Journal of Applied Mechanics Vol.11, pp.399-410
Laboratory
Model Test and Numerical
(August 2008)
Analysis
JSCE
of Bearing
Capacity
of Rigid Strip Footing
on Slope
LiangLU*,Katsuhilco
ARAI**,ZongjianWANG*and RyujiNishiyama***
*Nonmember
, Doctoral
Program,
University
of Fukui(Bunkyo
3-9-1,Fukui910-8507)
**Fellow
, Dr.Eng.,Prof,Dept.ofArchitecture
andCivilEng.,University
of Fukui
***Nonmember
, M. Eng.,Kozo-sekkeiCo., LTD
This paper focuses on the estimation of bearing capacity of rigid strip footing on slope by performing
a number of
laboratory model tests and the numerical analysis. The laboratory model tests, including unreinforced and reinforced subsoil,
are carried out using three types of subsoil. A numerical procedure is proposed which is based on a smeared shear band
approach and a modified initial stress method, employing
Mohr-Coulomb
yield criterion with a simple plastic flow rule.
The proposed procedure is capable of estimating not only the bearing capacity for natural subsoil, but also under complex
conditions, for example, reinforced subsoil considering stiffness and deformation of materials. In most cases, a fairly good
agreement is obtained between the experimental and analytical results.
Key Words: bearing capacity, slip surface, shear band, laboratory model test, slope
1. Introduction
of many researches, the accurate description of localization
phenomenon
in soils is still open to question. For instance, the
When constructingstructureson slope,engineersmust solve
complicated bearing capacity problem. Some analytical
methodsare availablefor estimatingthe bearingcapacityon
slope. The most fundamentalapproach is to apply the limit
equilibrium method or the circular slip surface method
consideringthe bearing capacityproblemas a slope stability
bifurcation
problem.The limitequilibriummethodrepresentskinematical
conditionsonlyby usingthe mechanicallyacceptableshapeof
a slip surface,and evaluatesthe materialpropertiesonlyby its
final strength.It does not allow consideringthe stiffnessand
deformationof materials,which seemto play an importantrole
for evaluating earth reinforcementmethod, and which may
affectthe globalcollapsemode.
Due to these defects, some methods are proposed, for
instance, finite element method, and limit analysis method
which is foundedon the upperand lower boundtheorems in
to the peripheral
plasticity.Many researchesindicatethat classicalfiniteelement
method does not necessarilyprovide a reasonablecollapse
mechanism1),2).
Subjectedto Mohr-Coulombmaterial,the limit
analysishas not completelyovercomethe difficultythat the
limittheoremscannotbe provenwithoutthe normalityrule in
Narita8) applied the limit equilibrium analysis using log-spiral
plasticity,and that the normalityrule may not hold for the
material,although it is known that the analysis provides a
suitablesolutionin most casesdespiteof the difficulty.In spite
performed the model tests of strip footing on slope under
•\ 399•\
analysis that tries to simulate
actual localized
deformation seems to give a promising view, while the analysis
may not give reasonable solutions for complicated boundary
value problems
such as bearing
capacity3),4). This may be
because in bearing capacity problems it is not easy to duplicate
the rotation of principal stresses from the below part of footing
region. Adaptive
finite element
method
appears to require a lot of numerical efforts and to contain a
certain numerical difficulty in some cases5),6).
In recent decades, a number of analytical methods
and
experiments are performed to the bearing capacity problems on
slope. Kusakabe et al.7)carried out the model tests considering
various
factors
affecting
bearing
capacity
on
slope
and
presented a practical solution based on the upper bound
approach. They compared the solution with the results by
circular
slip surface method,
slip surface
and concluded
lower bound
that
the
approach,
analysis
etc.
somewhat
overestimates the bearing capacity. Shields et al.9) carried out
large-scale experiments on loaded slopes of sand and compared
the
results
with
some
computational
results.
Sugano"
various loading conditions and compared the result with the
bearing capacity on level ground. Kimura et al.11)performed the
unit:mm
unit: mm
(a) saturatedclay
Fig. 1 Soil container
(b) mountainsandA and B
Fig. 2 Distribution of earth pressure sensor
centrifuge model tests and showed a good agreement with the
results
of other's
tests on
a prototype
scale. Yoo)
and
Mostafa13) practiced the laboratory model tests and FE analysis
Case 1
for estimating the bearing capacity on slope reinforced
by
geotextile,
the
and
ascertained
the
accordance
between
Case 1
experimental and analytical results. Their main concern is the
suitable distribution
and properties of geotextile to establish
Case 2
Case 2
maximum
reinforcing
effect rather than the simulation
of
ultimate bearing capacity behavior.
In this paper, laboratory
model
tests are carried out to
evaluate the actual behavior of soil and collapse mode of soil
stratum on slope under footing pressure, and to study practical
evaluation
of the bearing
capacity. Then,
we propose
Case 3
Case 3
a
numerical procedure which is based on a smeared shear band
approach14) and a modified
numerical
procedure
initial stress method15 16). The
assumes Mohr-Coulomb
Fig. 3 Layout of geotextile for three cases
yield criterion
with a simple non-associated plastic flow rule and attempts to
Table 1 Material properties
provide an appropriate bearing capacity which is supported by
an explicit collapse mechanism represented by stress yield
condition. The calculated results by the proposed procedure are
compared with those obtained from the laboratory model tests,
to examine the numerical characteristics and applicability of the
proposed procedure.
2. Laboratory
Model
Test
2.1 Test Equipment
Fig. 1 shows the steel soil container used in the laboratory
model test, which is 700mm wide, 100mm thick and 550mm
high. One of the lateral sides has a composite glass plate with
grids of 5cm size so that the deformation of subsoil can be
observed obviously. Earth pressure sensors are installed in the
subsoil as shown in Fig. 2, to measure the vertical earth pressure.
Two dial gauges are used to measure the average settlement of
loading plate. Many markers are settled on the side wall of soil
container to observe the deformation of subsoil. Considering
the friction between soil stratum and soil container, we use a
thin rubber membrane
smeared with a thin layer of silicon
•\ 400•\
Table2 Particlesizedistribution(%)
grease
on
pressure
the
surface
is applied
using
loading
on base
of
loading
Cylinder.
of
condition
sidewalls
to a rigid
a Bellofram
undersurface
of
A
soil
plate
sand
plate,
container.
with
150mm
paper
so that
Loading
in width
is glued
it may
footing pressure (kPa)
onto
simulate
the
rough
friction.
(a) saturated clay
2.2 Soil
Materials
Model
two
tests
types
are canied
of
and properties
weight
used
and
particle
v: Poisson's
is given
tests
settlement
for
in Table
shear
the
for
types
three
of sandy
as
(b) mountian and A
B which
tests,
c
E is back-calculated
model
test.
of reinforcement
cases
c,ƒÓ
soil are
sand
in the
effect
plate,
modulus, ƒÁ:
2. In the model
test.
and
parameters
of loading
mountain
observed
discussing
performed
Two
clay
soil
E: Young's
A and
a direct
the
B: width
ratio.
sand
by
footing
also
where
for saturated
1 lists
parameters,
mountain
obtained
the
model
are
and
called
are
Table
strength
size distribution
and ƒÓ
from
sand.
of geotextile,
: Mohr-Coulomb
unit
out respectively
mountain
shown
The
material
in Fig.
3, where
(c) mountian sand B
Case
1
is
for
reinforcement
subsoil
two
the
natural
material,
with
one
layers.
by
sorts
geotextile
geotextiles
dotted
sand,
2.3 Test
Procedure
Case
of
lines
are
is
applied,
the
that
Fig.4 Footingpressure-settlement
relationship(experiment)
with
case
is also
in Table
1, two
non-woven
the
no
reinforced
each
shown
and
saturated
E is determined
is
3 is for
for
3. As
there
for
Case
geotextile
to
where
which
2
and
in Fig.
corresponding
mountain
in
of geotextile
placement
illustrated
of
while
layer
The
subsoil
woven
clay
by a tensile
and
the
test
(a) saturated clay (footing pressure 39kPa)
Foundation
material
subsoil
as shown
is thoroughly
is made
in Table
remoulded
greater
than
17.6%,
silt of 77.6%
the
respectively
1. For the
at water
liquid
limit,
and
content
which
clay
of three
saturated
of
types
clay,
of 65%
is
The
slurry
considerably
composed
15.3%.
of soil
clay
of
clay
sand
slurry,
of
earth
(b) mountian sand A (footing pressure 137kPa)
pressure
sensors,
prescribed
layer
height
of
loading
gauge
plate
which
plane
consolidation
for
After
all
setting
a little
reaches
has
one
day
the
49kPa.
In
tenns
of
distinguished
from
mountain
that
way
of step
the
mountain
sand
Fig.
3 (b). As
earth
pressure
is
put
to
above-mentioned
sensors
and
two
the
types
or consolidated
which
the
status
soil
as
for
on
and
the
then
final
is
the
of gauges.
3 (a),
soil
a
used
is put
the
each
put
(c) mountain sand B (footing pressure 147kPa)
is
Fig. 5 Verticalearthpressuredistribution(experiment)
pressure
consolidated
consolidation.
sand,
clay,
until
is
on
for
to
thickness
in Fig.
steps
that,
plate
the
loads
load
density
saturated
and
the stable
shown
placed
necessary
The
geotextile,
of primary
by the
the
width
to ensure
than
Under
until the completion
same
are
Note
it is
container.
and
so on,
in tum.
geotextile,
the
soil
gauges
by
and
container
or
of
higher
consolidated
geotextile
of soil
setting
horizontal
height
matters,
of
height
420mm
procedure
geotextile
sandy
pressure.
overall
height
of
are put
soil
are
Different
is
370mm,
as
shown
of saturated
clay,
into mountain
in
the
sand
•\ 401•\
in turn. The consolidationof mountainsand A is performedby
four stepwiseloadsof 9.8kPa,19.6kPa,29.4kPaand 34.3kPa,
in which each loadingstep is kept constantfor 3 minutes for
well consolidationof subsoil.For mountainsandB the loading
stagesof consolidationof subsoilare 24.5kPa,49kPa,73.5kPa
and 98kPa.The firstthreestagesare kept for 2 minutesand the
final is kept constantfor 5 minutes.After the subsoilhas been
consolidated,the portionof subsoilnear one edge is removed
fromthe subsoiland a modelslopeis carefullyformed.
The loadingtestsof saturatedclay are performedby putting
the loadingpressureat an incrementof 4.9kPa to the loading
plate until the subsoilreachesfailure,while mountainsand A
withthe incrementof loadingpressureof 9.8kPaand mountain
sandB with49kParespectively.Becausethe soilcannotdeform
immediately,each loadingpressurestage is stipulatedto keep
constantfor 5 minutes.The settlementof loadingplate, earth
pressureof soil layerand tensionof geotextileare measuredat
the intervalof one minuteuntil the next loadingpressurestage.
Photographsare taken4 minutesaftera new loadingpressureis
applied,in orderto recordthe movementsof the displacement
markers. From these photographs, the magnitude of
displacementsinthe slopeis obtained.
2.4Test Results
The observedfootingpressure-settlementrelationshipsare
givenin Fig.4 for thethreetypesof subsoil.As seeninFig.4(a),
the relationshipsbecome remarkablynon-linearat a certain
footingpressureand may reach the bearingcapacitywith this
turning point while tests are continueduntil large settlement
takes place.The resultsof saturatedclayin Fig. 4 (a) showthat
the earthreinforcementtends eitherto restrictthe settlementof
subsoil or to improve the bearing capacity.Comparedwith
Case 1 of naturalsubsoil,in which the subsoilcollapseswhen
footing pressure reaches 44kPa, Case 2 with one layer of
geotextileor Case 3 withtwo layersof geotextilehas a higher
failure load of 49kPa or 54kPa respectively.Because the
placementof geotextilein Case 2 is much deeper than the
regionof collapsemode in Case 1 as shownlaterin Fig. 16(a),
the geotextile little improves bearing capacity.While the
geotextileof upper layer in Case 3 largelyimprovesbearing
capacity,for its placementrestrictsthe formationof collapse
mode.Fig. 4 (b) showsthe experimentalresultsfor mountain
sand A. Concerningmountainsand B shownin Fig.4 (c), the
settlementis restrainedby geotextile,while no clear collapse
point is observed. It is difficultto evaluatequantitativelythe
effectof geotextileon bearingcapacityfor mountainsandA and
B. Thesephenomenawill be discussedlater in the numerical
analysisof testresults.Fig.5 showsthe monitoredverticalearth
pressures and illustratesthe comparisonamong three cases
definedabove.The figureshowsthat the earthpressureon the
centerlineof loadingplate increasesevidentlyin any instances
and decreasesto zeroon the sideof slope.For the saturatedclay,
near the left side of soil containerthe earthpressure in three
casesare almostthe same,exceptthe regionbelow geotextile.
However,for mountainsand A and B, the earth pressurenear
lateralsidewallhave a littlechange.The detaileddiscussionis
made laterinthe numericalanalysis.
•\ 402•\
3. Numerical Procedure
3.1 Outline
The proposedprocedurecan be used to estimatethe bearing
capacity directlyby a developmentof collapse mode. The
conditionsto get such a collapsemode are as follows: 1)
Assumean activewedge below footing,2) Treatthe yielding
mass as a stratifiedmaterialresultingfrom the smeared shear
band approach,and 3) Performrigorouslythe nonlinearFE
analysisbased on the modified initial stress method17).The
proposed procedure employs a simple constitutivemodel
whichrequiresa smallnumberof materialparameters,so that it
may be appliedto practicaldesignwork.
3.2 YieldCriterion
To relate the proposedprocedureto conventionalstability
analysis, Mohr-Coulomband Coulomb yield criteria are
employedrespectivelyto plane strain soil mass and friction
interfacebetween structureand soil. For the frictioninterface
we employthe thin layerfiniteelementas shownin Fig.618).
Mohr-Coulomb:
(1)
(2)
where, ƒÐx, ƒÐy
and
shear
and Ąxy: stress
stresses
3.3 Constitutive
(1)
illustrates
strain
rule
relationship
to
or plastic
as shown
for coulomb
Coulomb
the relationship
vector
interface
and ƒÐt and ƒÑst: normal
in Fig.
6.
Relationship
Stress-strain
Subjected
in friction
components,
{yst}.
potential
We
interface,
between
employ
Qc
defined
shear
a simple
interface
Fig.
stress
7
schematically
vector
non-associated
{Ąst} and
flow
as19)
(3)
where g: a hypotheticalparameterwhich is not cited actually,
because Qc is used only by its differentialform.For the thin
layerelement,as shown in Fig.6,the elasto-plasticstress-strain
relationshipis givenas15)
(4)
where•@
(a) shear band
Fig. 6 Coordinates in interface element
where
{ƒÂƒÐ} and
increments
(see
stress-strain
matrix
component
of [Dep].
(2) Stress-strain
When
that
Fig.
we
8 (a).
meaning
our
main
In
7),
[Dep]
to
this
paper,
surface
concern
the most
of shear
band
size
a shear
of a slip
is to
and
[D]:
elasto-plastic
elasto-plastic
coordinate
s-t
for Mohr-Coulomb
a finite
observe
employ
and
referred
relationship
shearing
often
{ƒÂƒÃep}
: stress
Fig.
Fig. 7 Stress-strain relationship
the
of soil
band
term
occurred
get
fundamental
element,
expression
strain
and
here,
elastic
and
it is well
surface
band
in the yield
a practical
Fig. 8 Shear band formation
dij:
material
or slip
shear
(b) stratified material
design
known
as shown
is used
as
element
Fig. 9 Direction of shear band
the
Since
procedure,
of inclination
(a) two slip surfaces (b) direction of slip surface in an element
in
we
of angle
as
(5)
Pietiuszczaket al.14)proposed the smeared shear band
approachwhich evaluatedthe averagestress-strainresponseof
solidand shearband Whenthe thicknessof shearband exceeds
a certainthickness,the yieldplanestrainelementbecomesclose
to the stratified or cross-anisotropic aterial, as shown in
Fig. 8 (b). Modifiedthis smeared shear band approach and
combined a plastic stress-strainrelationshipwith a simple
plastic flow rule as shown in Fig. 7 for Mohr-Coulombsoil
mass,to get a collapsemodeanalogousto Fig. 10,the proposed
procedureassumesthe yieldsoil elementas a stratifiedelement
and considersits elasto-plasticmatrix[Dep]to be equalto that
as givenby Eq. (4).
(3) Selectionof shearband
Generallya set of two shearbands or slip surfacesA-A' and
B-B' are possible for a finite soil element accordingto the
principalstress state as shown in Fig. 9 (a). Considcringthe
formationof activewedge, we assume the shear band B-B'
definedin Fig. 9 (a) within the activewedge in Fig. 10, and
assumethe shearband A-A' outsideof the activewedgeas seen
in Fig. 10. The directionof A-A' or B-B' line in Fig. 9 (a) is
generallydeterminedas (seeFig.9 (b))
Fig.
and Į:
Note
angle
that
stress Ąst
along
of
the
compressive
above,
yield
surface
after
also
happens
to
when
applying
Such
a
tensile
stress
stress
along
stress
is positive
A-A'
line
from
here
in Fig.
vertical
and
axis.
that
9 (a) and
shear
positive
state
is assumed
The
linear
yielding.
make
the
the
shear
relationship
finite
a constraint
along
with
stress-strain
produces
respect
side
the
relationship
in
value
Fig.
this
to ƒÐt, which
7
problem.
exceedingly
To avoid
in Fig.
along
decrease
to a boundary
elements20).
right
to move
stress Ąst
sometimes
for some
to move
State
a stress
movement
introduce
state
principal
of Loading
stated
stress
of slip surface
line.
3.4 Definition
we
major
is negative
B-B'
As
10 Isolation
high
confusion,
compels
a
7.
(7)
Eq. (7) means that normal stress perpendicularto the slip
surfacenever decreases.The finite element,in which stresses
violateEq. (7), is calledtensileelementhereafter.
:A-A' line
: B-B' line
3.5 Modified Initial Stress Method
(6)
The original initial stress method is based on an iterative
where 13:inclinationangle of shearband from horizontalaxis,
•\ 403•\
procedure.
From
mathematical
viewpoint
it is a special
application
of
applying
constitutive
does
even
not
by
finds
procedure.
directly
Fig.
, yield
of
determined
shear
loading
not necessarily
the
stress
because
stress
vector
the
mode
state
{ƒÐI}
state
strain
strain {ƒÃep}.
{ƒÐA}, and
a collapse
initial
{ƒÐ0}, total
the
stress
use
0
is
(a)
it throughout
other
methods
as shown
in s-t coordinate
Yield
determine
principal
Fig. 11 Modified initial stress method
iterative
of
et al 16). To
(a) initial stressmethod (b) initial stress method at a loading stage
method,
equilibrium
stress
major
stages,
provide
initial
Nayak
yield
stress
of plastic
10,
difficulties
stress
without
actual
initial
are
in Fig.
These
elasto-plastic
by
band,
using
succeeding
Firstly
{ƒÃe}, and
isolated
by
the
are
as piecewise
initial
stresses
stress
{ƒÐE}, virtual
strain
{ƒÐA} is
direction
defines
results
displacement
material.
initial
the
system,
as illustrated
a modified
{ƒÐA}, actual
stress
{ƒÃ}, elastic
stratified
the
11 (a)
stress
{ƒÐB}, elastic
stress
the
introducing
and
the nonlinearity
mode
with
subdivision
stress
treats
the collapse
assuming
avoided
which
create
numerical
element
of
method
When
together
the
finite
distributions
method.
method
above,
by the
This
though
are
stress
described
affected
observe&
linear,
Newton-Raphson
initial
model
unreasonable
often
the
modified
original
considerably
and
the
the
in Fig.
saturated
clay
(ƒÓ=0)
do
10.
is
(8)
Referring
to Eq.
(4), component dij
component
of
elastic
matrix
components.
This
means
interface
and
in [Dep]
is the
except
the
[D]
same
as the
third
row
(b)
both
in
application
the
Fig.
the
of Eq.
mechanical
11 (b),
method
yield
the numerical
of
initial
the stress
stage,
and ƒÐt0 in Eq. (8)
strain
(8) reduces
loading
is given
both ƒÐs0
plane
meaning
in which
present
that
state
the
has
basic
elements.
effort
to yield
in the
A (ƒÓ=15.9•‹)
clarifies
Referring
attained
equation
sand
The
and
stresses.
mountain
vanish
to
state
initial
at
stress
as
(c) mountain
Fig.
sand
B (ƒÓ=27.76•‹)
12 FE meshing
(9)
where, {ƒÐE}={ƒÐ}n-1+[D]
for
calculating
[K]:
area
global
constraint
184
{r}i: residual,
components
stiffness
of the element,
{ƒÃe} in Eq. (9)
The
[B]
strain
matrix,
and
nodal
{ƒÐf}: load
suffixes
and Fig.
given
from
[B]i:
increment
i and j denote
11 is calculated
matrix
displacements,
vector,
element
Aj:
number.
as [D]-1({ƒÐA}-{ƒÐ}n-1).
ƒÐ
by Eq. (7) is represented
as
(10)
Fig.
where {ƒÐA}={ƒÐA}i
element
has
element
yielded
(10)
linear
are
possible
simultaneous
to
or {ƒÐC}={ƒÐ}in-1
yielded
at the
present
at the preceding
equations
directly
equations.
with
solve
For
respectively
loading
stage.
respect
Eqs.
instance,
(9)
stage
Since
both
to unknown
and
(10)
Eq. (9) for
when
the
n or when
the
Eqs.
(9)
{ƒÐst0},
as
finite
a
set
element
with
respect
13 Verification
to unknown
of shape
{ƒÐst0}j given
of active
wedge
as
and
it is
(11)
of
i
•\
404•\
where||3
denotes
solving
Eqs.
numbers
both
additional
an
of
loading
stage
by
{ƒÐE} and
elements
in which {ƒÐE}
elements
in which
elements,
calculate
preceding
stress
(10).
6)
state.
find
the
an
determined
at 5). When
elastic
including
procedure
found
7) Based
variables
on
use
to
total
and
tensile
neither
new
yield
results
results
and
{GE}
and
(a) saturated clay
direction
Eqs.
(9) and
elements
by
{ƒÂf} and
{ƒÐst0}
elements,
and
tensile
elements.
nor
the
inclination
yield
tensile
at 6)
yield
and
or tensile
the
tensile
the
band
both
yield
yield
final
For
tensile
of
new
finite
and
solving
new
the
3)
shear
and
by
{ƒÐB}, settlements,
4. Numerical
find
yield
finding
the yield
from
{ƒÐast0} by
analysis
the
until
(7).
a
{ƒÂf},
{ƒÐA}, calculate
and
{ƒÐst0} subjected
elements
state
Concerning
(6). 5) Determine
Again,
Find
{ƒÐA} both
stress Į,
during
increment
criterion,
Eq.
finite
1) Performing
load
yield
violate
stress
4)
performing
this
the
following
yield
steps
as follows.
as the firstmethodangle a is given by Eq. (5), regardingthe
verticalfootingpressureas the majorprinciplestress.However,
for practicaldesign, the minimumbearing capacitymust be
When
constant
the
the
numerical
11 (b). 2)
violate
yield
Thus
actual
{ƒÂƒÃ}in Fig.
principal
determine
using
etc.
the
determining
The
stresses
vector,
assume
and ƒÐt0.
are summarized
analysis
by Eq.
of
must
for
elements.
calculate
angle ƒÀ
we
is required
tensile
of the major
component
(10),
unknown Ąst0
and
elastic
third
and
iteration
elements
typical
the
(9)
Repeat
(b) mountain
element
calculate
sand A
is
necessary
so on.
discussions
4.1 Preparations
The
numerical
laboratory
sub-soils
plate
procedure
model
used
is
elastic
in our
modeled
12
beam
modulus
set
between
G is given
The
material
considering
meshing
in which
the
footing
elements
provides
much
solutions
To
requires
our
a
to
12,
principle
peripheral
observed
angle ƒ¿
we
solutions
global
assume
is
region.
mode,
shape
also
al.7)
the
of the active
wedge
assumed
the
Eq.
wedge
wedge
duplicate
part
the
(5).
active
These
specified
of
for
as
shown
footing
wedge.
researches
(b) mountain sand A
in
rotation
below
For
of
to
researches
wedge
by
into
procedure
the
developing
active
taken
stresses
solutions.
experimental
active
conventional
proposed
active
to
below
employs
with
of
(a) saturated clay
When
procedure
initial
the
subsoil.
1.
are not
the conventional
Many
the
than
stresses
shear
of
proposed
isotropic
difficult
from
actually
similar
with
the
it
stresses
et
give
in which
Table
capacity
its
and
as shown
E and
in
the
initial
collapse
because
using
Fig. 14 Effect of shape of active wedge
loading
by
elements
given
bearing
or
(c) mountain sand B
three
A=0.012m2
subsoil,
stresses,
anisotropic
A lot of analyses
Kusakabe
initial
Thus
get
by
to the
of
represented
area
and
are
lower
consideration.
comparing
footing
as E/2(1+u)
in which
and
cross
Interface
parameters
anisotropic
is applied
FE
E=2.1•~107kPa,
6, are
above
shows
of inertia I=1.44•~10-7m4.
in Fig.
validity
proposed
Fig.
analyses,
by
modulus
moment
Fig.
tests.
the
have
footing.
(c) mountain
example,
determined
by
may
the
Eq. (5). In our
verify
Fig.15 Footingpressure-settlement
relationship
(Case 1,analysis)
analysis,
•\
sand B
405•\
e=0,
e=1.5cm,
h
1/3h
qu2=35.89kPa
qu1=39.521kPa
(a) saturated clay
e=0,
(a) saturated clay (q=49kPa)
h
e=1.5cm,
qu1=85.76kPa
1/3h
2=75.96kPaqu
(b) mountain sand A
e=0,
e=0,
h
1/3h
(b) mountain sand A (q=137kPa)
qu2=93.21kPa
qu1=115.4kPa
(c) mountain sand B (q=147kPa)
(c) mountain sand B
Fig. 16Yieldregion(Case1,analysis)
49kPa)
Fig. 17Majorprinciplestressfield(Case1,analysis)
(a) saturated
clay (q=
(b) mountain
sand A (q=137kPa)
(b) mountain sand A
(b) mountian sand A
(c) mountain
sand B (q=147kPa)
(c) mountain sand B
(c) mountain sand B
(a) saturated clay
(a) saturated clay
Fig. 19Footingpressure-settlement
relationship Fig.20 Footingpressure-settlement
relationship
Fig. 18Displacementfield
(Case2, analysis)
(Case 1,analysis)
(Case3, analysis)
height of active wedge as denoted in Fig. 13.
selected by changing the shape of active wedge to ensure th
is
Because the proposed procedure cannot perfectly duplicate
conservative. In our analysis, as the second method the shape of
the strain localization behavior, it cannot produce an infinite
active wedge is assumed to depend on variation of its height
and eccentricity. Fig. 13 gives the patterns of active wedge
plastic shear flow of subsoil. It determines the bearing capacity
by the stress condition of soil mass. Because of the assumption
assumed in our analysis. Based on the FE meshing shown in
of the smeared shear band for yield element, it may create an
Fig. 12 with some changes for shape of active wedge in Fig. 13,
estimated collapse mode by the distribution of yield element.
the proposed
As an example, Fig. 16 shows the yield region given by the
bearing
capacity
given
procedure
by
the
proposed
procedure
provides the relationship
between
It demonstrates that the calculated results are affected by the
proposed procedure, where a solid line in some finite elements
represents the direction of shear band or slip surface, and that
variation of active wedge
e
these elements have yielded. Since the shear band means the
represents the eccentricity of active wedge and h denotes the
slip surface in each yield element, the line connected by shear
footing pressure and settlement for Case 1 as shown in Fig. 14.
in a certain extent, in which
•\
406•\
band is thoughtto correspondto a continuous
slipsurface.It can be seenfromthe figurethat
the yield elementsconnectto form a collapse
mode when currentfootingpressurereaches
the denotative value. This means that the
bearing capacityis determinedas the value
when a global collapse mode is created
initially.In Fig. 16, qu1representsthe bearing
capacitywhen the activewedgeis deteimined
by Eq. (5) and qu2is the minimumsolutionof
variousshapesof activewedge. In this paper,
we will calculatethe bearingcapacityfor all
shapes of activewedgeas shown in Fig. 13,
and then selectthe minimumsolutionas the
bearingcapacity.
e=1.5cm,
1/3h
e=0, 1/3h
qu2=36.6kPa
qu2=69.51kPa
(a) saturated clay
(a) saturated clay
e=1.5cm.
1/3h
1/3h
qu2=111.5kPa
(b) mountain sand A
(b) mountain sand A
e=0, 1/3h
e=0, 2/3H
qu2=165.72kPa
qu2=284.82kPa
(c) mountian sand B
4.2 Case 1 (natural subsoil)
e=1.5cm,
qu2=80.86kPa
(c) mountian sand B
Fig.21Yieldregion(Case2, analysis) Fig.22Yieldregion(Case3, analysis)
(1) Saturatedclay
The solutionscalculatedby the proposed
procedureare comparedwith the results of
experimentsas shown in Fig. 15 (a), where
the bearingcapacityqu1and qu2are defined
aboveand whereanalyses1and 2 correspond
to qu1and qu2respectively.Fig. 15 contains
also the upper bound solutionpresentedby
Kusakabeet al.7)Fig. 16 (a) showsthe yield
region or collapsemode by the proposed
procedurecorrespondingto the value of qu1
and qu2.The collapsemode is supportedby
the principalstressfiled shownin Fig. 17 (a)
and the displacementfield shown in Fig.
18(a)which appearmechanicallyreasonable.
Note thatFigs. 17and 18showtheresultonly
for qu2.As shownin Fig. 15(a), the proposed
(a)Case1
(b)Case2
(c)Case3
Fig.23 Collapsemodeby shearstraindistribution(experiment)
proceduredoes not providea clear turningpoint of settlement
butgivesthe bearingcapacitycloseto thatof experimentby the
way as statedbefore.The reasonwhy the experimentalresultis
slightlylargerthan the analyticalvaluesmay be attributedtothe
influenceof side frictionbetween subsoiland container.The
proposedprocedureoverestimatesthe bearingcapacityonly for
the saturatedclay thanthe solutionby Kusakabeet al.
(2) Mountainsand
For two typesof mountainsand,the propertiesare given in
Table 1. As shown in Figs. 15 (b) and (c) correspondingto
mountainsand A and B respectively,the bearingcapacityqu1
and qu2are defined in the same way as stated before,which
dependon the collapsemode representedby the yieldregionas
shown in Figs. 16 (b) and (c). The collapsemode is supported
by the principalstressfieldshowninFigs.17(b) and(c) and the
displacementfield shown in Figs. 18 (b) and (c). As seen in
•\
(a)Case2
(b)Case3
Fig. 24 Displacementfieldin reinforcedsubsoil(analysis)
407•\
Figs. 15 (b) and (c), the determinedbearing
capacity is in good accordance with
experimentalresult and that by Kusakabeet al.
Differentfrom the saturatedclay,the mountain
sand does not form an explicittiming point of
settlementbothin analysisand experiment.
(3) Discussions
The collapsemode providedby the proposed
procedureis createdby consideringthe weight
of subsoil, stiffness of footing and subsoil,
frictionbetweenfootingand subsoil,and stress
concentrationat the edge of rigid footing,most
of which are ignored in the limit equilibrium
approaches.The yieldregion in Fig. 16tendsto
distributemore deeplybelow footingthan that
assumedin the conventionallimit equilibrium
analysis.This is because the verticalpressure
must reach lower subsoil due to the vertical
equilibriumcondition,and becausethe pressure
makes lower subsoilyield. From Fig. 18, we
observe little deformationof lower subsoil.
Fig. 16 shows that the collapse mode
corresponding to qu1 is close to that by
Kusakabeet al.Thisis becausethe activewedge
for qu1given by Eq. (5) is the same as that
assumed by Kusakabe et al. However, the
collapsemode for qu2is shallowerthan that for
(a)Case1
(b)Case2
(c)Case3
Fig.25 Displacementfield(experiment)
saturated clay (q=39kPa)
mountian sandA (q=78kPa)
mountian sandB (q=147kPa)
(a) Case 1
(b) Case 2
qu1.This is because the correspondingactive
wedgeto qu2is shallowerthan that of quiso that
the obtainedglobalcollapsemode shrinksalong
with such an activewedgeas shownin Fig. 16.
The laboratorymodel tests cannotprovidethe yieldregion as
stated above, but it visualizesa tendencyof collapsemode
representedby the shearstraindistributionwhich is calculated
from the monitoreddisplacementof subsoil.Fig.23 showsthe
observed collapse mode represented by the shear strain
distribution.As shown in Fig. 23 (a) the large shear strain
distributealongthe collapsemode showninFig. 16.Fig.25 (a)
shows the monitoreddisplacementfor Case 1, which gives
similartendencywith the calculateddisplacementfieldshown
in Fig. 18. Fig. 26 (a) comparesthe verticalearth pressures
calculatedby the proposedprocedurewiththatmonitoredinthe
model tests. For upper subsoilthe calculatedearth pressureis
(3) Case 3
Fie. 26 Vertical earth pressure distribution
saturatedclay (q=39kPa)
saturatedclay (q=39kPa)
mountain sandA (q=137kPa)
mountain sand A (q=137kPa)
mountain sandB (q=196kPa)
mountain sandB (q=196kPa)
generallyin good agreementwith monitoredresults.For lower
subsoil,the calculatedearthpressureis a littlehigher than the
monitoredone becausethe foundationsubsoilis assumedto be
elasto-plastic
continuuminthe numericalanalysis.
(a) Case 2
4.3 Case 2 and Case 3 (reinforcedsubsoil)
(1) Saturatedclay
(b) Case 3
Fig. 27 Distribution of tensile force of geotextile
•\ 408•\
Figs. 19 (a) and 20 (a) illustratethe reinforcementeffecton
the bearingcapacityin Case2 or Case3 comparingwithCase 1
(naturalsubsoil).FE meshingand soilparametersare the same
as those in Case 1 exceptthe geotextileis placedas shownin
Fig. 3. The propertiesare given in Table 1. As shown in
Fig. 19(a) for the saturatedclay the reinforcementhas no large
improvementin Case 2. 1) The place of geotextilecannot
restrict the development of collapse mode as shown in
Fig. 21 (a), namely,the collapsemode is createdin the upper
subsoil than the location of geotextile.2) The stiffness of
non-wovengeotextileused is not high.However,in Case 3 we
can observelargerimprovementon bearingcapacitythan Case
2 because the geotextile placed on the upper subsoil can
effectivelyrestrictthe collapsemode as shown in Fig. 22 (a).
Similar to the experimentalresults, the proposed procedure
evaluatesnot onlythe reinforcingeffectof geotextileonbearing
capacitybutalsothe restrictionof settlementof foundation.
(2) Mountainsand
Figs. 19(b), (c) and Figs.20 (b),(c) showthatthe geotextile
can improvethe bearingcapacitygreatlyfor each of mountain
sand, which agrees well with the experiment. This
improvementis more evidentin Case 3 than in Case2. Sucha
result can be explainedby the collapsemode as shown in
Figs.21 (b), (c) and Figs.22 (b) and (c). The collapsemode is
observedto be restrictedto a narrow limit and to restrainthe
unconstrainedplasticflowby geotextileespeciallyby the upper
geotextile.It is alsonoticeablein thesefiguresthatthe geotextile
of two layersrestrictthe developmentof collapsemode to a
greatextent The figuresshow the reductionof settlementafter
failureof foundationfor any casebesidesthe improvementof
bearingcapacity.Withthe increasein stiffness,the controlsof
geotextileon settlementbecome clear. This tendency agrees
qualitativelywiththeexperiments.
(3) Discussions
In all casesthe bearingcapacityin reinforcedsubsoilis larger
than that in natural subsoil,because the collapsemode is
restricted by geotextile especially in Case 3. As seen in
Figs.21 and 22, the yieldelementsare mainlyconcentratedon
the region upper the geotextile.As the experimentalresult,
Figs.23 (b) and (c) providethe yield regionrepresentedby the
shear strain in Case 2 and 3. Comparedwith that in natural
subsoil on the same footing pressure, the shear strain in
reinforcedsubsoildecreasesevidentlyfor the restrainedeffect
of geotextile.The shearstrainzonesin Case 2 are concentrated
onthe regionupperthe geotextile,and the largestshearstrainis
producedat the regionbelow the loadingplate.The trend that
the shearstrainin Case3 extendswidely,resemblesthe collapse
mode createdby the yield elementsshown in Fig.22. Fig. 24
shows the displacementfield for these two cases. Compared
with Case 1 (Fig. 18), the vertical displacementdistinctly
•\ 409•\
decreasesdue to geotextileexcept for the saturatedclay.The
reason why the verticaldisplacementof saturatedclay has no
the distinct tendency for reinforced subsoil is due to two
aspects:1) Differentloadingpressureis appliedto each case,
while for mountainsand the same footingpressureis given.2)
Non-woven geotextileused for the saturatedclay has little
effectfor restrainingthe verticaldisplacement.The horizontal
displacementsin Case 2 or Case 3 decrease evidentlyin all
cases, particularlybelow the geotextile.This means that the
geotextile mainly restricts horizontal displacement. The
monitoredverticaland lateraldisplacementsin Fig.25 have a
fairly good agreementwith the calculatedthose, under the
footingbaseand on the edgeof slopein mostcases.Figs.26 (b)
and (c) showthe verticalearthpressuresin Case 2 and Case 3.
In Case 2 the calculatedresults agree fairly well with the
experimentalresults.In Case 3 the monitoredearthpressureat
centerline is considerablylarger than the calculated one
especially for the mountain sand, because the subsoil is
particulatemedia which induces the concentrationof stress.
Fig. 27 shows the distributionof tensile force acting on
geotextile.Thereis a littledifferencebetweenthe calculatedand
monitoredresults for mountain sand. This is because the
stiffnessof geotextileis estimatedfrom the assumptionthat the
stress-strainrelationshipis linearelastic.For the saturatedclay,
the largedifferencebetweenthem is attributedto the difficulty
to evaluatethe stiffnessof non-wovengeotextile.
5. Conclusions
This paper performed the laboratory model tests for
simulatingthe bearingcapacityof rigid stripfootingon slope
and proposeda numericalprocedurefor estimatingthe bearing
capacityconsideringstiffnessof materialand collapsepattern.
The procedureaimsto fill a gap existingbetweenconventional
stabilityanalysisand classicalFEM. The procedureemploys
Mohr-Coulomband Coulombyieldcriteriarespectivelyfor soil
mass and friction interfacebetween soil and structure. By
assuminga shearband for yield elementand by employinga
modified initial stress method, the procedure provides a
collapsemechanismanalogousto a slip surface assumed in
conventionalstabilityanalysis.At the collapsemode created,a
stressyieldcriterionis satisfiedas well as alongthe slip surface
supposedin conventionalstabilityanalysis.Such a definitionof
collapsemode is differentfrom most applicationsof FEM
which tend to expressthe collapsemode by the distributionof
shearstrainor displacement.
The procedureproducesa collapse
mechanismas assumed in conventionalstabilityanalysis,and
that the mechanismis supportedby a displacementfield and a
stress field. This characteristicindicates the possibility of
applyingthe procedureto the stabilityanalysis which takes
8) Narita,K. and Yamaguchi,H.: Bearing capacityanalysis
of foundationson slopes by use of log-spiral sliding
surfaces, Soils and Foundations,Vol.30, No.3, pp.
144-152,1990.
9) Shields,D.H.,Scott,J.D.,Bauer,G.E.,Deschenes,J.H.and
Barsvary,A.K.: Bearing capacity of foundationsnear
slopes,Proc.9thICSMFE,Vol.1,pp.715-720,1977.
10) Sugano, Y., Okamura, M. and Furukawa,N.: Bearing
capacity and displacement characteristicsof stripe
foundations on slope and its applicationto seismic
displacementprediction,Proc. of 42nd annual symp.on
theJapanese GeotechnicalSociety, pp.677-678,2007.
11) Kimura,T., Kusakabe, O.and Saitoh,K.: Geotechnical
modeltests of bearingcapacityproblemsin a centrifuge,
Geotechnique,
Vol.35,No.1,pp.33-45,1985.
12) Yoo, C.S.: Laboratoryinvestigationof bearing capacity
behaviorof stripfootingon geogrid-reinforced
sand slope,
Geotextilesand Geomembranes19,pp.279-298,2001.
13) El Sawwat M.: Behavior of strip footing on
stiffness and deformation of material into consideration.
Most case studies show that the bearing capacity obtained by
the proposed procedure agrees fairly well with the laboratory
model tests. These results suggest that the proposed procedure
provides an appropriate solution for general problem, and that
the procedure can be applied quantitatively to experimental or
actual bearing capacity problems, because it requires some only
fundamental
soil parameters
Mohr-Coulomb
strength
such as elastic modulus
parameters.
Some
case
and
studies
demonstrate that the proposed procedure can be applied to the
subsoil
reinforced
stiffness
and
by
geotextile
deformation
reinforcement
because
of
it considers
materials.
The
the
effect
of
on bearing capacity is based on restricting the
development of collapse mode, which leads to an increase in
the bearing capacity. All the calculated results show that the
earth reinforcement improves not only the bearing capacity but
also the settlement after failure, which is well compatible with
the monitored
results. Comparisons
with experimental
and
other analytical results show the possibility that the proposed
geogrid-reinforced
sand overa softclay slope,Geotextiles
and Geomembranes25, pp.50-60,2007.
14) Pietruszczak,S. and Mroz,Z.: Finiteelementanalysisof
defamation of strain-softeningmaterials,International
Journalfor NumericalMethodsin Engineering,Vol.17,
procedure gives realistic predictions and provides a useful
engineering tool for the design of foundation on slope.
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(ReceivedApril 14, 2008)
•\
410•\