157
SOIL MECHANICS AND PLASTIC ANALYSIS OR LIMIT DESIGN*
BY
D. C. DRUCKER and W. PRAGER
Brown University
1. Introduction. Problems of soil mechanics involving stability of slopes, bearing
capacity of foundation slabs and pressures on retaining walls are often treated as problems
of plasticity. The soil is replaced by an idealized material which behaves elastically up
to some state of stress at which slip or yielding occurs. The shear stress required for
simple slip is often considered to depend upon the cohesion and linearly upon the normal
pressure on the slip surface. In more complete plane investigations an extended Coulomb's
rule is used,** (Fig. 1)
R = c cos <p — <7x "jr sin <p, (1)
A
where i? is the radius of Mohr's circle at slip, the maximum shearing stress [{cx — <x„)2/4 +
rlv]1/2; c is the cohesion; c cos <p is the radius of Mohr's circle at slip when the mean
normal stress, (<rx + <ru)/2, in the plane is zero, and <p is the angle between the tangent
to the Mohr's circles at slip and the negative a axis.
Fig. 1. Mohr-Coulomb hypothesis.
The purpose of this note is to discuss the implications of assuming the soil to be a
perfectly plastic body. The validity of this assumption is not at issue. No account is
taken of such important practical matters as the effect of water in the soil or of the
essentially different behavior of various constituents such as clay and sand. All that is
sought is a theory consistent with the basic assumption. Comparison of the predictions
of such a theory with the actual behavior of soil will give an indication of the value of
the idealization.
*Received Nov. 19, 1951. The results presented in this paper were obtained in the course of research
sponsored by the Office of Naval Research under Contract N7onr-35801 with Brown University.
**K. Terzaghi, Theoretical soil mechanics, John Wiley and Sons, 1943, pp. 22, 59.
158 D. C. DRUCKER AND W. PRAGER [Vol. X, No. 2
2. Yield function and stress-strain relation. A yield function which is a proper
generalization of the Mohr-Coulomb hypothesis (1) is
f = aJ1 + Jl/2 = k, (2)
where a and k are positive constants at each point of the material; Jx is the sum of the
principal stresses:
J\ — <Tl 4" 02 + 03 = fx + <Ty + = 0" 11 + a2 2 + 033 = 0,'i
J2 is the second invariant of the stress deviation:
J2 = hijSa = — 0k)2 + (0» — 02)2 + — 0x)2] + r2xy + t\z + t]x .
The stress deviation is defined as — a,-,- — (Ji/3) , where <5;, is the Kronecker delta,
zero for i j and unity for i = j. For example, Sn = sx = %[ax — \(<tv + crj], s12 =
SXy Txy •
As is well known, the yield surface, / = k, in principal stress space is a right circular
cylinder for the Mises criterion (a = 0). For a > 0, the surface is a right circular cone
with its axis equally inclined to the coordinate axes and with its apex in the tension
octant. A generalization of (1) as a modified Tresca or maximum shear stress criterion
is 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 corresponding
to the yield function (2) is
c'i = X df/daa , (3)
where e"' is the plastic strain rate and X is a positive factor of proportionality which
may assume different values for different particles. Substituting the expression (2) for
/, we obtain
t'l = XM„- + su/2 J1/2]- (4)
A very important feature of Eq. (4) is that the plastic rate of cubical dilatation is
t'u = 3 «X. (5)
Equation (5) shows that plastic deformation must be accompanied by an increase in volume
if a 0. This property is known as dilatancy.**
3. Collapse or limit design theorems. It has been shown previously*** that when
the limit load is reached for any body or assemblage of bodies of perfectly plastic ma-
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 problems
of elasticity, that changes in geometry are negligible.
As we shall deal only with collapse or limiting states, e'; equals t',- , the total strain
rate.
The major limit theorems for stress boundary value problems are:
(i) collapse will not occur if any state of stress can be found which satisfies the
*R. v. Mises, Mechcmik der plastischen Formaenderung von Kristallen, Z. angew. Math. Mech. 8,
161-185 (1928).
**M. Reiner, A mathematical theory of dilatancy, Am. Journ. of Math. 67, 350-362 (1945).
***D. C. Drucker, H. J. Greenberg, W. Prager, Extended limit design theorems for continuous media
Q. Appl. Math. 9, 381-389 (1952).
1952] SOIL MECHANICS AND PLASTIC ANALYSIS OR LIMIT DESIGN 159
equations of equilibrium and the boundary conditions on stress and for which f < k
everywhere; and
(it) collapse must occur if for any compatible flow pattern, considered as plastic
only, the rate at which the external forces do work on the body equals or exceeds the
rate of internal dissipation.
The theorems are valid when frictional surface tractions are present if there is no
slip or if the frictional forces are known.
Discontinuous stress or flow states are permissible and are generally convenient for
computational 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 €33 , e'l3 , t'23 (or t'z , y'zi , y'vt in en-
gineering notation) vanish. Therefore, from (4), s13 and s23 (txz , tuz) are zero, and
S33 = —2aJ\/2. (6)
Thus,
J1 = §("n + 022) — 3aJl/2, (7)
and
J2 =
2 (7") + T'» /(! - 3t*2)- (8)
Substituting (7) and (8) into the yield function (2), we find
/ = 3a + (i _ = k
3a <JX + tr,,
+
(1 - 3a2)1/2 (1 - 3a2)1/2 2
Equation (9) becomes identical with Eq. (1) if we set
'(^j +
(9)
Jc 3a .
c ~ (1 - 12ccT* ' (1-3a2)1/2 ~ Sln ^ (10)
and hence
(1 - 12a2)1/2
(1 - 3a2)1/2 " C0S^-
A clearer picture of the meaning of the plasticity relations is obtained by considering
a plane velocity pattern as shown in Fig. 2. The upper portion is supposed to translate
Fig. 2. "Simple" slip.
160 D. C. DRUCKER AND W. PRAGER [Vol. X, No. 2
as a rigid body while the lower part is fixed. If the transition layer between is made
thin enough, Fig. 2 represents simple slip. The essential feature is that e'x = 0. As e' is
zero, €y cannot be zero. In fact, from Eq. (5), t'„ = 3aX and the upper shaded region
not only moves to the right but it also must move upward. Such volume expansion ac-
companying slip is a well known property of reasonably packed granular material. It
is an interesting and in fact a necessary result of plasticity theory that if the yield point
in shear depends upon the mean normal stress, slip is accompanied by volume change.
This necessary relation is an essential difference between a plastic material and an
assemblage of two solid bodies with a sliding friction contact.*
Figure 1 shows clearly that the shearing stress on the slip surface is not the maximum
shearing stress R but is R cos <p.
It is interesting to observe that a is not completely free because 12a2 cannot exceed
unity. As shown by Eqs. (10) this is equivalent to restricting sin tp to be no more than
unity.
5. Rate of dissipation of energy. The rate of dissipation of mechanical energy per
unit volume is
D = (Tijt'ij = «r„ X = X/ = A*. (11)
dan
As the dissipation is to be computed from a plastic strain rate pattern, it is desirable
to express X in terms of strain rate. It follows from (3) that
' ' _ \2 d/ df _ ^2/o 2 1 1
^ » J ^ t J A 13(2 "I" gy
ova OGij \ 2.
(12)
Substituting in (11)
kje'jt'j,) c cos <p
D = (#+V2T = (iW [2tf + 2£"2 + 2e''2 + + + 7"],/2 (13)
which for the plane strain case may be written as
D = C COS <p[(e'x - e'vf + T^]1/2 (14)
For purposes of calculation it is convenient to consider a transition layer as in Fig. 2
to be a simple discontinuity. However, for a 7^ 0 a discontinuity 8u' in tangential velocity
must be accompanied by a separation or discontinuity 8v' in normal velocity. A transition
layer of appreciable thickness therefore must be present in a soil while there is no need
for such a layer in a Prandtl-Reuss material where a = 0.
The rate of dissipation of work per unit area of discontinuity surface is
Da — tSu' + a8v' (15)
where r is the shearing stress and a the normal stress (tension is positive) on the surface.
The same result can be obtained by taking the limit of tD, Eqs. (13, 14) as t, the thickness
of the transition layer goes to zero while te'12 = t\y'rv = §5u' and te'22 = te'u = Sv'. It
follows that
c'
8v' = 8u' = J 2^1/2 8u' = 8u' tan <p. (16)
Zti2 (1 12a )
*D. C. Druoker, Some implications of work hardening and ideal plasticity, Q. Appl. Math. 7, 411-418
(1950).
1952] SOIL MECHANICS AND PLASTIC ANALYSIS OR LIMIT DESIGN 161
Also
D- = (1 -%r "e (17)
These results apply equally well to a curved surface of discontinuity because there
is no essential distinction locally as the thickness of the transition layer approaches zero.
6. Dilatation. The rate of cubical dilatation, ej, , which accompanies plastic deforma-
tion is given by Eq. (5). Substitution of (12) leads to
(oj y/2
1/2
(1 + 6a )
(18)
Sin y /rw r \1/2
(1 + sin2 <p)l/2
For plane strain, we obtain the rate of dilatation
A' = + < = (i +Tin^)1/2 (2e'2 + 2<2 + 7")1/2 (19)
or
A' = sin <pl(e'x - e'v)2 + y'xl}1/2 = I" sin v, (20)
where T' is the maximum shear rate.
In terms of principal strain rates for plane strain, t[ , e'2 , where > t'2
<1 = tan2 (45 + o)- <21)
1 — sin <p \ z/
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,
the surfaces of discontinuity, idealizations of discontinuity layers, are planes and circular
cylinders for a Prandtl-Reuss material, a = 0. When a 9s 0, a plane discontinuity
surface is still permissible but the circle is replaced by a logarithmic spiral which is at an
CENTER OF ROTATION
fc, NWSCONT,NU,TY Vs^LOG SPIRAL
RIGID
RIGID I
Fig. 3. Rigid body "slide" motions.
angle <p to the radius from the center of rotation. The circle is not a permissible surface
for rigid body "sliding" because a discontinuity in tangential velocity requires a separa-