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´ Osman, A. S. & Bolton, M. D. (2005). Geotechnique 55, No. 6, 435–447
Simple plasticity-based prediction of the undrained settlement of shallow circular foundations on clay
A . S . O S M A N * a n d M . D. B O LTO N * A kinematic plastic solution has been developed for the penetration of a circular footing into an incompressible soil bed. In this solution, the pattern of deformation around the footing is idealised by a simple plastic deformation mechanism. Strain-hardening behaviour and nonlin

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Osman, A. S. & Bolton, M. D. (2005).
Ge´ otechnique
55
, No. 6, 435–447435
Simple plasticity-based prediction of the undrained settlement of shallowcircular foundations on clay
A. S. OSMAN* and M. D. BOLTON*
A kinematic plastic solution has been developed for thepenetration of a circular footing into an incompressiblesoil bed. In this solution, the pattern of deformationaround the footing is idealised by a simple plastic defor-mation mechanism. Strain-hardening behaviour and non-linear stress–strain characteristics are incorporated. Thisapplication is different from conventional applications of plasticity theory as it can approximately predict bothstresses and displacements under working conditions.This approach therefore provides a uniﬁed solution fordesign problems in which both serviceability and safetyrequirements are based directly on the stress–strain be-haviour of the soil. The design strength that should limitthe deformations can be selected from the actual stress– strain data recorded from a carefully speciﬁed location,and not derived using empirical safety factors. Thevalidity of this design approach is examined against non-linear ﬁnite element analyses and ﬁeld measurements of foundations on clay under short-term loading.
KEYWORDS: clays; deformation; design; footings/foundations;plasticity; theoretical analysis
Une solution plastique cine´matique a e´te´de´veloppe´e pourla pe´ne´tration d’une assise circulaire dans un lit de solincompressible. Dans cette solution, la forme de la de´for-mation autour de l’assise est ide´alise´e par un simpleme´canisme de de´formation plastique. Nous incorporons lecomportement de de´formation-durcissement et les carac-te´ristiques de contrainte-de´formation non line´aire. Cetteapplication est diffe´rente des applications convention-nelles de la the´orie de plasticite´car elle peut approxima-tivement pre´dire les contraintes et les de´placements enconditions de travail. Cette approche offre donc unesolution uniﬁe´e aux proble`mes de conception dans les-quels les besoins de commodite´et de se´curite´sont base´sdirectement sur le comportement contrainte-de´formationdu sol. La force nominale qui devrait limiter les de´forma-tions peut eˆtre se´lectionne´e d’apre`s la contrainte re´elle – donne´es enregistre´es depuis un emplacement soigneuse-ment spe´ciﬁe´et non de´rive´es en utilisant des facteurs dese´curite´empiriques. Nous examinons la validite´de cetteapproche de design par rapport aux analyses d’e´le´mentsﬁnis non line´aires et aux mesures sur le terrain desfondations sur de l’argile sous charges a`court terme.
INTRODUCTIONDesigners have to check that shallow foundations willneither penetrate the soil subgrade in a bearing capacityfailure, nor settle excessively. Bearing failure is checked using plasticity theory, whereas settlement is usually checked using elasticity. Conventionally, the calculations for settle-ment in saturated clay are divided into two components:immediate settlements due to deformation taking place atconstant volume, and the consolidation settlement accompa-nying the dissipation of pore water pressure (Skempton &Bjerrum, 1957). Excessive total or differential settlementsare a main cause of unsatisfactory building performance.Although this is sometimes due to unexpected consolidation,the inadequacy of linear elasticity to describe the earlier phase of undrained settlement leads to signiﬁcant uncertain-ties. This paper proposes a resolution of the latter problem.Circular shallow foundations are usually designed on the basis that the net imposed load under working conditionsshould not exceed the ultimate net imposed load that would cause collapse, divided by a safety factor. Estimation of thecollapse load in the geotechnical design of circular footingsis well established from plasticity theory. Several techniques based on stress characteristics, and on upper- and lower- bound theorems of limit analysis, have been used to calcu-late bearing capacity factors for circular footings.Levin (1955) presented an upper-bound solution for the problem of the indentation of a smooth circular punch on ahalf-space of a perfectly plastic material that obeys Tresca’syield criterion. In this solution, the geometry of Hill’s planestrain mechanism (Hill, 1950) was used to simulate contin-uous axisymmetric displacements. Shield (1955a) presented a complete solution for a smooth circular footing on a purely cohesive soil, and Eason & Shield (1960) extended the same solution to the case of a perfectly rough footing.Cox
et al.
(1961) solved a number of cases of the circular surface footing on
c
–
ö
weightless soil. Cox (1962) extended these solutions by including the soil weight. Houlsby &Wroth (1983), Kusakabe
et al.
(1986) and Tani & Craig(1995) considered the bearing capacity problem of a circular footing on cohesive soil whose undrained strength varieswith depth. Shield (1955b) presented upper- and lower- bound solutions for the average bearing pressure at failureof a circular footing on a semi-inﬁnite layer of elastic- perfectly plastic cohesive soil resting on a rough rigid base.Chen (1975) solved the problem of indentation of a circular cylinder of ﬁnite dimensions by a ﬂat-ended rigid circular footing.Butterﬁeld & Harkness (1971) considered the stepped mobilisation of shear strength in rigid plastic Mohr–Cou-lomb material under strip loading. Shear strain is assumed to be concentrated in the slip lines, and the soil is modelled asrigid until the instant it yields completely. Although thistechnique was able to calculate the proportional displace-ment within the plastic mechanism, it was unable to relateground displacement to shear strain in the soil. In currentdesign practice a reduction factor on peak strength of thesoil is introduced to account for the need to predict displace-ment. However, it is not possible to relate the shear strain(based on the reduced strength mobilisation) to ground displacement other than empirically.Linear elasticity is often used to calculate displacements.
Manuscript received 17 August 2004; revised manuscript accepted 27 April 2005.Discussion on this paper closes on 1 February 2006, for further details see p. ii.* Schoﬁeld Centre, Department of Engineering, University of Cambridge, UK.
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However, the stress–strain behaviour of soil is highly non-linear from very small strains (Jardine
et al.,
1984; Burland,1989; Houlsby & Wroth, 1991). The ﬁnite element analysesconducted by Bolton & Sun (1991) for centrifuge tests on a bridge abutment showed the importance of using a non-linear elasto-plastic model in order to predict properly thedisplacements and stresses on the abutment. Jardine
et al.
(1986) showed that the non-linearity of stress–strain behav-iour has a dominant inﬂuence on the form and scale of thedisplacement distribution around shallow foundations.Bolton & Powrie (1988) and Osman & Bolton (2004) proposed a new design approach for retaining walls based on the theory of plasticity and the mobilisable soil strengthconcept. The proposed design method treats a stress path ina representative soil zone as a curve of plastic soil strengthmobilised as strains develop. Strains are entered into asimple plastic deformation mechanism to predict boundarydisplacements. Hence the proposed mobilisable strength de-sign (MSD) method can satisfy both safety and serviceabil-ity in a single step of calculation. This paper presents anMSD solution for the bearing capacity problem of a shallowcircular footing in undrained conditions.PLASTIC DEFORMATION MECHANISM FOR CIRCULAR FOUNDATIONSThis solution is based on plasticity theory. However,instead of a rigid perfectly plastic material, which is gen-erally used in simple plastic analyses, strain-hardening be-haviour is incorporated. In this solution, the ground displacement is related to shear strain in the soil. The keyadvantage of this technique is that the settlements of circular footings can be predicted directly from the stress–strain dataof a triaxial test on a characteristic soil sample. Thissolution therefore provides a rational procedure for selectingdesign parameters, because the strength that limits theground deformation is chosen with respect to the stress– strain behaviour of the soil and the level of acceptabledeformations under working conditions.The well-known Prandtl mechanism (Fig. 1) for planestrain indentation is used ﬁrst, but only to create a boundaryfor the continuous displacement ﬁeld that is taken to exist beneath a circular punch. Within this boundary there arethree zones of distributed shear. These regions are required to shear and deform compatibly and continuously with norelative sliding at their boundaries. Strains and compatibledisplacements are developed according to the stress incre-ment and equilibrium condition. Outside this region, the soilis taken to be rigid. This represents the rapid increase of soilstiffness away from the near-ﬁeld plastic deformation.It is possible to satisfy compatibility by considering thekinematics of each of the three soil zones.Figure 2 shows the active zone OAF. If there is no volumechange, the following condition should be satisﬁed
@
u
@
r
þ
ur
þ
@
v
@
z
¼
0 (1)where
u
and
v
are the radial and vertical displacementrespectively,
r
is the radial distance from the centreline of the footing, and
z
is the depth below the ground surface.Axisymmetry conditions imply that
u
¼
0 at
r
¼
0. If thevariation of
v
is independent of
r
, then the radial displace-ment
u
can be given by
u
¼À
r
2
@
v
@
z
(2)The vertical displacement
v
might be assumed to be given by
v
¼
a
1
z
2
þ
a
2
z
þ
a
3
(3)where
a
1
,
a
2
and
a
3
are constants.Here,
v
has a maximum value of
ä
at
z
¼
0 and decreasesto zero at
z
¼
D
/2. If displacements in the adjoining fanzone are assumed to have no radial component with respectto the fan centre, and if there is no slippage between thezones, then tangential displacement along the boundary OFin the active zone must be zero.Therefore the radial and vertical components of displace-ments in the active zone can be given by
u
¼À
4
ä
D
2
zr
þ
2
ä
Dr
(4)
v
¼
4
ä
D
2
z
2
À
4
ä
Dz
þ
ä
(5)In each of the remaining zones, the soil is assumed to move
OF A45°
D
/2
r
,
uz
,
v
ä
Fig. 2. Active zone OAF
z
,
v r
,
u
PassiveFan ActiveFanPassive
D
ä
R
Fig. 1. Plastic deformation mechanism for shallow circular foundation on clay
436
OSMAN AND BOLTON
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along lines parallel to the outer rigid boundary (Fig. 3) withdisplacements decaying with 1/
r
in order to satisfy theincompressibility condition. Table 1 shows the componentsof the displacement in each zone.
Determination of strains
Strains can be found from the ﬁrst derivative of displace-ments. Applying axisymmetric conditions
å
r
¼À
@
u
@
r
ª
r
Ł
¼
0
å
Ł
¼À
ur
ª
Ł
z
¼
0
å
z
¼À
@
v
@
z
ª
zr
¼À
@
v
@
r
À
@
u
@
z
(6)The principal strains are given by
å
1
¼
12
À
å
Ł
þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
å
2
Ł
þ
ª
2
rz
À
4
å
r
å
z
q
å
2
¼
å
Ł
å
3
¼
12
À
å
Ł
À
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
å
2
Ł
þ
ª
2
rz
À
4
å
r
å
z
q
(7)where
å
1
and
å
3
are the major and the minor principal strainrespectively, and
å
2
is the intermediate principal strain.
Average shear strain
As all displacements are proportional to the verticaldisplacement
ä
of the foundation, and all spatial dimensionsare proportional to the diameter
D
of the foundation, itfollows by dimensional reasoning that all strain componentsare proportional to
ä
/
D
.The engineering shear strain
å
s
can be deﬁned as thedifference between the major and minor principal strains(equation (7))
å
s
¼j
å
1
À
å
3
j
(8)The average shear strain mobilised in the deforming soil can be calculated from the spatial average of the shear strain inthe whole volume of the deformation zone (Fig. 1). This procedure gives
å
s,mob
¼
Ð
vol
å
s
d
vol
Ð
vol
d
vol
¼
M
c
ä
D
(9)in which
M
c
can be shown to take the value of 1.35 (Osman,2005).
Hypothesis
The displacement pattern beneath circular footings isidealised by the plastic deformation mechanism shown inFig. 1. This displacement ﬁeld links the average shear strainmobilised in the soil,
å
s,mob
, to the normalised footingsettlement
ä
/
D
(equation (9)). The shear stresses in the soilare related to the external loading of the footing by the usual bearing capacity coefﬁcient (
N
c
)
ó
mob
¼
N
c
c
mob
(10)where
ó
mob
is the average applied bearing pressure, and
c
mob
is the shear stress mobilised in the soil.A relation between applied bearing pressure and thedisplacement of the footing can be established if the relation between shear stresses and shear strains can be obtained,such as from a carefully chosen undrained triaxial test.The compromise of the new approach is therefore tocouple together an equilibrium solution based on the mobili-sation of a
constant
shear stress,
c
mob
, with a kinematicsolution based on the creation of an
average
mobilised shear strain,
å
s,mob
. Fig. 4 illustrates the method of estimating theload–settlement curve directly from a stress–strain curve.This makes it clear that the non-linearity of the representa-tive stress–strain curve is to be taken as identical to that of the
normalised
load–displacement curve of the foundation.Plasticity theory is used to obtain the linear transformationof the axes through the normalisation factors
M
c
and
N
c
.Of course, in a real footing problem, soil elements would differ in their past stress history, and in their response to differ-ent stress paths induced by the new loading. Different elementswould have different non-linear stress–strain responses and
z
,
v r
,
u
FPGO
Fig. 3. Displacement ﬁeld in the fan and passive zonesTable 1. Displacement ﬁeld for shallow foundation on clay
Zone Radial displacement,
u
Vertical displacement,
v
Active zone
À
4
ä
D
2
zr
þ
2
ä
Dr
4
ä
D
2
z
2
À
4
ä
Dz
þ
ä
Fan zone
V
1
r
1
r z
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
D
2
À
r
2
þ
z
2
s
V
1
r
1
r D
2
À
r
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
D
2
À
r
2
þ
z
2
s
Passive zone1
ﬃﬃﬃ
2
p
V
2
r
2
r
À
1
ﬃﬃﬃ
2
p
V
2
r
2
r V
1
¼
4
ﬃﬃﬃ
2
p
r
1
D
2
ä
,
r
1
¼
D
2
À
1
ﬃﬃﬃ
2
p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
D
2
À
r
2
þ
z
2
s
,
V
2
¼
4
ﬃﬃﬃ
2
p
r
2
D
2
ä
,
r
2
¼
3
D
4
À
r
2
À
z
2
UNDRAINED SETTLEMENT OF SHALLOW CIRCULAR FOUNDATIONS ON CLAY
437
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would mobilise different shear stresses. Nevertheless, for the purpose of obtaining a simple calculation, it was considered best to employ a unique representative stress–strain curve. Aweighted average approach is adopted to select a representativeshear strain that mobilises the required shear strength. Theusefulness of this hypothesis will be demonstrated in thefollowingsections.
Consistency of the proposed plastic deformation mechanism
To demonstrate the plausibility of the chosen mechanism,limit analysis calculations will now be carried out using the proposed displacement ﬁeld to derive an upper bound to thecollapse load, taking the soil to be ideally plastic withconstant strength
c
u
. The results are to be compared withexisting plasticity solutions.For a Tresca material, the upper-bound calculation iscalculated from the following equation (Shield & Drucker,1953)
ó ð
4
D
2
_
ää
¼
_
W W
¼
_
D D
¼
2
ð
vol
c
u
j
_
åå
1
j
d
vol
þ
ð
s
c
u
j
˜
v
j
d
s
(11)where
_
W W
is the work done by the vertical load
ó
on thefooting,
_
ää
is the downward incremental displacement of thefooting,
_
D D
is the total energy dissipation,
c
u
is the undrained shear strength,
s
is the surface domain,
˜
v
is the jump indisplacement increment across the discontinuity, and
_
åå
1
isthe largest principal plastic strain increment.As the distributed shear zones in Fig. 1 have been proved to shear and deform compatibly and continuously with norelative sliding at their boundaries, there is no displacementdiscontinuity. Accordingly,
ð
s
c
u
j
˜
v
j
d
s
¼
0The largest principal plastic strain increment
_
åå
1
can becalculated from the displacement increments at collapsefollowing equations (7) and (8) rewritten in terms of incre-mental plastic strains and incremental displacements atcollapse. As is the case with the classical solutions, withwhich a comparison will shortly be made, changes of overallgeometry due to ﬁnite deformation will be ignored.The bearing capacity factor
N
c
is calculated by equatingthe energy dissipation and the work done. The
N
c
valuecalculated using this technique for a smooth circular footingis 5.86. The computation is detailed in Osman (2005). Thisvalue is only 3% higher than that calculated by Shield (1955a) and Houlsby & Wroth (1983), which was 5.69.Although this close correspondence cannot be taken as proof that the selected displacement ﬁeld is adequate, its consis-tency is encouraging. It must, however, be recognised thatalternative plastic mechanisms have been proposed by othersseeking solutions to the bearing capacity of circular founda-tions. For example, Kusakabe
et al.
(1986) followed Levin(1955) in adapting Hill’s (1950) plane strain mechanism for solution in axial symmetry. Although Levin’s solution offers
N
c
¼
5.84 for the bearing capacity of a smooth circular punch, which is slightly better than the authors’ solution of 5.86, Levin’s displacement ﬁeld (Fig. 5) gives zero displace-ment at every point beneath the centreline of the footing,which seems physically unreasonable. The selection of amechanism that is adequate to represent both equilibriumand kinematics can be taken only following an independentveriﬁcation using ﬁnite element analysis.COMPARISON WITH FINITE ELEMENT (FE) ANALYSISA series of axisymmetric ﬁnite element analyses has been performed to predict the displacement of a circular footingin the short term in which undrained soil conditions areassumed. Accordingly, excess pore pressure is not allowed to
000
c
mob
Stressstrain curve
M
c
ä
D
0
ó
N
c
ó
0
ä
D
Loadsettlement curve
å
s
,mob
Fig. 4. Calculation procedure in the MSD method
z
,
v r
,
u
PassiveFan ActiveFanPassive Active
D
Fig. 5. Levin’s mechanism for shallow smooth circular founda-tion on clay
438
OSMAN AND BOLTON

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