

be@

= -n

Log spiral zone ofc-0 soils--The extension of th e aboved iscuss ion to the more general case of a log spiral zone ofso i l s is evident. A picture of six r igid t r iangles, at an equal angleA8 to each 9ther, is shown in Fig. 8a. I t is found that th e energyd is si pa ti on in a log spiral zone of c-0 so i l is equal to the energydissipated along the sp i ra l discontinuity surface, which is :

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or

D = c Sr V de 6= c S (r - e x p e t a n ~ ) (V expetan0) de0001D = 2 C Va r o c o t ~ (exp2@Dtan0 - 1) (11)

where V , r , and8 are shown in Fig. 8a. A detailed d iscuss ion ofo 0th e lo g spira l zone is given in the Appendix.

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3 . THE STABILITY OF VERTICAL CUTS

3.1 Limit Equilibrium Analysis

The comparison between l imit equil ibrium and plast ic l imi tanalysis can be i l lus t ra ted by evaluatirig the s tab i l i ty of so i l in aver t ica l bank. The height a t which an unsupported vert ical cut, asi l lus t ra ted in Fig. 9, wil l collapse due to th e weight of the soilwi l l be defined as the cr i t i ca l height, H . The conventional a-crnalysis ( l imi t equilibrium) wil l be examined f i r s t and then comparedto the method of l imit analysis .

I t is common practice to evaluate this problem by the equilibrium method. The fa i lure surface i s assumed to be a plane inclineda t an angle e to the horizontal (See Fig. 9) and Coulomb's law offailure i s applied. From Eq. (1) ,

' f c + cr tan0

The distr ibut ion of a and T along the failure plane is unknown, buti f s is the length of the shear plane:

I ' f ds = I c ds + I a tan0 ds = cs + tan0 I a ds

Equilibrium requires that :

I ( J ds = WeaseI ,. ds = W sine

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(12)

(13)


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Substi tut ing Eq. (13) into Eq. (12) yields

W defines th e uni t weight of the so i l , w, multiplied by the volume of

where

W sine Hcr= C --'-e + Wease tan0S1nw HZW = cr2 tane

(14)

the so i l mass above the s he ar p la ne . I f e i s minimized then:

Equation (14) then yields the cr i t i ca l height

(15)

(16)

Assuming a curved surface instead of a plane for the s l id ing surfacewil l reduce the cr i t i ca l height s l ightly . Fellenius22 used this con-clition and found the cr i t i ca l height to be

H =cr 3.85e 1 1W tan(L;TT + 20) ( 17)

In this analysis, Coulomb 's l aw is only satisf ied along the assumedfailure plane. I t is not shown, nor known, i f Coulomb's law of failureis violated at other points. A val id solut ion requires that equilibrium,

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compatibili ty (and boundary condit ions) and the s t ress - s t ra in relat ion-ship be s a t i s f i ed . Here only equi l ibr ium has been s a t i s f i ed . Fromthe l imi t theorems i t i s known th at th is solution is not a lower boundsince only equil ibrium is sa t is f ied , and not the yie ld cr i te r ion . Thesolut ion is merely one of many solutions tha t satisfy equil ibrium. I tis not known whether or not it is unique.

3.2 Plast ic Limit AnalysisThe l imi t a na ly sis o f this problem, f i r s t performed by

Drucker and prager 6 , involves determining a lower bound on the collapseload by assuming a s tress f ield which sa t is f ies equil ibrium and doesnot violate the yield cr i te r ion a t any point . An upper bound is ob-tained by a veloci ty f ie ld compatible with the flow rule in which therate of work of the external forces equals or exceeds the rate of in-ternal energy d i s s i p a ~ i o n .

Star t ing f i r s t with the upper bound , a mechanism (veloci tyf ield) is selected as shown in Fig. 10. As the wedge, formed by th eshear plane which makes an angle with the ve r t i c a l , sl ides down-ward along the discont inui ty surface, there is a separation veloci ty,Vsin0, from the discontinuity surface. The rate of work done by theexternal forces i s the ver t ica l component of the veloci ty multipliedby t he weight of the so i l wedge:

1 -2-2wH t a n ~ V cos(0 + B)cr

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(18)


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While th e rate of energy dissipated along the discontinuity surfacein Eq . (7):

Hc ~ V cos0c o s ~ (19)Equating the rate o f e xte rn al work to th e ra te of internal energy dis-sipation gives:

H < 2c cos0cr - w s i n ~ c o s ( + ~ )

I f 0 is minimized then

and

(20)

(21)

(22)

This i s th e same value obt ai ned by th e method of l imit equil ibrium.This implies that th e l imit equilibrium method solution presentedpreviously is an upper bound, unless the solution is exact .

A lower bound can be obtained by constructing a s tressfield composed of three regions as shown in Fig. 11a. Region I , inthe bank i t se l f , is subjected to u nia xia l compression, which increaseswith depth. Region I I is under biaxial compression and region III isunder hydrostatic pressure (IT =cr) . Fig. l lb shows correspondingx y

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Mohr circles for each r ~ g i o n . Failure occurs when the circle repre-senting Region I meets the yield curve. Therefore,

= c cos0 + sin0and (23)

hence, th e lower hound solution i s only one half the v alu e given byth e upper bound solution. 3e 1 1I f the average is used, Her = W- tan(4TI + 20),then the maximum error i s 33%.

6As indicated by Drucker and Prager , the upper bound may beimproved by choosing for a discontinuity surface a logarithmic spiralrather than the plane used here (See Fig. 10). The lower bound maybe improved by the choice of other s tress f ie lds , probably quite com-p1ex and which may involve tensile stre ss es in th e soi l .

3 .3 Soi l Unable to 'Take TensionIn the laboratory so i l may exhibit th e abi l i ty to res is t

tension. In th e f ie ld , however, th e presence of water, or tensilecracks near the surface, may destroy th e tensile strength of the so i l .Hence, the tensi le strength of soil is not re l iable and i t may be neg-lected. This is a conservative idealization. The Coulomb yield cri-terion i s modified by th e tension cut-off as shown in Fig. 1 in whichthe requirement of zero tension i s met by the c i rc le termination as

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cV cos0 (30)

which, for a cohesionless soi l (c=O), is zero. Equating the rate ofinternal and external work yields:

The l imit load or th e collapse load is obtained by maximizingo which yields:

and

p = ~ W H 2 tanO cot(O +

1 2 211Pan = ZWH tan ("4TI - 20)

(31)

(33)

For th e p as siv e earth pressure case, th e mechanism is quitesimilar to the active earth pressure case excep t. that since the motionis upward, the veloci ty, V, of the soi l wedge is reversed, althoughs t i l l directed a t an angle 0 to th e discontinuity surface. The rateof internal and external work are found in the same manner as for theactive earth pressure, and:

1 1o = -T T + -04 2

and1 2 2 1 1Ppn = 2'wH tan (L;TT + 2'0)

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(34)

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The upper bound is then minimized to obtain th e passive ear thpressure.

To obtain lower bounds, Fig. 16 shows a discont inuous equi-l b 1 2 7 . h h k h h h h hr1um so ut10n 1n W lC 18 a parameter w 1C is C osen sue t a tth e Coulomb yield cr i te r ion wil l be sa t is f ied . The Mohr 1 s circ les inthe figure shows c le ar ly t ha t two extreme values of k are possible whichfurnish th e needed lower bound s olu tio ns o f th e la te ra l earth pressureproblem. From th e f igure, one obtains:

for active earth pressure

for passive ear th pressure

(36)

k (37)

Equilibrium is then determined

p = JH kwy dyo (38)The results indicate that th e upper and lower bounds for th e

active earth pressure and th e passive earth pressure, respectively,gree, indicat ing the exact solution.

The case considered is a special case of th e general problem,where is the backfi l l angle, 0 th e wall f r ic t ion angle (6


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for th e active ease

for th e passive case

P tan6 V cos(0+0)an

P tan6 V cos(O-0)pn

(44)

(45)

where P and P denote the normal to th e wall components of the ae-an pnt ive and passive earth pressures, respectively,

Equating the rate of internal energy dissipat ion and the rateof external work yields

-21WH2 s inO cos(0+0) - cH cos0c o s ~Pan = - e - o - s ( ~ O + ~ ~ - ) sin(0+0) + tan5 cos(0+0)

and

c o s ~ iWH2 sino cos(O-0) + cH cos0Ppn = - c - o - s ( - O + ~ ~ - ) sin(O-0) - tan6 cos(O-0)

(46)

(47)

These equat ions a re valid for a ver t ica l wall , 6


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28equilibrium is then :

pP = b 2h= c(2n + ~ ) + wh (49)

or

p = 6.28c(l + 0.32 ~ + 0.16 :h) (50)

Two other assumptions concerning possi bl e sur faces o,f failureare given in Fig. 19b,c f or purposes of comparison28 The f i r s t con-s i s t s in assuming th e c en te r o f ro ta tio n'O at th e edge of the foundation,as shown in Fig. 19b. The choice of this fai lure mode for a l imit equi-librium solution would raise several questions. What is th e normal andshear stress distr ibut ion along th e circular shear plane? Also, what isthe normal stress distribution along th e vert ical separation between th equarter circle and the tr iangular region? The resul tant of the shearstress along this l ine passes through 0, but th e normal stress resul tantis not defined. A commonly accepted method in l imit equil ibr ium analysisis to assume the quarter circle rotat ion produces uni-axial compressionof th e so i l i t bears against , as shown in Fig. 19b. The maximum compres-sian that may be sustained by th is so il is (2c + wh) since 2c is thegreatest allowable difference between principal stresses and wh corre-sponds to a hydrostat ic pressure which is in equilibrium with the sur-charge at the top surface. (The pressure due to th e weight of th e soi lwithin tne region is neglected). Summing moments about 0 of a l l forces,yields:

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for h=O:

pp = b = c (n + 2) + wh 5.l4c + wh (51)

which is nearly 20% less than the value of 6.28c obtained in Fig. 19a.

p 5.l4c (52)

An even lower l imit equilibrium solution can be produced byconside ring the scheme shown in Fig. 19c, which assumes a tr iangularso i l block under th e punch. Equilibrium of forces shows that :

p 4c + wh (53)

As with other l imit equilibrium solutions, it is not known which solu-t ions should be chosen. I f i t can be shown that the stress f ie ld inregion I can be extended throughout the so i l and s t i l l satisfy equi-l ibrium and th e yield condition, then this is a lower bound. Unfortu-nately, i t is not known in the present case whe ther th e stress f ieldcan be extended in th is manner.

5.2 Plas t ic Limit Analys is --Upper BoundsTo obtain an upper bound of the same problem by l im i t analys is ,

the chosen mechanisms as shown in Fig. 20 are quite s im ilar. In Fig. 20ai s a r igid body rotat ion about 0, with shearing t ~ r o u g h the surcharge,which i s of depth h. I f the angular velocity is a, th e rate of work

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uP 5.78c

The two solut ions jus t i l lus t ra ted are actually approximations of th ewell known Hil l (See Fig. 21c) and Prandtl (See Fig. 21d) solut ions.Both yield the following

up = (2 + TI) c

Since the solut ions for a perfectly rough and a perfectly smooth punchare identical , the roughness of the punch does not affect the bearingcapacity.

T his, then, i l lus t ra tes th e power and simplicity of the l imitanalysis approach. Although th e Hill and Prandtl solut ions are notvery di f f icu l t , they might not be the f i r s t to come to mind. The rota t ing so i l mass (pu = 6.28c) or the two wedges (pu = 6c), however, arechoices that readily come to mind. Yet, they are only 22% and 17% inerror , respectively. Moreover, the bearing c apac it y o f the rotatingso i l ~ s s may be reduced by shif t ing the center of rotation. I f thecenter i s shifted to 0 ', as i l lustrated in Fig. 22a, the rate of ex-ternal work is :

1 P(rcose - 2b)a (60)

where r is the radius of the discont inui ty surface and e the angle be-tween the face of the punch and the l ine AD'.

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The rate of internal work is given by

(n-28) r c (61)

and the resul t ing upper bound solution is :

2pU = c(n-28) rreose - !b2

(62)

The upper bound may be reduced by minimizing the solution with respectto the variables e and r :

opU 0or =and (63)

'opu 006 =

'Hence:

and

b = rcosecr (64)

Equation (64) s ta te s th at the location of the cen ter of rotat ion 0' is

b = r [2cos8 - (n-28 )sin8 ]cr cr cr

directly above 0 as shown in Fig. 22b. Equating Eq . (65) yields

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(65)


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case = (TI-ze )sinecr cr cr

Introducing t hi s r el at io n into Eq. (62) yields

(66)

u PP = b = 4csin2ecr (67)

e can be determined from Eq. (66). This is found to be:cre = 23.2cr

and

(68)

up 5.53c (69)

which agrees with Fellenius ' solution22 .uAlthough the upper bounds, p =6.28c, 6.14c, 6c , 5.78c, 5.53c,

are not ext remely close to the actual answer (2+n)c, they are obtainedquickly and easi ly. Each can p rovide useful information when the exactanswer is not available and each can prove even more valuable when th esoi l being loaded i s inhomogeneous and so not easily amenable to exact

. solution.

5.3 Plast ic Limit Ana1ysis--Lower BoundsTo obtain a lower bound of th e bearing capacity problem by

l imit analysis, choose a simple s tr es s f ie ld , which gives 2c as a lowerbound, as shown in Fig. 23a. Since a hydrostat ic pressure has no e f fec t

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on plast ic yielding for soi l s w ~ t h 0=0, th e simple add it ion o f thehydrostatic pressure 2c+wh as shown in Fig. 23b to Fig. 23a increasesthe ultimate load to (Fig. 23c):

p = 4c + wh

which is an improved lower bound, but not a very good one.

(70)

Now, a l i t t l e physical intui t ion wil l raise th e lower boundto a very useful point. Exper ience has shown that the load on so i lnspreadstf, or is carried by an even greater area, the deeper one goes(See Fig. 24a). Therefore, conside r the ' s t ress f ie ld, shown in Fig.24b, consisting of two inclined "legsn of 2c each. Where they meet,under the punch, there is a ver t ica l component of 3c and horizontalcomponent of c. I f th e stress field of Fig. 23b are superposed ontothe field of Fig. 24b, th e stress f ield shown in Fig. 24c resul ts andi t is eas il y ve ri fi ed that th e s u p e r p o ~ i t i o n of these stress fieldsdoes not violate the yield condition anywhere. Therefore:

.tp = 5c + wh

This lower bound may be improved through a more judicious choice of

(71)

stress f ields. Valuable techniques for constructing lower bounds of30this "truss- l ike" nature have been developed by Drucker and Chen .

I t was shown in thei r paper that the familiar Prandtl f ield may beimagined physically as a load supported by inf ini tely many supporting

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legs. A picture of 9 ver t ica l and incl ined legs at 10 to each otheris shown in Fig. 25 . As the number of legs grows, th e stress in eachdecreases. In the l imi t , the Prandtl field beneath the footin g isrecov ered. Figure 24c shows c le ar ly t ha t the three legs 's ap-proximation for the Prandtl field gives the answer suff ic ient ly ac-cu rate for pract ica l purpose.

In order to avoid obsc uring the basic points, the discussion,up unti l now, has been lim ite d to a cohesive so i l (0=0) so that th eCoulomb's yield cri ter ion of s oi l p la s ti ci ty is reduced to th e familiarTresca yield cri ter ion of metal plas t ic i ty . The extension of the aboved iscuss ion to inq1ude c-0 so i l is evident. The detai ls of such develop

21ment has been discussed by Chen .

5.4 'Limit Analysis of Three Dimensional ProblemsIn this section, upper and lower bounds are obtained for th e

ultimate bearing c apac ity o f square and rectangular footings on a co-hesive soi l (0=0).

A simple fai lure mechanism for the footing is shown diagram-matically in Fig. 26. A-E-E'-A' is the area of footing and th e down-ward movement of th e footing i s accommodated by m o v e ~ n t of the materialas indicated in Fig. 26a. In Fig. 26b is shown the plan and se ctio n inFig. 26a, and i t can be seen tha t th is mechanism is an extension intothree dimensions of a simple modification of th e two-dimensional Hil l ' s

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mechanism in Fig. 2lc.

The internal dissipation of energy on th e discontinuous 8ur-face A-B-C-D-D'-C'-B'-A ' and in the radial shear zone E-B-C-C'-E'-B'is :

2 ba + 2 b TTcv - cv- -aJ2 J2 2 (72)

Energy dissipated on the two end surfaces A-B-C-D-E-A and A'-B'-C'-D'-E I_A' is :

b h TT b 22cv--+2cv 42J2 J2

Energy dissipated through the surcharge is :

( 73)

c ~ ah (74)J2

while the rate of external work is Pv /J2 - wabhV /J2 . Hence, the re- '_sul t ing ult imate bearing capacity is :

upU = L =ba b whc ( 5.14 + 2.52 - +-)a c (74)

Thus, for a square footing, for which b=a, Expression (74) gives th evalue 7.66c + wh and a value of 5.77c + wh is found for a rat io b/a = 4.This expression tends to the value of 5.14 c + wh for rectangles whose

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length is great compared with thei r breadth, in agreement with thefamil iar Prandt l ' s solution for the two-dimensional footing.

Cons idering nex t th e lower bound, a three-dimensional s tressfield for the rectangular footing is shown in Fig. 27, which is a direct extension of the two-dimensional stress field in Fig. 23b and Fig,24b. The s tre ss fie ld of Fig, 27b establishes 3c as a lower bound forthe ultimate bearing pressure. I t is easi ly ver if ied that superposit ionof the s tr es s f ie ld of Fig, 27a on that of Fig, 27b does not lead tos tre ss es in excess of the yield l imit . Thus, 5c + wh is a lower boundfor the ultimate bearing pressure in th e three-dimensional rectangularor square footing, The failure mechanism of Fig. 26 and th e s tr es s f ie ldof Fig, 27 , therefore, show that the u ltim ate b earin g p ressu re for asquare footing l ies within 21 percent of the v alu e 6.33c + wh,

Further improvement in the upper bound would require th e moreelaborate fai lure mechanism discussed by Shield and Drucker3l . I t wasfound that the value 5,71c + wh is a bet ter upper bound for the squarefooting then those obtained previously.

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6. CONCLUSIONS

Although a so i l with cohesion c and in ternal f r ic t ion 0is not modeled accurately by a th eo ry o f perfect plas t ic i ty , i t isof interes t to note that many classica l s olu tio ns o f s tabi l i ty problems in so i l mechanics can be obtained by J im it a na ly si s technique.Within th e framework of the idealizations, th e l imit analysis approach i s rigorous and the techniques are competitive with those ofl imit equilibrium, in some instances being much simpler. I t shouldbe kept in mind, however , that too much should.not be expected of theperfect plast ic assumption for such a complicated het erogeneous substance as so i l . Predictions based on this assumption on other problems in so i l mechanics may be misleading. Nevertheless, i t seemsclear that th e perfect plast ic assumption is suff ic iently good fors tabi l i ty problems in so i l mechanics.

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7. ACKNOWLEDGMENTS

The resul ts in this paper were obtained in th e course ofresearch supported by the Ins t i tu te of Research of Lehigh Universityand The National Science Foundation under Grant GK-3245 to LehighUniversity.

The authors would l ike to thank Professor J. W. Fisherfor his review of t he manusc ript , and are indebted to Miss J. Arnoldfor typing t he manuscr ip t and to Mr. J. Gera for th e drafting.

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8. TABLE AND FIGURES

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TABLE 1 COMPARISON OF LATERAL EARTH PRESSURE SOLUTIONSBY METHODS OF LIMIT EQUILIBRIUM AND LIMIT ANALYSIS

angle of wall backf i l l ACTIVE EARTH PRESS. PASSIVE EARTH PRESS.internal friction angle k = k secD k = k secOf r ic t ion angle a an p pnLIMIT LIMIT LIMIT LD1IT0 8 13 EQUILIBRIUM ANALYSIS EQUILIBRIUM ANALYSIS

10 0 0 .704 .704 1.42 1.4210 0 .634 .635 1.65 1.7320 0 0 .490 .490 2.04 2.04

10 .569 .566 2.59 2.5910 0 .426 .446 2.52 2.6410 .. 507 .. 531 3.53 3.7020 0 .350 .426 2.93 3.5210 .. 430 .516 4.62 5.5930 0 0 .333 .. 333 3.00 3.0010 .. 374 .372 4.08 4.0920 .441 .439 5.74 5.7810 0 .290 .307 3.96 4.1510 .334 .350 6.03 6.3120 .. 401 .420 9.94 10.4120 0 .247 .297 5.06 6.1510 .282 .338 9.05 10.9120 .334 .413 19 .. 40 23.3740 0 0 .217 .217 4.60 4.6010 .238 .237 6.84 6.8520 .266 .266 11 .. 06 11.1010 0 .195 .204 6.63 6.9210 .215 .223 11.53 12.0820 .. 242 .253 24.00 25.36

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FrictionPlastic

Simple TensionCrack

Fi g. 1 P l a s t i c S t r a i n Rate i s Normal t o YieldCurve for P e r f e c t l y - P l a s t i c Theory, butP a r a l l e l to Axis for F r i c t i o n a l Theory

(I)

tn Displ.. // 8r----- - - --... . ,I r - - - - --------, IlJDENSE SOILS

Fi g. 2 A Simple Physical Model

- 5 0 -


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I r;DiSPlel - - - - ,........r ~I I 'f I_ . J . - __ _ -1._

~ ~ ~ ..... \.:, .. \: ...f.: : , ~ ~ ~SoilPLASTIC SHEARING

c I > ~ O(0)

FRICTION SLIDING(b)

1-

Fig. 3 Difference Between Coulomb Sliding andCoulomb Shear

(0 ) (b)

Fig. 4 Dilatation Accompanies Perfect ly-Plast icHomogeneous Shearing

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............................... .................... ~ IIII

(0)

.. '< Y- .... - - - - Y/2 ~..........--------.

Fig. 5 Homogeneous Shearing Zone with DifferentShapes of Mode

Fig. 6 Narrow Transition Layer

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B

o

(b)Fig. 7 Rigid Triangles Give Radial Shear Zone in Limit(a ) Energy Dissipated Along th e Rad ial Line O-B

is the Same as Dissipated Along th e Line A-B(b) Vn=V2=V1=Vo=V True for a ll Values of 88

o." 1!.9y--,/ \ ,r O / ~ \ \ '/ ~ c9


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Actual Failure Surface

H

wWeos

Fig. 9 Limit Equil ibrium Solution for the Stabil i tyof a Vertical Cut

H Rigid

Fig. 10 Limit Ana lysi s Failure Mechanism for theStabi l i ty of a Vertical Cut

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........_-- - '............-............---.. - - - - - - - -----

- x

CTX= TXY=O

I

H

crx =w ( y- H)

(a.) An Equilibrium Solution

Im lIII

CTy=CTX= w(y-H)y

( b) Mohr Circles of (a)

T

c

Fig. 11 Limit A n a l y s - ~ $ Stress Field for theStabil i ty of a Ver t ica l Cut-55-


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c

H

------ -"\ \ w=y. \ ~ \,-- Rigid\ 0. \ ~ W .\ \ : --Tensile Crack\ \\ \ Y\\8 \y\\ r,- \

-------\ y A\ .1L+epep \ 2(f-T)Fig. 12 Rotational Mechanism Containing a SimpleTension Crack and the Homogeneous ShearingZone of Fig. for Soil Unable to TakeTension

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Movable Wall. .

-----a- InwardII

Outward - IP n ~ I1 _(a) Vertical Section Through Bin

I nwa rd - -......

Wall at RestPanoOutward

(Active) - -

Collapse Load ( Passive)Ppn - - - ~ - - = - - - - ......UnrestrictedPlastic FlowContained Plastic Flow

..

WALL MOVEMENT( b) Load Displacement Relationshi p

Fig. 13 Results of Retaining-Wall Tests

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T s

H pw,.

F

p

Fig. 14 Limit Equil ibrium Solution for Lateral EarthPressure (Coulomb's Solution)

T S

Rig-idH p

...

RigidV

Fig. 15 Limit-Analysis Sol ut ion fo r Active EarthPressure

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H p Kwy1:~ ~ ~ , . . . . . - - - - _ ....------ - - -----

+ W ( Y ~ ) ,:II - f -W(Y-H)w(y- H) Ity

(a) An Equilibrium Solution'

T

- - - - I ~ - . , . - - - . - . . . . - - - . . - - - - - - _ _ . - - . . . - C T

(b) MOH R Circles of Region 1: in (0)Fig. 16 Limit Analysis Stress Field for LateralEarth Pressure

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Fig. 17 P l ~ n e Sliding for Lateral Earth PressureProblems in General

T

H- p

RigidRigid

(0) Activen - .

(b) PassiveFig. 18 Active and Passive La te ra l Ea rth Pressuresfor a Vertical Wall

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h wh)P=6.28c (1+0.32 b +0.16 C

o t

R. ( I ) wh..-- 2 b(2c +wh) ~ 2c+wh

P=5.14c +whb

(b)

_ ) , W ~~ 2 C + W hT =C ~ 45 0 .h. (I ); I ~ 2;I ~ ~ - - - - b(2c+wh)(J " / ~ .

. / ~;I '1.....-------(c)

Fig, 19 L im it E q u il ib r iu m Modes o f Fai l ur e,-61-


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pU =6.28c (I + 0.32 ~ + 0.16 ~ )h

Rigid(0) .

u cbp = 6 ~ 1 4 c + b +wh~ - . - . . : . . . - - - - 0 E

. . " . . , . . . . . . . . . . , . . , . , . . , . . . , . . . , . . , . . , . . , . ~ ~ I \ +V

Rigid

h

h

RigidB

(c)Fig. 20 Limit Analysis Upper Bounds on t he Bea ring

Capacity of S o i l s ~ 0=0-62-


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(d) Prandtlls SolutiontU =5.78 beb

pU =5.78 be

b

/~ - - - - - - - - . . . . . _ .......... - - ' - - ~ : ; . For All(0) An Approximation to the Mode of Failure in (c)

P=(2+ 1r) be

(b) An Approximation to the Mode of Failure in (d)

tp = (2 + '77") be/ / /Ji l 77/~ ~ 1f2v For All

(c) Hill's Solu tio n

Fig. 21 Rigid Blocks Sliding to Approximate Slip-LineFie lds

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Rigid

b

pU+

(a)

pU =5.53 beb

Rigid

Fig. 22 Rotating Soil Mass with Center at O' forPurpose of Minimizing Upper Bound

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2c

- x

wy+ 2C+W Y

o

t-+-y =UxpL =(4c+wh)b =2c+ wyHydrostaticPressure

UNI-AxialCompressionI I(0)

o

wy~ + W Y

4c+wy2C+WY--- (b)

=o

(c) =(a ) + (b)Fig. 23 Limit Analysis Lower Bounds on th e Bearing Capacity, StressField (c) is th e Simple Addition of the S tr es s F ie ld s (a ) and (b)

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( b)

p

F F30(0)

(c) = (b) + Fig. 23bFi g. 24 L i mi t A na ly si s Lower Bounds: Stress F ie ld (c ) i s th e Resulto f Adding the Stress Field s i n (b ) and i n Fig. 23b

wy~ 2 C + W Yc+wy

~ C + W Yc+wy+-66-


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tpL =5.122cb-....,...,."""'--- ............

Fig. 25 Supporting Legs Give Prandtl ' s Solution inLimit

- 67-

---0-- 2c


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A

(0) Fai1ure Mechanism For Rectangular Footing

h

bAI E'.,..,..,...,..,..,....,..,..---......----. 0 I

a

A E o(b) Plan and Section In (0)Fig. 26 Three-Dimensional Failure Mechanism for BearingCapacity Porblem, 0=0

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2c+wy

(0) Extension of Fig. 23 b3c

Hydrostatic. Pressure

(b) Extension of Fig. 24 bFi g. 27 Three-Dimensional Stress F i e l d fo r Bearing

Capacity Problem,

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9 . APPENDIX

Energy dissipation in a log spiral zone of c-0 soi ls :

The extension of the d iscuss ion o f energy dissipation to azone of radial shear, which includes th e more general case of a logspiral zone, for c-0 soi ls can also be developed. A sl ip au mustalways be accompanied by a separat ion 8v a s d is cuss ed in Section 2.There is no need to cons id er th e separat ion when the shear strength ofa soi l is due only to the cohesion. Consider th e six r igid t r ianglesa t an equal angle to each other as shown in Fig. 8a and th e corre-sponding compatible veloci ty diagram for two t yp ic al t ri ang le s A-O-Band B-O-C which are shown in Fig. Sb. I f the central angle issuff ic ient ly small, one may write:

VI (1 + tan0)

v = V (1 + ~ e t a n ~ )n n-1The velo ci ty i n th e nth triangle O-E-F is :

v = V (1 + tan0)Dn 0

-70-

(75)

(76)


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where V is th e in i t i a l velocity. Clearly, the log spiral zone i soobtained as a l imiting case when the number of the r igid t r ianglesgrows to inf ini ty . Hence, as n ~ 00 , Eq. (76 ) becomes:

V (1 + tan0)n = V (1 + e t a n 0 ) ~ V exp(etan0)o 0 n 0or (77)

V V exp(etan0)a

where V is the velocity a t any angular location, e, along th e spiraland which agrees with the va lue ob tained by Shield32 .

From Equation.(7), the rate of energy dissipation along atypical radia l l ine, say O-B, is :

(78)

where au appears a s V168 . Similarly, the dissipation along the spira lsurface A-B is :

where the length of A-B is ( r 2 ~ e / c o s 0 ) and 6u= VI cos0. Again, th edissipation along a radial l ine i s the same as along the spiral surfacesegment provided the central angle i s small. Thus, the expressionfor energy d is si pa ti on in th e lo g spiral zone wil l be ident ical with

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the expression along the spiral surface which can be obtained by inte-grating Eq. (79) along the spiral surface r=r exp8tan0:o

c J rV de1 V c o t ~ ( e 2 ~ t a n 0 _ 1 )= 2c 0 r 0 VJ

which agrees with the value obtained by Haythornthwaite33 .

-72-

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b

cD

HkakPp

prvwwOvOu

O ! , ~ , e , o

a,wy.Ymax.e

10. NOMENCLATURE

punch width

cohesiondissipat ion functionrate of dissipation of e nergy per uni t areaforceheightcoeff ic ient of active ear th pressurecoeff ic ient of passive ear th pressurebearing capacity peF uni t areaload, forceradiusveloci tyweight per unit volumeto ta l weightdiscontinuous normal veloci tydiscontinuous tangential veloci tyangular parameterangular v e l o c ~ t yengineering shear stra in ra temaximum ra te of engineering shear s tra inangle of wall f r ic t iondistancenormal stra in rate

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t tensile principal component of th e p l a s t ~ cstrain rate tensor

cr normal s tress'f shear s tress0 angle of in ternal fr ic t ion8 angular parameter defining zone of radial shear.

-74-


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11 REFERENCE S

1. Terzaghi, K.THEORETICAL SOIL MECHANICS, New York, John Wiley and Sons,1943.2. Drucker, D. C.

ON STRESS- STRAIN RELATIONS FOR SOILS AND LOAD CARRYINGCAPACITY, Proceedings, Firs t International Conferenceon the Mechanics of Soil-Vehicle Systems, Turin, I ta ly ,Edicioni Minerva Technica, pp . 15-23, June 1961.3. Sokoloviskii, V. V.STATICS OF GRANULAR MEDIA, First Edition, New York, Oxford,

1965.4. Drucker, D. C.

A MORE FUNDAMENTAL APPROACH TO STRESS-STRAIN RELATIONS,Proceedings of the Firs t United States National Congressof Applied Mechanics, pp. 487-491, June 1951.5. Drucker, D. C., Prager, W., Greenberg, H. J.EXTENDED LIMIT DESIGN THEOREM:S OF CONTINUOUS MEDIA, QuarterlyApplied Mathematics, Vol. 9, pp. 381-389, 1952.6. Drucker, D. C., Prager, W.SOIL MECHANICS AND PLASTIC ANALYSIS OR LIMIT DESIGN, QuarterlyApplied Mathematics, Vol. 10, pp. 157-165, 1952.7. Drucker, D. C.

LIMIT ANALYSIS OF TWO AND THREE DIMENSIONAL SOIL MECHANICSPROBLEMS, Journal of the Mechanics and Physics of Solids(Oxford), V ~ l . 1, No.4 , pp. 217-226 , 1953.8. Shield, R. T.

ON COULOMB'S LAW OF FAILURE IN SOILS, Journal of the Mechanicsand Physics o f Solid s, Vol. 4, No.1 , PP. 10-16, 1955.

9. Dais, J . L.AN ISOTROPIC FRICTIONAL THEORY FOR A GRANULAR MEDIUM WITH ANDWITHOUT COHESION, Ph.D. Thesis, Applied Mathematics, BrownUniversity, May 1967.

10. Brinch, HansenEARTH PRESSURE CALCULATION, Danish Technical Press, 1953.11. Drucker, D. C.

COULOMB FRICTION, PLASTICITY, AND LIMIT LOADS, Journal ofApplied Mechanics, Vol. 21, Transaction ASME, Vol. 76,pp. 71-74,1965.

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12. Drucker, D. C.CONCEPTS OF PATH INDEPENDENCE AND MATERIAL STABILITY FORSOILS, in "Rheologie e t Mecanique Des Soils", ProceedingsIUTAM Symposium, Grenoble, April 1964, Edited by J .. Kraand P. M. Sirieys, Springer, PP. 23-43, 1966.

13. De Jossel in De Jong, G.LOWER BOUND COLLAPSE THEOREM AND LACK OF NORMALITY OF STRAINRATE TO YIELD SURFACE FOR SOILS, Rheology and Soil Mechanics,IUTAM Symposium, Grenoble, France, pp. 69-75, 1964.

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