A note on failure mechanisms in soil beneath foundations

Engineering Geology Elsevier Publishing Company, Amsterdam - Printed in The Netherlands

A NOTE ON FAILURE

MECHANISMS

IN SOIL BENEATH

FOUNDATIONS

B. DENNESS Engineering Geology Unit, Geophysics Division, Institute of Geological Sciences, London (Great Britain) (Accepted for publication June 21, 1972)

ABSTRACT Denness, B., 1972. A note on failure mechanisms in soil beneath foundations. Eng. Geol., 6:203-211. A small-element analysis is presented whereby surfaces of sliding beneath foundations may be determined. It is shown that, for a uniform cohesive soil under undrained conditions, these surfaces compare favourably with those observed in practice beneath loads that Terzaghi considered to be the ultimate bearing capacity. The minimum load to cause local shear is shown to be similar to that proposed by Terzaghi. An explanation is suggested for the observation of humping next to failed foundations without the necessary observation of a slip surface there in practice.

INTRODUCTION Earth pressure theories have been proposed for various soil conditions for centuries. For instance, Coulomb (1776) c o m p u t e d the passive earth pressure of ideal sand on the simplifying assumption that the entire surface of sliding consists o f a plane through the lower edge of the loaded face bounding a wedge of sand behind it. This plane was independent of the applied loads or the strength of the sand. Referring to the stability of loose earth, Rankine (1857) proceeded to derive his active and passive state theory from which the overall shear patterns within a deforming mass might be determined based on a knowledge of the stress distribution and the angle of internal friction of the material. This was another great step forward and could be considered as the basis of the present study which seeks to implement much of Rankine's theories in conditions of very variable soil with real strength/deformation characteristics. Prandtl (1920) developed a bearing-capacity hypothesis, originally set up for metals, which considered the curved surface bounding the failure zone beneath a rough continuous foundation to be a logarithmic spiral. The surface of sliding depends on neither soil cohesion nor applied load, but is related to the internal friction angle of the soil. To derive the ultimate bearing capacity Prandtl resorted to a limiting equilibrium approach; that is, he chose a potential failure surface based on the friction angle theory, and analysed the equilibrium of the whole section bounded by that surface. In the parallel context of slope stability Frontard (1922) proposed a theory which

204

B. DENNESS

disregarded the existence of a zone of transition between the Rankine states which are thought to prevail in the vicinity of the upper and lower limits of a sliding mass. The discontinuous shear pattern thus assumed lead to excessive errors despite the complicated computations. Jaky (1936) assumed that such a slide occurs along an arc of a toe circle intersecting the slope at the same angle as a real surface of sliding. Terzaghi (1943) pointed out that such an assumption was unjustified and went on to summarize further limitingequilibrium slope-stability solutions based on simplifying assumptions. These include the methods of Fellenius (1927) and Taylor (1937), based on the assumption that the surface of sliding is circular, and that of Rendulic (1935) who proposed replacing the real surface of sliding by a logarithmic spiral. Recently Morgenstern and Price (1965) and Nonveiller (1965) proposed slope-stability analyses which permit the consideration of complete noncircular surfaces of sliding. Unfortunately an assumption relating to the distribution of normal pressure around such a surface is required for solution and this is not always readily available for a predetermined potential failure surface in a real soil. Janbu (1954) and Bishop and Morgenstern (1960) had already advanced methods by which such solutions could be similarly derived for circular sliding surfaces. One of the major problems inherent in most techniques that consider real slip surfaces is that the stress distribution in the material around the locus of the potential failure surface must be known or estimated. Unfortunately, for natural slopes the topography often causes this distribution to be somewhat complex, though not perhaps insoluble. Denness (in preparation) considers a small-element analysis of a special case in which these limitations are in part overcome. The analysis of failure under drained conditions requires knowledge of the component of stress perpendicular to the locus of potential failure at any point so that the frictional component of resistance may be calculated. Furthermore, the shearing resistance mobilised along the failure surface often varies with deformation from a peak to a residual value, here termed brittle behaviour, thereby causing further changes in the stress distribution. SMALL-ELEMENTSOLUTION FOR BEARING CAPACITY ON COHESIVE SOILS The principles involved in solving the slope-stability problems are similar to those proposed here for the case of normal loading of a flexible foundation, under undrained conditions, on the horizontal surface of a soil which exhibits little or no drop in strength after straining to the peak value and approximates to an ideal elasto-plastic medium. In practice, these conditions are commonly fulfilled, for example, by the rapid filling of oil storage tanks on estuarine deposits, or of grain silos, etc., on soils with similar characteristics; Fig. 1 illustrates a typical, simplified strength/deformation characteristic of such materials. In this special case the resistance along all the failure surfaces is almost constant with respect to strain after the peak value has been reached and the surfaces have been initiated so that the initial stress distribution may be assumed to remain constant, provided the pre-failure stress/strain relation approximates to a linear function and the strain at failure is small.

F A I L U R E M E C H A N I S M S IN S O I L B E N E A T H F O U N D A T I O N S

205

Failure Stress

Strain Strain at beginning of catastrophic failure Fig. 1. T y p i c a l s t r e s s / s t r a i n c h a r a c t e r i s t i c s f o r a n o r m a l l y c o n s o l i d a t e d alluvial soil.

At this point it is convenient to introduce the concept of "boundary" and "internal" failure in the context of self-generating failure. Boundary failure, or a boundary shear locus, refers to the plane below which there is no further failure. Internal failure, perhaps comprising a large number of internal shear loci, refers to the overall plastic deformation within the mass encompassed by the boundary failure. These definitions are illustrated in Fig.2. 2a

~ " ~ "- Boundary shear

~ Internal shear

Fig.2. T y p i c a l set o f s h e a r p l a n e s ( b o u n d a r y a n d i n t e r n a l ) b e n e a t h a flexible f o u n d a t i o n . Cu : 3 0 0 0 k g / m 2 , 3' = 2 0 0 0 k g / m 3 , K 0 = 0 , 5 , a = 3.0 m , z = 0.

It is later shown that the orientation of file shear surfaces, both boundary and internal, at any point on these planes is a function of applied load, depth and lateral position, with all other parameters remaining constant for a given case. The internal shear planes emanate from any point on the base of the foundation and generate in two complementary directions until reaching the boundary surface or an area in which the deviator stress is

206

B. DENNESS

lower than the shear strength as shown in Fig.2. The deviator stress below the boundary shear locus is lower than the shear strength of the soil owing to the restraint offered by neighbouring soil in the same way that the deviator stress at great depths of burial beneath an unloaded surface is insufficient to cause failure. The two-dimensional stress distribution in a semi-infinite, ideal elastic solid beneath a strip footing of width 2a subject to a pressure q, as depicted in Fig.3, has components:

Ox=-q__[a-sinacos(a+26)] ;

Oy=q_ [o~ + sina cos (a + 26)] 7"f

rxy = q sin a sin (a + 26) 7r 2Q

~/x~\\\'~. ~

\ 7/,&,-
/

< (x,y)

Fig.3. Configuration for the determination of the stress distribution induced by uniform pressure at the surface of a real soil. To introduce these components into a small-element calculation, static equilibrium of forces is determined by resolving along the potential failure surface, as shown in Fig.4. Stress components due to overburden and water table are also taken into consideration. A failure surface is defined as a line on which the maximum deviator shear stress has the value Cu in Ko (earth pressure at rest) conditions. At other orientations it would be found that Ko ~ Kp (average) (or locally Ko > Kp (average)) in order to maintain r < Cu. Resolving forces parallel to any shear plane: [(')'y Jr Oy) Z~kX"4- Txy Ay]

sin 13

- ( [ K o [ 3'y - Tw (y -- z)}

+ ,yw Cv - z ) -

Ox] A y + 7 x y ~ )

COS ~J= Cu . z~s

where 3' = bulk density of the material; K o = coefficient of earth pressure at rest; Kp = coefficient of passive earth pressure; Cu = undrained shear strength of material; 7w = density of water; z = depth to water table.

FAILURE MECHANISMS IN SOIL BENEATH FOUNDATIONS

I,

2Q

i

r

& , ..,, ,,., ,.

x

207

,,,,

~,/~,\\\x4/// [(cry +~y}Ax+'TxyAy ]

1 [-"x" KJ~y-~ ly-~)l ,~,,,(y-z)]Ay + 1:xy Ax

Fig.4. Forces acting on a small element in a real cohesive soil beneath a surface load.

N.B. I f y ~
Ko7w (Y - z) = 0

The solution of the above general momentary equilibrium equation in terms of tan/3 (= Ay/,~c), and also the solution for the specific starting slope, tan/30, are presented in the Appendix. It is interesting to note that tan/3o, and hence/3o, is real as long as q >~~rCu. Therefore, i f q < 7rCu, no real self-generating failure can occur. It is readily seen that the potential failure loci will vary with any of the parameters involved, including q. However, as catastrophic failure, or the unbalance of moments around the boundary failure surface, is of primary concern in design it is required that the value of q be determined that causes this to occur; this is termed the ultimate value o f q . By taking moments about (0,0) this value of q may be determined for any given set of parameters. This procedure is currently under investigation and charts of the critical value of q are in preparation for a wide range of the parameters. It is seen that a fundamental advantage of this method is that it incorporates all the soil and stress parameters introduced for each element encountered. EXAMPLE

Particular solutions for the following soil parameters, load and foundation dimensions give the boundary failure surface loci shown in Fig.5: Cu = 5000 kg/m 2 ; 3' = 2000 kg/m a ; Ko = 0.5 ; a = 8 m; and z = 0 m. For illustration purposes only, Cu, "7 and Ko are taken as constants. It is interesting to note that the failure surface corresponding to a value o f q = 5.7 Cu which is the ultimate value of bearing capacity for general shear according to Terzaghi (1943), is very similar to

208

B. DENNESS

7/2~\\\\X~//

q, : 31TC u ~ _ _ q =5

C

u

~

q = 5Cu

Fig.5. Loci of potential boundary slip surfaces in a cohesive alluvial soil beneath a strip surface load.

Cu = 5000 kg/m2 , 3' = 2000 kg/m 2 , K0 = 0.5, a = 8.0 m, z = 0. the failure surface taken by Terzaghi and to that observed in practice. It may also be noted that Terzaghi suggests a value o f q = 3.8 Cu at which local shear would be induced. From the present analysis it is seen that local shear would commence at q = rrCu, the lowest value at which real shear could take place. Since the conditions are symmetrical about the vertical centre line, the solution is also theoretically symmetrical. Thus a "mirror image" set of failure surfaces may be represented in Fig. 5, as inset f o r q = 5 Cu. It is thought that the failure will be of a slow plastic form, with local shearing near the foundation, for values o f q less than the ultimate value because the major displacement of material to allow the formation of a full boundary shear zone is not permitted until that value. Values o f q greater than the ultimate value give loci that are obviously not of practical significance, as failure occurs at q = qultimate. It is also interesting to note that the loci of the shear surfaces do not emerge at the ground surface in any case. It is thought that this may explain the "humping" of soil by the side of a failed foundation, without the necessary observation of a slip surface. It is not necessary for there to be a deviator stress great enough to create a sheared surface in order that there should be enough deformation to cause this "humping"; stresses lower than the failure stress may induce sufficient deformation. CONCLUSIONS The analysis permits the derivation of the surfaces of sliding beneath foundations

FAILURE MECHANISMS IN SOIL BENEATH FOUNDATIONS

209

based on local stress and strength conditions within the soil and the dimensions of the foundation. The major assumption is that deformation does not appreciably alter the stress distribution in a soil which exhibits no loss of strength when sheared past its peak strength in undrained conditions. The example given shows that the boundary slip surface determined from this analysis, for the bearing pressure that Terzaghi considered to be the ultimate value, i.e. q = 5.7 Cu, is very similar to a real bounding slip surface in a uniform soil. It is independent of an internal angle of friction in undrained conditions. It was also found that the minimum pressure beneath which any form of real shear could take place, i.e. q = 7r Cu, compares favourably with the pressure at which Terzaghi considered local failure to occur, i.e. q = 3.8 Cu. Theoretically, no slip surface emerges at ground surface in any case. It is considered that this may explain the humping of soil adjacent to a failed foundation without the necessary observation of a slip surface there in practice. ACKNOWLEDGEMENTS The author is indebted to the Natural Environment Research Council of Great Britain for financial support and to Dr P. G. Fookes of the Engineering Geology Division at Imperial College for general encouragement during the preparation of some of this work. The author is also grateful to Dr M. W. Green of the Mathematics Department of the University of Dundee for his criticism of the analysis. He would also like to thank Dr J. W. Bray of the Mining Department at Imperial College, University of London, for his constructive criticism. A special acknowledgement is due to Mr. P. Thomas of the Mathematics Department of the University of Dundee who prepared and executed a series of computer programmes in connection with this work. REFERENCES Bishop, A. W. and Morgenstern, N., 1960. Stability coefficients for earth slopes. Gkoteehnique, 10: 129-150. Coulomb, C. A., 1776. Essai sur une application des rbgles des maximis et minimis ~ quelques probl~mes de statique relatifs ~ l'architecture. Mere. Acad. R. (Paris), 7. Denness, B., in preparation. Possible stabilization of a vertical excavation/cliff face based on an analysis of a self-generating slip plane. Fellenius, W., 1927. Erdstatische Berechnungen. W. Ernst, Berlin. Frontard, M., 1922. Cycloides de glissement des terres. C R. Acad. Sci., 174: 526-528. Jaky, J., 1936. Stability of earth slopes. Proc. Int. Conf. Soil Mech. Found. Eng., 1st, Cambridge, Mass., 2: 125-129. Janbu, N., 1954. Stability analysis of slopes with dimensionless parameters. Harv. Soil Mech. Ser., 46. Morgenstern, N. and Price, V. E., 1965. The analysis of the stability of general slip surfaces. Gkotechnique, 15: 79-92. NonveiUer, E., 1965. The stability analysis of slopes with a slip surface of general shape. Proc. Int. Conf. Soil Mech. Found. Eng., 6th, Montreal, 1965, 2: 522-525.

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B. DENNESS

Prandtl, L., 1920. Uber die Harte plastischer K~Srper. Nachr. K. Ges. Wiss. GOttingen, Math. Phys.

Klasse. Rankine, W. J. M., 1857. On the stability of loose earth. Philos. Trans. R. Soc. Lond., 147: 9 - 27. Rendulic, L., 1935. Ein Beitrag zur Bestimmung der Gleitsicherheit. Bauingenieur, 16: 230-233. Taylor, D. W., 1937. Stability of earth slopes. Boston Soc. Civil Eng., 2 4 : 1 9 7 - 2 4 6 . Terzaghi, K., 1943. Theoretical SoilMechanics. Wiley, New York, N.Y. pp. 118-134.

APPENDIX Solution of the general momentary equilibrium equation in terms of tan j3, and also the solution for the specific starting slope, tan Go.

[ (mY + Oy) ~x + rxy ,xy ] sin ([Ko {~/y- Tw (y z)} + ~ w ( y - z ) - o

x ] Ay + "cxy Ax \) cosl3 = Cu.As

.'.

[ (mY + Oy) cos/3 + rxy sin t3 ] sin - ( [Kol'~Y--'rw(Y--z)}i+~w~V--Z)--axlsin~+rxyCOS#)COs~=Cu

.'.

[ ( 1 - K 0) {~/y-~wO'-z)}+ (Oy+Ox) ] sin/3cos#+~'xy(Si n2 ~3- cos 2 ~)=Cu

.'.

[ (l-K0)

{3,y-~/w(y-z)}+ 2qbr sinc~cos(c~+ 26)] sinl3cost3 +(q/n) [sinc~sin(c~+ 26)(sin 2 t~ cos 2 t3)] =Cu (1)

From Fig.2 it is seen that: sin6 =

x/d:; cos6 =y/d2; sin(c~+f)=(2a-x)/dl; cos(~+6)=y/d~

But: d~ = ( 2 a - x ) ~ + y2 and d~ = ( - x ) 2 + y2 .'. sin6 = - x ( x 2 + y : ) - ~ / 2 ; a n d c o s 6

=y(x ~ +y~)-~/2;

sin (c~+6) = ( 2 a - x ) ( [ 2 a - x ] : + y 2 ) l# ; and cos (a +6 ) = y ( [ 2 a _ x ] 2 +y~)-I h But: cos (c~ + 26) = cos (c~ + 6) cos 6 - sin (c~ + 6) sin =y([2a_x]2

+ y2)-~#.y(x2 + y2)-~/2-(2a-x) ( [ 2 a - x ] ~ + y2)-~/2.(-x)(x2 + y2)-~/2

= {y2 + ( 2 a - x ) x }

{ ( [ 2 a - x ] 2 + y 2 ) ( x ' +y2)} -,/2

Similarly: sin (~ + 26) = sin (c~ + 6) cos 6 + cos (a + 6) sin = (2a_x)([2a_x]~

+y~)-~/2.y(x: +y:)-~/2 + y ( [ 2 a _ x ] ~

= 2 y ( a - x ) { ( [ 2 a - x ] ~ +y~)(x ~ + y ~ ) } -~/~ Also from Fig.2: 2a = d~ sin ~ sin (90 ° + 6) 2a 2a .'. sinc~=d~--.sin(90 ° + 6 ) = ~ cos6 = 2t/ ( [ 2 a _ X ] ~ + y 2 ) - 1 / 2

. y ( x 2 q_y2)-l/~

= 2ay { ( [ 2 a - x ] ~ +y~)(x ~ + y ~ ) } - ' / ~

+y~)-i/~.(_x)(x ~ +y:)-~/~

FAILURE MECHANISMS IN SOIL BENEATH FOUNDATIONS

211

:. Eq.1 becomes:

+-~-.2ay

((1-Ko)[-ty-Tw(y-z)] {[(2a-x) 2 +y']

{[(2a-x) 2 + y ~ ] ,Ix'

+y']l

{Y'

-'/~"

[x' +y=]}-'/2)sint3cost3+(_~._q. 2ay{([2a-x]'

1

+x(2a-x)

+y')(x 2 +y')'-'['

•

2 ( a - x ) y {([2a - x] 2 + y= ) (x = + y , )} -1/,) {sin' /3 - cos' j3) = Cu '"

( ~ u K°)

{~/Y-~/w(Y-z)} +

.'. tan, O(Cucr

+ Cue

4qay IY' + x (2a-x)} )tan :{,([2a-x] = + y=)(x: + y ' ) } 13

Cue { ( [ 2 a - x ] ~ + y ' ) ( x '

4qa(a-x)y'

{([2a-x] = +y')(x'

+y=)}

4qay (y~ + x ( 2 a - x ) l l

+

-1)+tan~3(

+y')}

+ 1

(tan s J3-1) = 1 + t a n st3

(1-K°)[~/y-~wf3-z)] Cu

(

+y~)(x ~ +y~)}+ 1

)

= 0 (2)

Putting:

4qa (a - x) y2 -I=A Cur { ( [ 2 a - x ] 2 + y 2 ) ( x 2 +y2)} (1 + K o ) / 7 Y - ~ ' w ( Y - z ) } + Cu and:

x) y2 Cult {([4qa- ( x ~ - + y ' ) ( x 2 Ay

then: ~

= tan/3 =

4qay ty 2 + x ( 2 a - x ) } Cur

{([2~-x] ~ +

y~)(x ~

+y2)}

=B

+y')}+ I=C

- B +-(B' - - 4 A C ) q 2

(3)

2,4

Eq.3 is soluble for any point given by the coordinates (x, y). To determine the locus o f a potential failure surface it is necessary to determine the slope progressively for each small element. The boundary shear locus, which bounds the overall failure, starts from the point x = y = 0. If Ay is taken as a unit element then ~¢ = Ay/tan(3 so t h a t y = ~Ay a n d x = ~£xx. To establish the initial slope it is necessary to rearrange certain terms in eq.2 to avoid the simultaneous vanishing o f numerator and denominator, so that: ( tan2 /30

4a2q tan2 /3° _ 1 ) + tan/3o ( 8qa2 tan/3o ) 4a 2 Cue (1 + tan 2 /30 ) '4a 2 Cu~r (1 + tan 2 t3o ) 4a2q tan 2 j3 o -(4a2Cu~(l+tan~/3o)

": as x, y ~ O, y/x ~ Lxy/~x = tan/30

+1/

=0

i.e. tan 2 0o [q tan 2 /30 - Curt (I + tan 2 /3o )] + tan 0o (2q tan/30 ) - [q tan 2 t~o + Cu~r (1 + tan s/30 )] = 0 •". tan2 /30 [ta ns /3o ( q - Cue) - Cur] + 2q tan s /30 - [ tans /30 (q + CuTr) + Cue] = 0 •"- tan 4 0o (q - Cue) + tan s /30 (q - 2Cu~r) - Cult = 0 •". tans/30 = [ - (q - 2 Cu~r) + (q2 --4qCu~r + 4Cu2e 2 + 4qCue - 4Cu2~r2)q2]/2(q- Curt) i.e. tan s ~o = (2Cult - q + q)/2 (q - Cue) = Culr/(q - Cue) or - 1 The latter value is extraneous as it has no solution in tan/30. ~y / _ '" ~Z-~ ~'~(o|,o7) tan0° = [Culr/(q- Culr)] ~/2