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Technical Report No. 93

THE BEARING CAPACITY OF A FOOTING

ON A SOIL (PLANE-STRAIN)

by

R. T. Shield

GRADUATE DIVISION OF APPLIED MATHEMATICS

BROWN UNIVERSITY

PROVIDENCE, R.I.

August, 1953

All-93/10

All-93 1

THE BEARING CAPACITY OF A FOOTING ON A SOIL (PLANE-STRAIN)1

2 by R. T» Shield

Brown University

The purpose of this report is to outline a method of

showing that the Prandtl stress solution [ll to the plane-

strain problem of a flat rigid punch (or footing) is valid for

soils with cohesion whose angles of internal friction are less

than 75 degrees. Since a kinematically admissible velocity

field can be associated with the Prandtl stress solution [2],

limit analysis [3,1*] shows that the Prandtl value [l]

2,« p = c cot 9 [exp(* tan <p)tan (3 +*L) - 1] (1)

is an upper bound for the collapse value of the average pres-

sure over the punch, where c is the cohesion and 9 is the angle

of internal friction of the soil, A statically admissible ex-

tension of the Prandtl stress field into the rigid region is

found here, so that the value (1) is also a lower bound and

therefore the true value of the average pressure.

In Fig. 1, OE is the center-line of the punch OA

indenting the surface OD of the soil. The region OBCD is com-

posed of the Prandtl stress field of two regions of constant

state, OAB and AX, and a region of radial shear ABC. The Prandtl

The rosults presented in this paper were obtained in the course of research sponsored by the Office of Naval Research under Contract N7onr-3',*801 with Brown University.

2 Research Associate in Applied Mathematics, Brown University,

^Numbers in square brackets refer to the bibliography at the end of the paper.

All-93 2

field is continued to the line of stress discontinuity BSP which

is as yet unspecified. The stress field below this line is de-

termined from the following conditions, (i) ox and d are func-

tions of y only and x only respectively and i^ is zero, satisfy-

ing the equilibrium equations and the symmetry condition across

OE. (ii) In the im^odiate neighborhood of the line of discon-

tinuity, the material is in a plastic state of stress. These

conditions, together with the jump conditions [?] across the line

of discontinuity, are sufficient to determine the stresses in

the region BEFSB and to determine the line BSF. Figure 2 shows

the Mohr's circles for the states of stress on either side of

the line at a typical point S. It is found that, in region

BEFSB, o is a monotonic algebraically decreasing function of y

and Oy io a monotonic algebraically increasing function of x.

The yield condition is nowhere violated in this region if <p is

less than 75 decrees.

A perfectly plastir material may be considered as a

soil for which ?—* 0. Designating the shearing stress on a

plane of slip by k, it follows from the above that the value

(2 +n)k Is a lov/er an well as an upper bound for the collapse

value of the average pressure in the indentation of such a mater-

ial. A somewhat similar method has been used by Bishop [6] to

show the validity of the "compieto" solution to the v-notched

bar problem for a perfectly plastic material.

All-93

APPENDIX

The soil is assumed to be a plastic material in which

slip or yielding occurs in plane strain when the stresses satisfy

the Coulomb formula [7]

2(dx + °y)sln <P + ji(ox " °y)2 + Txy\ " - c cos 9 = 0. (2)

This equation and the two equations of stress equilibrium (in

which the weight of the soil is neglected) form a hyperbolic

system of equations for the determination of the stresses dx, 0

and x , The two characteristic lines are inclined at an angle xy

*A + ?/2 to the direction of the algebraically greater principal

stress and will be called the first and second failure lines,

with the convention that the direction of the first failure line

is obtained from the direction of the algebraically greater

principal stress by a clockwise rotation of amount itA + 9/2.

The angle of inclination of the first failure line to the x-axis

is denoted by 9.

It is convenient to put

(d~ - d,) p - ~± L > 0, (3)

2 sin 9

whore 0, and dp (d. < dp) are the principal stresses. From (2)

and (3) it can be shown [8] that

d = - p [1 + sin 9 sin(20 + 9)] + c cot 9,

d = - p [1 - sin 9 sin(2S + 9)] + c cot 9, ^

T = p sin 9 cos(2© + 9). xy

(»0

All-93 *••

The equations of equilibrium can be replaced by the relations [9]

•T cot op log p + 0 - const, along a firs* failure line, 7 (5)

•1 cot 9 log p - 0 = const, along a second failure line.

Across a line of stress discontinuity in a plastic

stress field, the normal and tangential components of stress must

be continuous across the line for equilibrium. The equilibrium

conditions across aline of discontinuity separating two plastic

stress fields a and £ can be written [5]

sin(Oa + ©. - 2Q + <p) + sin <p cos(©a - ©.) = 0,

p cos(2©a - 2Q + 9) = pb cos(20b - 2Q + <p)

(6)

where Q is the inclination to the x-axis of the normal to the

line at a current point of the line. Subscripts £ and jb. distin-

guish the values \;hich p and © assume on the two sides of the

line.

Referring now to Fig. 1, the surface AD is free from

traction and o = 0 in the constant state region ACD. Since

0 = •n/V - qp/2 in ACD it follows from the second of fiqs. (h) that

p has the value c cot p 1

(1 - sin 9)

in the region ACD. The first of Eqs. (5) then shows that at a

point G on the first failure line BCD,

p = ° COt * cxp {?. tan9 q - » - a)) , (7) (1 - sin 9) l 2 J

where a is the value of © at the point G,

All-93 5

It has been mentioned above that the material Just

below the line of stress discontinuity BSP is assumed to be in

a plastic state of 9tress, We shall denote by ^ and £ the two

plastic stress fields immediately above and immediately below

the point S respectively, where S is the point of intersection

of the second failure lino AG and the line of discontinuity.

Since the failure line AGS is straight, the value of p, pa, in

region j at S is also given by Eq, (7) and we have 0. = o* In

region jj at S, ux < o so that ©^ = %/k - 9/2. Also the normal

to the 1.ne of discontinuity at S is inclined at an angle

Q = K/2 - \J> to the x-axis, where 9 is the inclination of tho

lino to the negative x-axis. Substitution of these values into

the first of the jump conditions (6) gives

sin(j- + | + a + 2\|i) = sin 9 sin(£ + .? + a),

and the relevant root of this equation is

where ,n 9

;in p. = sin 9 sin(r + -5 + <*), 0 < p < 9. (9)

.Vith this value of 9, the second of tuJ jump conditions (6) gives

cos(r - a + >i - 3J) Pb = P ft cos(j| . a - ji - |) '

c Jl+sin29-2 sin9Cos(£ + 2 +a-ji)} > pb= .--i r~2 2 ^-exp {tan 9 (5 -<p -2a) ,(10)

(1 - sin 9)sLn 9 cos 9 l 2 J

where the value (7) for p., and Eq. (9) have been used. a

All-93 6

yrora Eq. (M-), the non-zero stresses in region Jj at the

point S are given by

d = - Pfc(l + sin 9) + c cot 9,

d = - p^d - sin 9) + c cot cp.

(11)

u

For equilibrium, d,, and 0 are taken to be functions of y only 'A y

and x only respectively in region BEFSB. The values of <*x and

d just below the line BSF are known from Eqs. (11) and (10) 30

that d and d can be found at any ooint of the region BEFSB. * y

It can be shown that p^ is a monotonic decreasing function of a.

It follows that just below the line BSF, a% and d are increasing

functions of a, and hence that in region BEFSB 0 is a monotonic

decreasing function of y and d„ is a monotonic increasing func-

tion of x.

This extension of the Prandtl stress field is permiss-

ible only if the yield condition is nowhere violated in region

BEFSB, i.e., if the expression on the left hand side of Eq. (2)

is less than or equal to zero at all points in the region. Be-

cause of the monotonic character of the stresses d and 0 , the

yield condition will not be violated anywhere in the rezion if

it is not violated at the point E at infinity on the y-axis. At

the point E, cx has an (algebraic) maximum value and d has an

(algebraic) minimum value and these values are

dx = - 2 c tan<* + *),

d = - c cot 9 exp(n tan <p)tan2(J - %) - X *, y L » 2 j

All-93 7

Substituting these values into the expression on the loft hand

side of Eq. (2) and setting the resulting expression less than

or equal to zero gives, after reduction, the inequality

exp(it tan 9) £ tan (5 + $).

The inequality is satisfied if the angle 9 is less than an angle

which lies between 75 and 76 degrees. Thus the yield condition

is nowhere violated in region BEFSfl if 9 is less than 75 degrees.

If the length of the line AS in Fig. 1 is denoted by

r, then we have

dr = rdCa + 9)tan(\Ji + 0 + 9), or

dr = rda tanC^P + | + & - £).

a = - nA - 9/2, determines the line it discontinuity. As a tends

to 71A - 9/2, the line tends asymptotically to a straight line

inclined at an angle TIA - 9/2 to the negative x-axis.

AU-93 6

Bibliography

1. "Ueber die Haerte plastischer Koerper". by L. Prandtl, Nachrlchten von der Koeniglichen Gesellscnaft der Wissens- chaften zu Goettingen, Matheraatisch-Physikalische Klasse, 1920, pp. 7*f-85.

2. "Mixed Boundary Value Problems in Soil Mechanics", by R. T. Shield, Quarterly of Applied Mathematics, vol. 11, 1953, pp. 61-75.

3. "On Plastic-Rigid Solutions and Limit Design Theorems for Elastic-Plastic Bodies", by D. C. Druuker. H. J. Greenberg, E. H. Lee, and W. Prager, Proceedings of the First U.S. National Congress of Applied Mechanics, June, 1951» PP» 533-538.

h, "Soil Mechanics and Plastic Analysis or Limit Design", by D. C. Drucker and W, Prager, Quarterly of Applied Mathemati cs, vol. 10, 1952, pp. 157-165.

5. "Stress and Velocity Fields in Soil Mechanics", by R. T. Shield, Brown University Report All-81 to the Office of foval Research, December, 1952.

6. Private coTiunication from J. F. w*. Bishop to W. Prager, October, 1952.

7. "Theoretical Soil Mechanics", by K, Ter-saghi, John V/iley and Sons, Inc., New York, .J. Y. , 19^3, p. 22.

8. "Statics of Earthy Media", by V. V. Sokolosky, I^datolsvo Akademii Nauk S. S. R. , Moscow, 19^2.

9» "Die Bestiramung des Drucks an gekruemmten Gleitflaechen, eine Aufgabe aus der Lehre vora Erddruck", by F. Ko'tter. Monatsberichte der Akidrmic der Jissonschaften zu Berlin, 1903, pp. 229-233.

AM-93

Fig. I An extension of the Prandtl stress solution for a footing on a soil

All-93 10

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