drucker dc_1952_soil mechanics and plastic analysis or limit design

8/4/2019 Drucker DC_1952_Soil Mechanics and Plastic Analysis or Limit Design

1/9

,;'~').:I

"

-1-'-;

-i

.

. . : : - . . . ; , :

157

SOIL MECHANICS Al'iD PLASTIC ANALYSIS OR LIMIT DESIGN*BY

D. C. DIWCKEH. A:iiJ W. PRAGERBrown Universitu

ti)IJt'~f!~~j1IIII

1. Introduction. Problems of soil mechanics involving stability of slopes, bearing,..!p:,cit.y of foundation slabs and pressures on retaining walls are often treated as problems... plast.icity. The soil is replaced by an idealized material which behaves elastically up~(/some state of stress at which slip or yielding occurs. The shear stress required fo r::mplc slip is often considered to depend upon the cohesion and linearly upon the normalIlTI, ..sure on the slip surface. In more complete plune investigations an extended Coulomb's; 1 I 1 l ' i s u se d, ** (Fig. I)

~. ..- ..~~' .: r . . .. .

" .'" . . . " .'~.~ \ ' .-'.~;, . . '.~~,. -; : ".'~-". . . '.f'" . . i.~ !' ,",'"

_R ( f ~ + 1 1 = c cos !p - 2 Sill !p, (1)

. ' .' .. ' . .: > ~ . "

. ;~:. . . .';

\1J.~reRisthe radius of Mohr's circle at slip, the maximum shearing stress [ ( 1 1 : 0 - ( f , Y / 4 . +T ; . j " " ~ ; c is the cohesion; c cos !p is the radius of Mohr's circle at slip 'When the mean" l I r m a J stress, ( 1 1 " , + 1 1 . ) / 2 , in the plane is zero, and < p is the angle between the tangent. :0 ihI' Mohr's circles n.t slip and the negative (i axis.

r

FIG.!. Mohr-Coulomb hypothesis.

The purpose of this note is to discuss the implications of assuming the soil to be all('rCE'C-tlylastic body. The validity of this assumption is not at issue. No account istaken of such important practical matters as the effect of -water in the soil or of the''''''Cntially different beha "lor of various constituents such as -clay and sand. All that is-ought is a theory consistent with the basic assumption. Comparison of the predictionsIjj such n theory with the actual behavior of soil will give an indication of the value of 'Ii - idealization.

: . . . ." ' .t: ' '.

IIfIf

"Received Nov. lU, 1951. The results presented in this paper were obtained in the course of research"II)II~l)red by the Office of Naval Research under Contract Kionr-35801 with Brown University,~K.Terzaghi, Thenre licai8(J17 mechanics , John Wiley and Sons, 1943, pp."22, 59.

.. . '...


2/9

,~~"'biiili'-liiIe..yp....Z... ~ - - - - - - - . . .ec,-- .. --~----~~-----~--------------- ~, ~~t-.-.. ..~I " , t- ; , t ; l

J.... r '.\.j",

~~~;~'j

~-.- :. . ..

, .I-.. ,~. . ~" . : ; .. . . , . ~. _ , '

t o . ~ .

- j

-f:

158 D. C. DRUCKER AND W. PRAGER [Vol. X. No.2

(2 )

(3 )

(4 )

(5 )

I ? : >

iIlrclII'\-

"nlDC

,lip""m('uc:i

2, Yield function and stress-strain relation, A yield function which is a proJl(.'rgeneralization of the Mohr-Coulomb hypothesis (1) isf = aJ 1+ J~/2 = k,

where a and k are positive constants at each point of the material; J1 is the sum of th eprincipal stresses:

J2 is the second invariant of the stress deviation:J2 = ! S i j S i j = l [ ( C T ~ - C T . ) 2 + ( C T . - C T . ? + ( C T . - C T , ) 2 ] + T ! . + T : . + T ! " , '

The stress deviation is defined as S it = C T a ; - (J1/3) Oij, where Ot; is the Kronecker delta.zero for i;c j and unity for i= j. For example, S 1 1 = = S ~ = n C T " , - ! ( C T . + C T , ) ] , 8 1 ~ = =S.,. = T",

As is well known, the yield surface, f = k, in principal stress space is a right circularcylinder for the Mises criterion (a = 0). For a > 0, the surface is a right circular conewith its axis equally inclined to the coordinate axes and with its apex in the tensionoctant. A generalization of (1) as a modified Tresca or maximum shear stress criterionis also permissible, but will not be discussed here. It leads to a pyramid instead of a cone.

According to the concept of plastic potential," the stress-strain relation correspondingto the yield function (2) is

where E~~ is the plastic strain rate and > .. isa positive factor of proportionality whichmay assume different values for different particles. Substituting the expression (2) forf, we obtain

A very important feature of Eq. (4) is that the plastic rate of cubical dilatation is

Equation (5) shows that p la .s ti c d efo rm a ti on mu st be accompanied by an increase in volllJlltif a ;C o . This property is known as dilataney.f"3, Collapse or limit design theorems. It has been shown previously that whenthe limit load is reached for any body or assemblage of bodies of perfectly plastic rna-terial, collapse takes place at constant stress. This means that at the instant of collapse.the strain rates are purely plastic. The assumption is made, just as in most problemsof elasticity, that changes in geometry are negligible. .

As we shall deal only with collapse or limiting states, E7; equals E~i , the total strainrate.. The major limit theorems for stress boundary value problems are:

(tJ collapse "ill not occur if any state of stress can be found which satisfies theR. v. Miees, . "' If ec h a ni k de r p l as ti sc h en Forma enderung lIOn KristaUen, Z. angew. Math. ~lech. a .

161-185 (H128).M. Reiner, A mathematical t~OTY of dilatancy, Am. Journ. of Math. 67, 350..3G2 {19-!5j.D. C. Drucker, H. J. Greenberg, W. Prager, Erterukd limn d ea ig n th .e or eT M fo r c old in uo u. IIItt/j"Q. AppJ. Math. 9, 381-389 (1952).

: 0 : .

.~, E

~

I

I


3/9

--,--SOIL MECHANICS AND PLASTIC ANALYSIS OR LI~nT DESIGN. q r ''OS of equilibrium and the boundary conditions on stress and for which f < k

~\'er~ iere; and(it) collapse must occur if for any compatible flow pattern, considered as plastic

nnly, the rate at which the external forces do work on the body equals or e:..ceeds therate of internal dissipation.The theorems are valid when frictional surface tractions are present. if there is noslip or if the frictional forces are known.

Discontinuousstress or flow states are permissible and are generally convenient forcomputational purposes.4. Plane strain. The yield function (2) will first be shown to reduce to the Mohr-

Coulomb rule in the case of plane strain. In this case E;3 , E:3 , E~3 (or E~ , ")';. , " y ; . in en-gineering notation) vanish. Therefore, from (4),813 and 8 23(T.. , T ) are zero, and

833 = - 2 aJ ~/2 Thus,and

(

Substituting (7) and (8) into the yield function (2), we findf = Sa (lz + (I. + (1 - 3,l)J~/2 = k2

C.E.\_.tion (9) becomes identical with Eq. (1) ifwe setand hence

A clearer pict.ure of the meaning of the plasticity relations is obtained by consideringu plane velocity pattern as shown in Fig. 2. The upper portion is supposed to translate

r.i,,Lx

FIG. 2. "Simpie" slip. .

. -"15 9

(6)(7)

~8).'

(9 )

(10)


4/9

-.~r

r0::~:!' .~,,'-:;

"o.~

. ~ ' I 'n,. ~ . ..;-. , , '". ~ D. C. DRuCKER AND W. PRAGER [Vol X, N6 0as a rigid body while the lower part is f ixed. If the transition layer between is mthin enough, Fig. 2 represents simple slip. The essential feature is that E~ = O. As

zero, E~ cannot be zero. In fact, from Eq. (5), E~ = 3a>. and the upper shaded renot only moves to the right but it also must move upward. Such volume expansioncompanying slip is a well known property of reasonably packed granular material.is an interesting and in fact a necessary result of plasticity theory that if the yield pin shear depends upon the mean normal stress, slip is accompanied by volume chaThis necessary relation is an essential difference between a plastic material andassemblage of two solid bodies with a sliding friction contact."Figure 1 shows clearly that the shearing stress on the slip surface is not the maximshearing stress R but is R cos f{J.It is interesting to observe that a is not completely free because 12a' cannot exunity . .As shown by Eqs. (10) this is equivalent to restricting sin f{J to be no more tunity.

5. Rate of dissipation of energy. The rate of dissipation of mechanical energyunit volume is

.~ , '", . ,~ .~.t o ! " . \,r..0 " . '.. .,'..i.t

As the dissipation is to be computed from a plastic strain rate pattern, it is desirto express Ain terms of strain rate. It follows from (3) that

Ubstituting in (11)k(' ') 1/2 _- D _.E;;E;; ._ C cos f{ J [2 ,2+ , , _ , 2 + 2 ,2+ ,2+ ,2+ , ' ] 1 1 2_ (Sa2+ 1/2)1/2 _ (1+ sin" f{J)1I2~" __ .E. 0 'Yu "t.. "tn,

which for t~e plane strain case may be 'written as

For purposes of Calculation it is convenient to consider a transition layer as in Fto be a simple discontinuity. However, for a ;C 0 a discontinuity o u ' in tangential velomust be accompanied by a separation or discontinuity o v ' in normal velocity. A transilayer of.appreciable thickness therefore must be present in 3. soil while there is nofor such a layer in a Prandtl-Reuss material where a =O.The rate of dissipation of work per unit area of discontinuity surface is

D.... ' 1 " 0 1 1 . ' + u O v 'where T is the shearing stress and a the normal stress (tension is positive) on the surThe same result can be obtained by taking the limit of tD, Eqs. (13,14) as t, the thickof the transition layer goes to zero while tE~2 = = lh~.= lou' a.ntl M 2 = tE ~ =.o v 'follows that

-D. C. Drucker, Some implicatioM of tcOrk hardening an d ideal plaaticity, Q. App1.Math. 7, 41(1950).


5/9

1 ! 1 5 ! ! )

AlsoSOIL MECHANICS A~D PLASTIC ANALYSIS OR LI:\IlT DESIGX

These results apply equally well to a curved surface of discontinuity because therej , . : no essential distinction locally as the thickness of the transition layer approaches zero.6. Dilatation. The rate of cubical dilatation, E:i , which accompanies plastic deforma-tion is given by Eq. (5). Substitution of (12) leads to

Sa (n_ ' , ) 1 1 2(1+ 002)1/2 -ijliij .sin ~ (n_, , )112= (1+ sin" ~)1/2 -ijEij

For plane strain, we obtain the rate of dilatationA ' , + ' sin ~ ( n _ ' 2 + n _ , 2 + 12)112~ = liz E. = (1 + sin" tp)1/2 '"z '".. ,.

orl ! . . ' = sin ~[(E~ - E~):+ , . ; : ] 1 1 2 =I"sin e,

where I" is the maximum shear rate.Interms of principal strain rates for plane strain, E ; , E ~ ,where E ; > E ;, , 1 + sin ~ 't: (45 + ~)EI = E: 1 _ sin ~ =.E2 an 2 .

7. Simple -discontinuous solutions. Any soil mass, in particular the slope shown in'Fig. 3, may fail by rigid body "sliding" motion in two simple ways, As is well known,

~surfaceS of discontinuity, idealizat~ons of discontinuity layers, are planes ~nd ci~c~arnders for a Prandtl..Reuss material, a = O. When a ; ; c 0, a plane discontinuitysurface is still permissible but the circle is replaced by a logarithmic spiral which is at an

FiG. 3. Rigid body "slide" motions.angle ~ to the radius from the center of rotation. The circle is not a permissible surfacefor rigid body "sliding" because a discontinuity in tangential velocity requires a separa-

- .

161

(17)

(18)

(19)

(20)

(21)


6/9

..~ *' .'~.'h:':!

,-,-

"';;'.-'i5fn"'diEmGl'- d -wseir'iWV;:,;,>~k&t~i;, 1.J. . '; - :1

\ :1, " : f~~i " j:~ l ..-;..r"f". . . .. (,,i,,ofr- .

:_

.~

r.,~

1 6 2 D. C. DRUCKER AND W. PRAGER [Vol. X, Ntion velocity. For a "slide" discontinuity the angle between the velocity vector anddiscontinuity surface is "', as is seen from Eq. (16).8. Example: Critical height of vertical bank. The computation of the critical he

H. of a vertical bank, Fig. 4, 'will serve as an illustration of the procedure consiswith plasticity theory. No attempt will be made here to get an exact plasticity soluto this problem for even if it could be obtained it would not be helpful for more comcated slopes. Instead, the upper and lower bound techniques of limit design will be u

I H tan ~T h'H HlSy = -w,/~x = t ' l fY = 0

A A = - W YL f - 4, ~ x y - - W(H~'J)G y = l S " 1. FIG. 4. (a) An equilibrium solution, (b) Plane, (c) Log. spiral compatible solutions.

Denoting the. specific weight of the soil by w, Fig. 4a shows a discontinuous equrium solution inwhich the maximum shearing stress is wH/2 at the lower ground lwhere the average normal stress is -wH/2. It should be remembered that the sfield need bear no resemblance to the actual state of stress, Equilibrium alonebe satisfied in addition to the yield condition (1) or (2) for H to be safe. From Eq.

wH wH .2"" =c cos'"+ 2"" sm '"or a . lower bound on the critical height H. is given by

H > H = 2c COS~ = 2ctan (45 + f)e - to 1 - sin II' w 2An upper bound on H. may be found by taking a . plane slide as a velocity patFig. 4b. Equating external rate of work to the dissipation Eq. ( 18 ) gives

WH2 tan f J ou' (+ R ) c H ! t c ,--cos", ~ =-IJU2 cos II' cosfj ,80 that

2c c os '"H. < H = - . R (+ R )- W sm ~ cos '" ~minimizing the right hand side "givesr{Jfj = = 45 - -2


7/9

..SOIL MECH ...;., ,\ICSA:\D PLASTIC A~ALYSIS OR LI~nT DESIGS 1 6 3

(25)usual theory furnishes the right-hand side of (25) as the critical height. Howevere is a factor of 2 between the upper bound (25) and the lower bound (22) for H .The upper bound can be improved by considering a rotational diseont inuity (log-

hmic spiral, Fig. 4c) instead of the translational type.9. A modified stress criterion. The large discrepancy between the lower and uppernds derived from reasonable stress and velocity fields (Fig. 4a and Figs. 4b, c) has aple explanation. An improvement in the static or equilibrium answer would requireing stresses which will result in tensile stress in the soil. Kothing in the Mohr-ulomb criterion, Fig. 1 rules out a moderate amount of tension. If itis felt that soilnot take any tension, then the yield criterion must be altered to take into accounticitly that

e r r ~ e r ~ ~ [ ( e r r ~ e r o ) ' + T ! ~ J / 2 = R (26)As before, if rigid body discontinuous solutions are considered, one strain rate, say, is zero. The relation between the normal and the shearing stress on a surface ofis now given by Fig. 5 instead of Fig. 1. The yield function and the r : e -

FIG. 5. Modified criterion for stress on discontinuity surface R o - c tan' (45 + ,,/2)lting velocity relations previously derived apply only for the straight line portion ofe limiting curve. .In general, the rate of dissipation, DA , given by Eq. (15), may be interpreted as

dot product of a stress vector having rectangular Cartesian components a, T withvelocity vector having components o v ', o u '. The stress vector is shown on Fig. 6 asor OQ and th e v e loc' it y v ec to r is normal 10 th e l im iti ng c un :e . The normality condition

lds because (J and T are the only stress components 'Whichdo work. * Considering theess vector OQ to be the sum of 00' and O'Q, we haveD = R { [ ( O U ' ) ' + ( O V , ) : ' 1 /2 - a v ' } = C O U ' tan(45 + ,,/2) (27)

.A 0 J tan(45 + 8/2),*D. C. Drucker, A moreJundamentol approach to plastic strur-atrain relations, to apptar in Proceed-gs, 1st U. S. National Congress for Applied Mechanics.

. .~.; .. , . ..:.. .' ; . : ' ; . . .ilt,r,tI

. ." ::.

. - e - . .:~~.~.. .. - .t ;~, ; . . .~ o f "t,~~,': . .,. '. ;i'~~~:,i '.r '.'


8/9

I

'.i, .,.. )

"

.,, .

161 D. C. DRUCKER AND W. PRAGER [Vol. X .

_,_,_ -FIG. 6. Rate of dissipation on discontinuity surface is crorl' + T iu

where6v' .tan 8 = Ou' ~ tan tp

R o = c tan (45 + tp/2)If Fig. 4b is modified by taking the angle between the velocity vector and thecontinuity line as 8 instead of tp, the results are not altered. However, if a verypicture is employed of rotation about a point below the lower ground level, Fig. 7upper bound is easily reduced to

H < H = 2.5 c tan (45 + f ) " . - w 2

IIIII" L . . . 30.~p

FIG. 7. Rotation about P, soil of zero tensile strength .


9/9

~ .~I'

SOIL MECHANICS AKD PLASTIC ANALYSIS OR Ll:\IIT DESIGX 165(lther reduction is possible with other discontinuous patterns but the concept ofSOl! -able to take tension clearly requires much additional study.( "::onclusion. It has been shown that. the implications of plasticity theory and

mit design for soil mechanics are indeed far reaching. Circular sliding surfaces are re -aced by logarithmic spirals in a material whose yield stress in shear depends uponrmal stress. Volume expansion is seen to be a necessary accompaniment to shearingLow critical heights are found for slopes when the soil is assumed to be

to take tension.

1 l . 4 - . h S f i # 4 f ' < . . " ,. t, ;. ?C . rtW.;: lZ . *

,f':~ x ' :.. .:... !..~.iUQE,r'..w . _ = A $ ;Q.. .

drucker dc_1952_soil mechanics and plastic analysis or limit design

Documents