An energy method for a linear-elastic perfectly plastic method utilising the von Mises yield criterion with associated flow developed in 2013 by McMahon and co-workers is used to compare the ellipsoidal cavity-expansion mechanism, from the same work, and the displacement fields of other research by Levin, in 1995, an

B T Mcma Hon Ã

M D B Olton

S K Haigh

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McMahon, B. T. et al. (2013). Géotechnique 63, No. 16, 1447–1450 [http://dx.doi.org/10.1680/geot.13.T.002]
1447
TECHNICAL NOTE
Optimal displacement mechanisms beneath shallow foundations on
linear-elastic perfectly plastic soil
B. T. MCMAHON, S . K. HAIGH and M. D. BOLTON
An energy method for a linear-elastic perfectly plastic method utilising the von Mises yield criterion
with associated flow developed in 2013 by McMahon and co-workers is used to compare the
ellipsoidal cavity-expansion mechanism, from the same work, and the displacement fields of other
research by Levin, in 1995, and Osman and Bolton, in 2005, which utilise the Hill and Prandtl
mechanisms respectively. The energy method was also used with a mechanism produced by perform-
ing a linear-elastic finite-element analysis in Abaqus. At small values of settlement and soil rigidity
the elastic mechanism provides the lowest upper-bound solution, and matches well with finite-element
analysis results published in the literature. At typical footing working loads and settlements the cavity-
expansion mechanism produces a more optimal solution than the displacement fields within the Hill
and Prandtl mechanisms, and also matches well with the published finite-element analysis results in
this range. Beyond these loads, at greater footing settlements, or soil rigidity, the Prandtl mechanism
is shown to be the most appropriate.
KEYWORDS: bearing capacity; clays; foundations; plasticity; settlement
INTRODUCTION
Centrifuge tests observing the behaviour of circular, shallow
foundations on kaolin clay were performed in Cambridge
(McMahon, 2012). The observed undrained displacement
field was established using particle image velocimetry (PIV)
(White et al., 2003), and was found at typical working loads
to resemble an ellipsoidal cavity-expansion mechanism more
closely than the classical Prandtl (1921) or Hill (1949)
mechanisms.
A theoretical, ellipsoidal cavity-expansion mechanism for
perfectly rough, shallow, circular foundations was developed
in McMahon et al. (2013). Figure 1(a) shows a comparison
between the undrained deformation field observed in a
centrifuge experiment and the ellipsoidal cavity-expansion
mechanism. It can be seen that these mechanisms match
well. McMahon et al. (2013) also present an energy method
for a von Mises material with associated flow in three
dimensions. Yield was determined using the von Mises yield
criterion
( 1   3)2 þ ( 2   3)2 þ ( 1   2)2 ¼ 2 q2u (1)
where qu is the undrained strength in triaxial compression.
This, along with an associated flow rule, implies that the
incremental strain vector remains parallel to the deviatoric
stress throughout the deformation process, providing a com-
putationally simple evaluation of work. In order to facilitate
comparison with previously published finite-element ana-
lyses, it was assumed that
qu ¼ 2cu (2)
This analysis procedure, together with the cavity-
expansion mechanism, permitted the production of a theor-
etical upper-bound load–settlement curve for rough, circular,
shallow foundations. The load–settlement curve for this
mechanism does not reach a limiting value, and as such can
be considered to lose its validity beyond normalised stress
values (f /cu) similar to that of the classical bearing capacity
factor of rough circular footings, Nc ¼ 6.05 (Eason & Shield,
1960).
The energy method developed by McMahon et al. (2013)
can be validated against the results of published finite-
element and finite-difference calculations on vertically
loaded, circular, shallow foundations. These results will be
compared with the analytical results later in the paper. A
summary of the parameters used in the analyses is shown in
Table 1.
McMahon et al. (2013) demonstrated that the ellipsoidal
cavity-expansion mechanism gives a good fit to the finite-
element data for values of f /cu between 4 and 6, but
deviates significantly from the results outside this range.
There are obviously other mechanisms that could be evalu-
ated within this framework to determine an optimal upper
bound. These will be evaluated in this paper.
Yadong et al. (2008) show displacement fields for a strip
footing on sand at five different footing pressures. It can be
seen from the data presented that the mechanism appears to
transform from a cavity-expansion type mechanism at small
loads to a mechanism at failure represented reasonably well
by the Prandtl mechanism. While this research involved
sands rather than the undrained clays discussed here, a
similar trend of changing mechanisms at different load levels
might well be observed.
ALTERNATE MECHANISMS
Levin (1955) developed a displacement field within the
Hill mechanism in an upper-bound investigation of the ulti-
mate indentation pressure of smooth, circular punches. This
Manuscript received 24 January 2013; revised manuscript accepted 19
July 2013. Published online ahead of print 10 September 2013.
Discussion on this paper closes on 1 May 2014, for further details
see p. ii.
 Department of Engineering, University of Cambridge, Cambridge,
UK.
displacement field is shown in Figure 1(b). As part of the
development of mobilisable strength design (MSD), Osman
& Bolton (2005) developed a kinematically admissible
mechanism for a smooth, circular footing, which utilised the
boundary of the classic Prandtl mechanism. By using the
work equation of Shield & Drucker (1953) for an ideally
plastic Tresca material, it was determined that Nc ¼ 5.86 for
this mechanism. The primary difference between the Prandtl
and the Hill mechanisms is the result of different active
zones, which result in no movement below the centre of the
Hill mechanism.
Klar & Osman (2008) further investigated the concept of
MSD by using a process of energy minimisation for linear
combinations of the elastic deformation field and those of
either Hill or Prandtl in a process called extended MSD
(EMSD). The Prandtl EMSD analysis was shown to be more
optimal than that using the Hill mechanism.
Klar & Osman (2008) provide an expression for soil
movements beneath a circular, shallow foundation in an
elastic analysis based on integration of the classical solutions
for elastic deformation beneath a point load. As this
mechanism extends to infinity and involves an integral,
making calculation computationally intensive, an approxima-
tion was generated by running a linear-elastic finite-element
analysis of a vertically loaded, rough, rigid punch in Abaqus.
The analysis used rigid boundaries at five foundation dia-
meters in both the horizontal and vertical directions. The
resulting deformation mechanism was exported for use in
the energy approach.
The displacement fields within the Hill and Prandtl
mechanisms, the cavity-expansion mechanism and the elastic
mechanism are now used in the energy approach of
McMahon et al. (2013) for soils of varying rigidity to
evaluate an appropriate upper-bound at different levels of
foundation loading. These are then evaluated against the
finite-element analysis and finite-difference results of Taiebat
& Carter (2000, 2010), Gourvenec & Randolph (2002) and
Klar & Osman (2008).
RESULTS
A mesh size of 0.2%D was adopted for the analysis of the
cavity-expansion, elastic, Hill and Prandtl mechanisms. As
discussed in McMahon et al. (2013), the use of a finer mesh
produced negligible changes and, therefore, did not justify
the significant increase in computation time. A hemispheri-
cal radius of 2D was used for the cavity-expansion mechan-
ism.
Analyses were performed on soils with G/cu values be-
tween 10 and 10 000. It was found that the load–settlement
behaviour fell on a single line when the normalised stress
f /cu was plotted against the soil rigidity G/cu multiplied by
the normalised footing settlement /D for each mechanism.
Figure 2 shows the results for the elastic, Hill, Prandtl and
cavity-expansion mechanisms. The classical solution of
Eason & Shield (1960) and the results from the finite-
element analyses for rough footings of Taiebat & Carter
(2000, 2010) and Gourvenec & Randolph (2002), and for a
smooth footing from Klar & Osman (2008) are also shown.
The elastic displacement field for smooth footings was also
evaluated, but the results differed negligibly from those for
the rough footing and so these results were omitted for
clarity. It should be noted that the elastic mechanism line in
Cavity expansion model
δ
Experimental
δ
0
D/2
D/2
D/2
D/2
D/2
D/2
3 /2D
3 /2D
3 /2D
3 /2D
0
0
D
D
D
D
Levin (1955) – Hill mechanism
δ
Osman & Bolton (2005) – Prandtl
δ
(a)
(b)
Fig. 1. Comparison of mechanisms: (a) centrifuge experiment and cavity-expansion model
(McMahon et al., 2013) mechanisms; (b) displacement fields of Levin (1955) within the Hill
mechanism and Osman & Bolton (2005) within the Prandtl mechanism
Table 1. Summary of finite-element and finite-difference analysis
results provided in the literature
Source G/cu Footing
roughness
Derived Nc
Taiebat & Carter (2000) 100 Rough 5.7
Taiebat & Carter (2010) 100 Rough 6.17
Gourvenec & Randolph (2002) 167 Rough 5.91
Klar & Osman (2008) – Smooth 5.63
1448 OPTIMAL DISPLACEMENT MECHANISMS BENEATH SHALLOW FOUNDATIONS
Figure 2 is similar to the elastic settlement expression for a
smooth footing of Davis & Selvadurai (1996)
 f
cu
¼ 16


D
G
cu
 
(3)
The results from the finite-element analyses show consis-
tent results with only slightly different ultimate bearing
capacity factor values. The elastic mechanism agrees very
well with the finite-element analysis results up to
(=D 3 G=cu)  0:7,  f=cu  4: Beyond this point, up to a
value of (=D 3 G=cu)  3,  f=cu  6, the cavity-expansion
mechanism gives the lowest upper-bound solution and pro-
vides a good correlation with the published finite-element
analysis results, in what could be considered the typical
footing working range. Beyond this value the Prandtl me-
chanism is more appropriate, although it must be noted that
this mechanism assumes a smooth foundation in contrast to
the rough foundation finite-element results.
For all values of soil rigidity and settlement, the Prandtl
mechanism provides a lower upper-bound solution than the
Hill mechanism. The Prandtl mechanism converges to an
ultimate bearing capacity of Nc ¼ 6.11, 4% greater than the
value Nc ¼ 5.86 that was calculated in Osman & Bolton
(2005). The Hill mechanism converges to an ultimate bear-
ing capacity of Nc ¼ 6.16, approximately 5% greater than
the value of Nc ¼ 5.83 presented by Klar & Osman (2008).
These small differences are due to the von Mises yield
criterion being used in the work presented here, rather than
the Tresca criterion assumed by both Osman & Bolton
(2005) and Klar & Osman (2008). In contrast to Tresca, the
von Mises yield criterion allows work to be done by the
intermediate strain component, resulting in a slight increase
in work done in deforming the soil. The elastic and cavity
expansion mechanisms do not give ultimate bearing capaci-
ties, showing continually increasing load with increasing
deformation due to their infinite extent. In the upper-bound
analyses presented here, the lowest upper bound at any
settlement would be deemed to be the most appropriate
mechanism for that settlement. At failure, the Prandtl
mechanism becomes dominant, but for working loads, defor-
mations might be more accurately reflected by the cavity-
expansion mechanism.
A design line based on the lowest curve was produced
and is shown in Figure 3, together with the results from the
published finite-element analysis and finite-difference data.
Also shown are the results of MSD and EMSD for the
Prandtl mechanism from Klar & Osman (2008) and the
elastic solution of Davis & Selvadurai (1996) (equation (3)).
The discrepancy between the elastic solution and the new
design curve for bearing stress f , 4cu is small and prob-
ably the result of footing roughness. Figure 3 shows that the
cavity-expansion mechanism matches well with the finite-
element analysis data in the range 4cu , f , 6cu: The
design curve, which embodies elastic analysis at small
strains, elastic perfectly plastic cavity-expansion at inter-
mediate strains, and ultimate bearing failure can, therefore,
be used to describe the whole load–settlement behaviour of
circular, shallow foundations on a homogeneous, linear-
elastic perfectly plastic soil.
CONCLUSIONS
The energy method for a linear-elastic perfectly plastic
soil employing the von Mises yield criterion with associated
flow presented by McMahon et al. (2013) was used with an
elastic mechanism and displacement fields within the classi-
cal Hill and Prandtl mechanisms to evaluate the optimal
mechanism at different foundation loads. A parametric in-
vestigation was performed which demonstrated that, when
the footing load was plotted against the rigidity index multi-
plied by normalised settlement, a single line was produced.
The elastic mechanism provided excellent agreement with
the published finite-element analysis in the small settlement
region, while the cavity-expansion mechanism provided good
agreement in the typical working region. The cavity-
expansion mechanism, however, fails to reach an ultimate
value, and beyond (=D 3 G=cu)  3,  f=cu  6 the Prandtl
mechanism represents a better solution.
0
1
2
3
4
5
6
7
8
0·01 0·1 1 10
σ f
u
/c
Rough footing mechanisms:
Elastic
McMahon . (2013), cavity expansionet al
Eason & Shield (1960)
Smooth footing mechanisms:
Levin (1955) – Hill
Osman & Bolton (2005) – Prandtl
Rough footing finite-element analysis results
Taiebat & Carter (2000)
Taiebat & Carter (2010)
Gourvenec & Randolph (2002)
Smooth footing finite-element analysis results
Klar & Osman (2008)
δ
D
G
cu
⎛
⎝
⎛
⎝
Fig. 2. Normalised footing stress (f /cu) plotted against (/D 3 G/cu) for four mechanisms, the
classical solution and published finite-element analysis results
MCMAHON, HAIGH AND BOLTON 1449
ACKNOWLEDGEMENTS
The first author would like to acknowledge the financial
support he received from the Cambridge Australia Trust
(Poynton Scholarship) and the Principals of UK Universities
(Overseas Research Students Awards Scheme) throughout
his studies in Cambridge.
NOTATION
cu undrained shear strength
D footing diameter
G shear modulus
Mc compatibility factor
Nc bearing capacity factor
qu undrained strength in triaxial compression
 footing settlement
f footing pressure
1, 2, 3 major, intermediate and minor principal stresses
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0
1
2
3
4
5
6
7
8
0·01 0·1 1 10
σ f
u
/c
Rough footing:
Elastic (Davis & Selvadurai, 1996)
Smooth footing:
Rough footing finite-element analysis results
Taiebat & Carter (2000)
Taiebat & Carter (2010)
Gourvenec & Randolph (2002)
Smooth footing finite-element analysis results
Klar & Osman (2008)
δ
D
G
cu
⎛
⎝
⎛
⎝
MSD (Klar & Osman, 2008)
EMSD (Klar & Osman, 2008)
Design line
Fig. 3. Design line for rough, circular, shallow foundations in comparison to elastic, MSD and
EMSD methods with published finite-element analysis results
1450 OPTIMAL DISPLACEMENT MECHANISMS BENEATH SHALLOW FOUNDATIONS


Optimal displacement mechanisms beneath shallow foundations on linear-elastic perfectly plastic soil

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Optimal displacement mechanisms beneath shallow foundations on linear-elastic perfectly plastic soil

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