201
Two Dimensional A~nal see
6 ~ 1 Int rod uct ion
Three series of analyses were performed. The f irst
s eries cons isted of parametric studies to examine the
effects of porosity, cyclic stresses and initial stresses on
the permanent displacements of a flexible mat.
The second series investigates the creation of
permanent displacements on the Qosterschelde Barrier due to
wave cyc 1 ic! load ing. Th e third s eries compares the
results obtained with the proposed model with results f rom a
physical model of the Barrier employing a centrif uge
obtained by Rowe and Craig �978! .
6.2 Method of Analysis: Limitations
The analyses are performed using the two-dimensional
plane strain computer program presented in Chapter V. The
data required are the two permanent deformation compliances
given by equations �. 54! and �. 55!, the geometry and
boundary conditions.
202
Several limitations are present in the method; these
are discussed next.
6 ~ 2 ~ l Limitations of the Method
Important limitations of the approach are:
a. The first step in the solution is the computation
of the elastic solution, including the weight of the
structure, to compute initial stress es. Depending upon the
appl ied boundary loads, elements in the foundation may
yield. If this is the case, the value of g~/po exceeds the
limiting value given by q /pc = tga. The author chose to
compute the permanent deformation compliances us ing the
value of a /p = tga for the yielded region. Though this is0
a reasonable ass umption where yielding is not widespread!
it does not include the redistribution of stresses produced
by yielding,
b. The cyclic shear stresses are computed using
elastic theory with the same set of elastic parameters used
to calculate the initial stresses, We expect dif f erent
values for the elastic moduli for cyclic loading, however,
we assone that the differences in relative values for moduli
are small enough to allow the use of elastic theory.
c. The superposition of the initial state of stress
and the cyclic stresses can lead to values of g~ +
hq~>~ !/ p~ + Ap c>c! which are larger than the shear strength
of the soil. No provision is made in the formulation to
203
prevent this, except that to limit q /p to tge!. The
cyclic loads are f ully applied to all foundation materials.
The proper way to do it would be to apply just the
value of Ag and ~p to reach failure, and tocyc Cgc
redistribute the rest to other elements. If some elements
are al ready y ield ed, all the cyc1 ic loads sho uld be
determined with other moduli However, the redistribution
of cyclic stresses due to yielding would unnecessarily
complicate the model at this stage of its developnent.
d. Laboratory res ults have shown that el ements with
very low cyclic stresses do not run permanent strain Marr
and Christian, 1981! . Based on those results, a lower
limit of hq /p equal to 0.05 was established. Elementscyc 0
with less than this level do not deform permanently. This
lower limit of cyclic shear stress ratio corresponds
approximately to a peak-to-peak shear strain of 0.008% for
average stress conditions.
e. The method of analysis assumes that isotropy is
valid and that results from cyclic drained triaxial
compression tests can be used to predict what happens under
other conditions. Results f rom dif f erent stress systems can
easily be incorporated in the linear viscoelastic model to
account for stress induced anisotropy. This has not been
done in the current version.
204
f. The method accepts only cycles of the same
amplitude. If the cyclic loads are applied as parcels of
dif f erent amplitudes, these must be converted to an
equivalent number of cycles. Such conversions have been
successfully used in earthquake engineering; Lee and Chan
l972!, Annaki and Lee l976! and Seed, et. al. l975! . A
second approach is to apply each parcel as an independent
uniform cyclic load, update the moduli and soil properties
at the end of the parcel and then apply the next parcel.
For analyses described herein a constant amplitude number of
cycles is applied. In order to compare results, the concept
of equivalent number of cycles is used.
development; how ev er, a v iscoplast ic
f eas ible.
formulation is
g . The method is l i near. Th is impl ies that the
compliances do not change due to changes in stress during
the anally s is . The c reation of permanent displacement
involves a redistribution of stresses with a consequent
change in. moduli, which is not included in the model. The
importance of this limitation depends on the magnitude of
stress redistribution with cycling. Including non-linearity
would complicate the model and method at this stage of
205
6.3 Flexible Foundation
Th es e analyses were performed to check the
reasonableness of results f rom the f inite element code and
to demonstrate the effects of cyclic stresses, initial
stresses, poros ity and . cyc1 ic load ecc ent ric ity on the
magnitude and pattern of permanent displacement.
Figure 6.1 portrays the geometry and properties for the
flexible mat case. Figure 6.2 presents the ultimate bearing
capacity Vesic, 1974! of the footing for different
combinations of horizontal and v crt ical loads. Th es e
horizontal and vertical failure loads are the basis for
selecting loads to use in the parametric analyses.
From Figure 6 ~ 2 a ratio of horizontal/vertical load of
approximately 0. 25 gives a maximum horizontal load of 5. 20
LT/mL] and a vertical load of 20.8 LT/mL] ~ The analyses
were performed with a horizontal load maximum of 5 [T/NL]
and a vertical load of 20 LT/ML]. This combination provides
a reasonable set of likely maximum design conditions. Table
6.1 presents the eight different load combinations used, and
the cases that incorporate the ef f ects of porosity and
eccentricity. Five hundred cycles of uniform amplitude were
applied to each case using a variable step size to integrate
the equations. All cases assane linear stress change during
the cycle increment.
206
Figure 6.3 presents the finite element mesh used for
the calculations.
6. 3. 1 Ef f ect of Cycle Number
Figures 6.4 to 6.6 show the accunulation of permanent
displacement beneath the loading surface as a function of
nmber of cycles for the eight cases presented in Table 6.1.
Figure 6. 7 depicts the permanent displacements beneath the
footing as a function of number of cycles for two different
porositiesi 394 and 43%, and Figure 6.8 portrays the permanent
displacement beneath the footing as a function of nenber of
cycles for two different eccentricities e/B = 1/6 maximum
allowable eccentricity! and e/B = 1/12 �/2 of maximum
allowable eccentricity! . All the results show larger
displacement at the edge of the footing as compared to the
center and more displacement for more cycles.
6.3.2 Effect of Initial Stresses
Figure 6.9 presents the permanent displacements beneath
the loading flexible mat for three different values of
V/Vmax af ter 10 and 500 cycles. The V/Vmax ratios produce
dif f erent initial before cycling! stress conditions. The
results show only a small ef feet of the initial stresses for
the ratios o f V/Vmax analyzed.
Figure 6. 10 depicts the ratio between the edge vertical
displacement and the center vertical displacement as a
f unction of nanber of cycles for three V/Vmax ratios, and
207
two hH/Hmax ratios. For AH/Hmax=0.1, igure 6.10 shows
relatively little ef f ect of V/Vmax and not a definitive
trend. The ratio of settlement between the edge and center
eventually reduces as a f unction of number of cycles
essentially due to stress redistribution.
For hH/Hmax=.05 the ef feet of V/Vmax is larger, but
again no def init.ive trend appears. The settlement ratio
decreases with nmber of cycles except for the case in which
V/Vmax=0.2 where it first increases and then decreases. For
the load combination of V/Vmax=0.2 and hH/Hmax=0.05, several
elements at the center of the footing develop cyclic shear
stress ratios Qzyc /p~ less than 0. 05 and do not deform,'
this produces an increase in the settlement. ratio ~ For
higher nmber of cycles stress redistribution eventually
leads to decrease in the settlement ratio.
S. 3. 3 Ef feet of C~cl ic Stresses
Figure 6. 11 port rays the ace ambulated permanent
displacement beneath the footing for three values of hH/Hmax
aft,er 10 and 500 cycles. An increase of AH/Hmax produces an
increase in the permanent displacement..
Figure 6. 12 shows the edge displacement/center
displacement ratio for diff erent values of hH/Hmax and two
V/Vmax rat. ios. Th e displacement ratio dec reas es with
cycling except for the case where hH/Hmax = .05 and V/Vmax
. 2, already explained in sect ion 6. 3. 2.
208
6.3.4 ~Porosit Effect
F igure 6. 13 shows the permanent d isplaceaent beneath
the footing for three porosity values after 10 and 500
cycles. An increase in porosity gives an increase in
accumulated permanent displacement.
Figure 6.14 presents the same data of Figure 6.13 in a
sl ightly dif f erent way, showing the edge displacement/center
d ispl ac ament rat io as a f unct ion of cycle number f or
dif f erent porosities and V/Vmax = 0.2 and AH/Hmax = 0.05.
Again the cutoff value of cyclic shear stress ratio Aqc>c /pp
produces an increase in the displacement ratio with cycling,
followed by a decrease. This decrease is not. present for a
porosity of 43% because the small ~qc>c /po values prevent
permanent displacements at the center of the footing. The
high porosity value gives high peak-to-peak shear strains at
the edge with the consequent increase in the displacement
ratio with cycling.
6. 3. 5 Eccentric Cyclic Loads
Figure 6 ~ 15 depicts the effect of the eccentricity of
the cycl ic loads on the acc umul at ion o f permanent
displacement beneath the flexible footing after 10 and 500
cycles. Eccentricity can cause considerable increase in
predicted cumulative displacement for both the center and
the edge of the loaded area.
209
Figure 6 ~ 16 shows the displacement ratio as a function
of number of cycles for three different load eccentricities
and V/Vmax = 0.2 and hH/Hmax = 0.05. The ratio decreases
with cycling except for the case in which the eccentricity
ratio is e/B = 0, as explained before. However the decrease
is less than found for initial stress and porosity.
6 ~ 3.6 Conclusions from Parametric Studies
The conclusions from the results of the parametric
studies are:
the computer program can be applied with a variety
of input data to obtain a consistent set of
results. i.e. the program works!
all the stress path parameters are important and
interact in a very complicated way that can not
easily be inferred without analysis.
the stress gath parameters that should
s ignif icantly af f ect the predicted magniteie and
patt em o f permanent d isplac ements i nit ial
stresses, cyclic shear stresses and porosity! do
sos
210
6.4 The Oosterschelde Barrier
The Netherlands government has the respons ibil ity to
design a barrier darn across the Oosterschelde inlet located
southwest of Rotterdam. The closure which links with dikes
must allow tidal flow during normal sea states and resist
storm tides and waves Narr and Christian, 1981! . The
chosen design consists of large gates resting on reinforced
concrete piers. Figure 6.17 portrays a pier section with
base plan dimensions of 25m wide and 50m long ~ The piers
will be constructed onshore, floated into position, sunk
Erosion of theinto a dredged trench, and ballasted.
f oundation materials will be prevented by placing a
protective cover or sill.
The foundation materials consist of f ine to medium sand
of uniform gradation over most of the closure. The upper
part has been deposited in the Holocene epoch and is
loose-to-medium dens e. The underlying sand, deposited in
the Pleistocene epoch, is medium dense to very dense. The
loose-to-med i um sand directly under each pier will be
densified and a protective layer of foundation slag placed
and densified prior to placing the pier.
The objective of this section is to predict the
permanent displacement of one of the piers for four
different load combinations and to compare the results with
predictions presented by Harr and Christian �981 ! .
211
Figure 6.18 shows the finite element mesh used far the
calculations and indicates material types and locations.
Table 6. 2 summarizes the elastic soil parameters used for
the foundation materials. The modulus for concrete was
adjusted to reflect the difference in moment of inertia
between the actual pier section and that of the solid
concrete section used in the analysis. The elastic moduli
for the loose Holocene sand, dens if ied Holocene sand and
dense Pleistocene sand were computed using the relationship
proposed by Narr and Christian �981! and reflect the
average stress and strain levels developed by loading.
Only Holocene and Pleistocene sands are assumed to
develop permanent strain from the cyclic loading. All other
materials are cons idered to behave elastically during
cyc l ing.
The analyses considered four different combinations of
static head loss and cyclic wave load at two different
densities for the Holocene sand directly under the pier.
Table 6.3 presents the loads for the four combinations and
Figure 6.19 illustrates these load combinations. Case A is
the combination of head loss and wave loading predicted from
hydraulic studies. In case B, the static load consists of
the design head loss plus one-half of the design cyclic
load; the cyclic wave load is one half of the design cyclic
load. In case C, the static head loss load is one-hal f o f
212
the design static load and the cyclic load consists of the
remaining half of the design static load plus the design
cyclic load. Xn case D, there is no static head loss load
and the cyclic load has the design value. Thus, case B has
the total load heavily biased toward the static component
and case C has the total load heavily biased toward the
cyc 1 ic component .
In all four cases, 600 cycles of maximum wave load were
applied to the caisson.
6.4.2 Results
Figure 6. 20 illustrates the accumulat.ion of permanent
displac anent with cycling for dens if ied. Holocene sand, and
Figure 6.21 presents the vectors of permanent displacement
with cycling for undensified Holocene sand.
Figure 6 22 compares cases A, B, C, D at N = 600 cycles
for the densified Holocene sand and undensif ied Holocene
s and.
Figure 6.23 presents a comparison at N = 600 cycles
between the densif ied and undensif ied Holocene sand for
cases A, B, C and D.
The conclusions from the results presented in Figures
6. 20 to 6 ~ 23 are the same ones obtained by Marr and
Christian �980!; they are:
213
The static horizontal force controls the permanent
dif f erential settlement and the permanent
horizontal displacement of the foundation both in
magnitude and direction.
The cyclic load controls the magnitude of average
permanent settlement.
�9S1!
Marr and Christian �98 l! computed the permanent
displacements of the Oostechelde Barriers for cases A, B, C
and D using the approach described briefly in Chapter I. In
order to compare Marr and Christian's �90l! results with
the method proposed in this thesis, it is necessary to
determine the number of cycles of "f ull" cycl ic load
equivalent to the four cases presented in Table 6.2.
Three elements are analyzed to compute the number o f
equivalent cycles for case neaber A design case!: one at
the heel, one at the toe and one at the center of the
foundation. Table 6. 4 presents the computation. Prom the
table we conc l uded that load combi nat ion A dens if i ed
Holocene sand! can be approximated by the follow ing nonber
of cycles of the maxim' wave:
214
6.1 cycles at the toe;
4. 5 cycles at the heel; and
3. 2 cycles at the center of the foundation.
We consider an average value of 5 cycles of the maximum
wave applies over the foundation for all conditions.
Table 6.5 compares the vertical settlement from Narr
and Christian l981! with the values f rom the linear
viscoelastic approach for the heel, center and toe of the
f oundat ion f or dens if ied and undens if ied Eiolocene sand.
Table 6.6 compares the horizontal displacement at the middle
of the caisson for the same conditions ~ Except for case 8,
the linear viscoelastic approach gives vertical settlements
which are one and oneself to two times higher than Marr
and Christian �981!. Zn all cases the horizontal
displacement obtained with the linear viscaelastic method is
two to four times higher than the values of Marr and
Christian �981! . Both approaches show similar patterns of
permanent displacement. The dif f erences between both
approaches can be partly explained by cons idering the
differences between the methods. Marr and Christian �981!
used the computer program FEECON Simon et. al, 1974! to
calculate the initial stresses before the application of the
repeated wave loading. In their approach they simulate the
excavation, densification, construction of the caisson and
the application of the head loss load in the computation of
initial stresses. Their computation includes the effect of
215
yielding and stress redistribution due to yielding. In the
linear viscoelastic approach the initial stresses are
computed using elastic theory and none of the construction
processes are modelled. In both approaches the stresses are
determined using an elastic analysis. Marr and Christian
�980! include no effects of stress redistribution due to
the accumulation of the permanent displacement. Finally,
even though both models are based on essentially the same
soil data, diff erences exist. in the way these data were
synthesized for input to the computer. Marr and Christian' s
approach f o1 low ed th e work o f hedberg � 9 77! and the
v iscoel ast ic approach us ed the work by kiadg e �9 7 9 ! ~
Keeping these differences in mind the agreement between both
approaches is good.
6.4.4 Conclusions from Oosterschelde Closure
The linear viscoelastic method predicts larger
vertical and horizontal displacements than the Marr and
Christian approach. These differences can be explained by
one or more of the following:
a! Treatment of cyclic shear stresses.
Narr-Christian's method, the value of the cyclic shear
stress ratio is 1 imited by the f ail ure envelope. En
the viscoelastic formulation the cyclic shear stress is
f ul ly appl ied to al 1 foundation materials�. Th is
dif f erence yields larger permanent strains in some
216
elements for the viscoelastic model.
b! Number of equivalent cycles. Both approaches us e
the concept of nunber of equivalent cycles to represent
the storm loading conditions but in a different way.
Prom available analyses it is not possible to isolateI
the importance of this difference.
c! Ef f ect o f mean shear stress on empirical
stress-strain relationships. In Harr-Christian include
an effect of the mean shear stress on both volumetric
and shear components of permanent strain. In the
viscoelastic approach only the permanent vertical
strain equation �.11! includes the effect of the mean
shear stress ratio. This difference yields a larger
ratio of horizontal displacement to vertical settlement
for the viscoelastic model.
d! Empirical stress-permanent strain relationships ~
The empirical equations used by the linear viscoelastic
approach equations 2. 10 and 2 ~ 11! repres ent a clos er
f it to the original data. It is dif f icult to assess
the import. ance of this difference.
e! Redistribution of stresses. Barr and Christian' s
approach does not consider the redistribution of
stresses due to the accumulation of permanent
d isplac ement.
217
Rowe and Craig �978! present results from f ifty-six
tests run on a centrifuge to predict the performance of the
Oosterschelde piers under cyclic loading. The objective of
this section is to compare the centrif uge results from four
tests with the predictions of the proposed linear
viscoelastic approach. Figure 6.24 depicts the centrif uge
"f ield" dimeric ions and the foundation materials. The
elastic moduli presented on Table 6.2 were used as material
properties. Conditions in all four centrif ug e tests were
identical ecept the weight of the structure which was
varied. Table 6 ~ 7 sumarizes test conditions. The analyses
were performed for 1000 cycles using a constant cyclic load
of 300 tonnes. Figure 6.25 illustrates the finite element
mesh.
In order to compare the centrif uge results with the
res ults f rosa the linear viscoelastic approach, it is
necessary to compute the number of equivalent cycles
corresponding to the cyclic load pattern applied by Rowe and
Craig �978! . Figure 6.26 depicts the cyclic loading
program applied to the centrif uge. Using the concept of
equivalent number of cycles, the load pattern presented in
Figure 6-26 was transformed into 300 cycles of + H = 300
tonnes with a constant head loss of H = 300 tonnes. The
number of 300 equivalent cycles corresponds approximately to
the average between elements at the toe, heel and center of
the foundation.
218
Table 6.8 compares the centrif uge results with the
results from the viscoelastic model for the center of the
foundation. The centrifuge results for horizontal
displacement are four to five times larger than the finite
element results . Th e permanent s ettl ement is approximately
three to four times larger. Despite the fact that the
results differ substantially in magnitude, the trend of the
results is similar. The horizontal displacement decreases
with the weight of the struct ure, and the settlement
decreases as the structural weight decreases ~
Table 6.8 also presents a comparison between the ratio
of horizontal/vertical displacements. The linear
viscoelastic model predicts the increase with decreasing
weight of the structure, measured in the centrifuge tests.
Ne think differences in soil properties account for the
smaller displacements predicted by the viscoelastic model
than measured in the centrifuge. Howe and Craig used a
dif f erent soil mixed with oil and compacted to achieve
compressibility thought to represent that of the Barrier
foundation. Th e permanent d isplac ement compl iances
parameters for the viscoelastic model come from laboratory
tests on one Oosterschelde sand which showed a relatively
high resistance to cyclic loading. Cyclic drained triaxial
tests on other Oosterschelde sands, similar in grain size to
the stiff sand, gave less resistance to cyclic loading.
Marr and Christian l981!, present predictions of permanent
displacement using parameters for this "soft" sand. In this
219
case, the residual deformat ions increase substantially.
displacement and decrease in settlement with decreasing
caisson weight.
b! d ispl ac ements p red ict ed with the 1 i near v iscoelas tie
method are three to f ive t imes smaller than the
centrifuge results. These differences can be explained
by the following a
unknown soil p rope rt ies us ed by Rowe and
Craig�978! make direct comparison of the magnitude
of displacement meaningless.
the Oos ter schelde actual ly cons ists o f s evera 1
sands of which one is Oosterschelde sand A. Limited
tests on qpnd C showed much larger permanent strains
than sand A for similar conditions. Permanent
displacements predicted with parameters from sand C
much larger, as shown by Ma rr andare
Christian�981! ~
Prom the comparison between the centrif uge
displacements and the displacements predicted with the model
we conclude:
a! both methods present similar patt erns of
displacements, with an increase in horizontal
220
6.6
Th ree s eries of two-dimensional analyses were
performed. The f irst series investigated the effects of
porosity, cyclic stresses and initial stresses on the
permanent displacements of a flexible mat. Results f rom
this f irst series, presented in non-dimensional form, showed
the importance of soil conditions and initial stress
conditions on the accuxnulation of permanent displacenents.
The second series investigated the accumulation of
d isplacements for the Oosterschelde closure, and compared
the results with the permanent displacements predicted by
Marr and Christian �981! . The linear viscoelastic method
predicts larger permanent horizontal displacement and
permanent settlement than Narr and Christian �981! . The
trends of accumulation of displacements are similar.
The third series compared the results obtained with the
viscoelastic model with those obtained using the centrif uge
f or the Oosterschelde barrier. The predicted horizontal
displac ements and sett 1 ements are th ree to f ive t imes
smaller than the displacements measured in the centrifuge.
Neverthel es s, th e v iscoel as tie model predicts th e s arne
pattern of permanent displacelnents as the centrifuge for the
f our cases a nalyz ed.
EffeCt Of porOsity: V/V = .2; H/H = .05; e/B = 0max max
n = 39m; n = 43%cl ' c2
Effect of eccentricity: V/V = .2; H/H = .05;max max
n = 4la e/B! = 1/12;c 1
e/B! = 1/62
LOJLD COMBINATIONS FOR FLEXIBLE FOOTING
TABLE 6-1
V = 20 T/mlmax
H = 5 T/mlmax
221
n = porosity = 41%c
eccentricity = e/B = 0
222
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Static Head Loss and C clic Nave Load Combinations
C clic Nave Load! faxijman LoadsCase
�0 N! Nwaber o~ Haves
Design Max head loss
Max wave load
76.8
70.2
1/2 Nax wave 35. 1
1/2 Max head loss, 38.4LargeNave
No head loss
Max wave load
0.0No
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225
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W t»tC»C4 0ft WCOCOCd +X . CdCd Cd C-IO>P
Cd
C4 OC>C» CII C»WC4 gdAt Cd
4 QCay Nl I»
0
O Cdc» a
NI X
l4X0
230
V~= 2Bq~
H
uesictl975! g�it ~,, ~ > it�8!st</>$;
f>= I for strip footing
tc; = i- Hw]s
BEARING CAPACITY FOR FLEXIBLE FOOTING
F IGURE 6-2
231 LLJ J
C9
O O UJKl
XLJJ
U LLJX LLJILJ
ILJI-
U
PI GURE 6-3
232
a Q
Q XCl
Z
Cl
UJ
8.7 S.e HS
Q X
Ih
X/B
PERMANENT DISPLACEMENT BENEATH FOOTING, hH/HSERIF@ V/V = 0.2!
max
O
M
ikey
18' Sdt 83. HD 83 H.+ 85 86 87 88 SS 18X/S
1'
1H'
8.8 SX 8% 8.3 H.< 8.5 8.6
1'
S.S 8.1 8.2 8.3 8.4 a5 a6 8.7 8.8 as
233
a
18
a
Cl
QX
a
ai 8Q a3 BA 8.5 aB 8.7 8.8 a9 185
X/8
PERMANENT DISPLACEMENT BENEATH FOOTING, hH/HSERIES V/V = 0. 1!
KRXFIGURE 6-5
IO
QX0!
8 8 ex m ez ai as ae e.~ B.e 8 9
X/8
BS a1 8Z 8.3 8.4 K5 8 $8.7 8.8 8.9
X/8
234
n
QX
R I-
D H I-LLt
1'
18
8.1 HD 8.3 H.+ 8.5 8.6 8.7 H.B 8.9 1Z
QX Ha
XfBPERMANENT DISPLACEMENT BENEATH FOOTING, M/H SERIES V~ = OP5!
FIGURE 6-6
18
1'
1H
H.H 8.1 HZ HB 8.4 8.5 8.6 8.7 H.B HB 1.8
235
Q XR
5 i518
8 8 au. aZ aa 8.+ 8.5 8+ 8Z 8 e 8+ 18
FIGURE 6-7
9 X
I-
bl
H D 5
18
ilia
8.1 8.2 8.3 8A S.S 8.6 8.7 8.8 8.9
PERMANENT DISPLACEMENT BENEATH FOOTING, POROSITY SERIES
236
IOXCl
lH'
n O
FIGURE 6-8
D H I�5
C K5
O
i5
H.l 8.2 8.3 HA 8.5 H.B 8.7 H.B HB 1.8
HA 8.5 8.6 8.7 H.e 8.9 L.H
X/B
PERMANENT DISPLACEMENTS BENEATH FOOTING, ECCENTRICITY
SERIES
237
1'
18 8 0 H.i 8.2 8.3 SA S.S 8.6 8.7 8 8 S.B 1.8X/B
1fiP
18' 8.8 8 1 SZ S.B SA 8.5 S.B 8.7 8 8 8.9 1.8X/EI
PERMANENT DISPLACEMENTS BENEATH FOOTINGS V/V SERZESIABX
FIGURE 6-9
238
NNBER OF CYCLESRATIO CF EDGE VERTICAL DISPLACEMENT TO CENTER VERTICAL
DISPLACENEHT, V/V SERIESBI& X
FIGURE 6-10
239
18 8.8 8 1 8Z 8.3 8.4 8.5 8.6 8.7 8.8 8.9 1 8X/8
18
ilia
18 88 8J. 8Z 83 8+ 85 86 8.7 88 8.9 1JVX/8
PE~NT DISPLACEHENTS BENEATH FOOTXNG, hH/H SERIESmax
F IGURK 6-1 1
240
i8 i$Pt4PSER OF CYCLES
f.5
t4PSER OF CYCLES
RATIO OF EDGE VERTICAL DISPLACEMENT TO CEHTER VERTICAL
DISPLACEMENT, hH/H SERIESHLRX
PIGURE 6-l2
241
i8
aP
J 8.8 8.1 8% 8.3 8A 8.5 8.6 8.7 8.8 8.9 2.8X/8
18
LliP
1' 8.8 8< 8> 83 8,+ 8.5 8~ Sr 8.e 8Z >8X/B
PERNtANENT DISPLACEMENT BENEATH FOOTING,
POROSITY SERIES
FIGURE 6-13
242 W A W Ei0 N
a 8 0 0 I-I
FIGURE
243
18 88 81 82 83 8.4 8.5 86 8.7 8.8 8.9 18X/8
18
1'
18 88 81 SX 8a e.a a5 a.a 8.7 8.e 8.9 18xre
PERMANENT DISPLACEMENT BENEATH FOOTING, ECCENTRICITYSERIES
FI GURE 6-l 5
5
0 0 0 MFIGURE 6-16
245
C
SlU
Ei
0 0
sexvAX53 FIGURE 6-1 7
246
UJ
Z LLjUJ
CO
CA
OR O
O Ch0
FalO
C4LLjK'
0 0
O UJO 4J
O O
FI GURR 6-l 8
247
CJ
U
FI GURE 6-1 9
QJ
O O X
LU
3
44CO
R
O R
248
DESIGN CASE
2 cNI
LARGE WAIE
DEHSIFIED HOLOCEOK
SAhS
+ IO CYCLES
O 60 CYCLES
Ch 600 CYCLES
CALCULATED PERMANENT DISPLACEHENTS FOR DENSIFIED HOLOCENE
SAND, NUNBER OF CYCLES SERIES
FIGURE 6-20
249
DESIGN CASE
NO T
2 cps
LARGE TOE
LARGE WAVE
UNDENSIFIED HOLO
SAND
+ 0 CYCLES
0 60 CYCLES
600 CYCLES
CALCULATED PERMANENT DISPLACEMENTS FOR UNDENSIFIEDHOLOCENE SAND, NUMBER OF CYCLES SERIES
FIGURE 6-21
250
OENSIFIED HOLOCENE SAND
UNDENSIFIED HOLOCENE SAND
2 asia
DESIGN CASE
0 NO TIDE
0 LARGE TIDE
+ LARGE WAVE
N=600 CYCLES
CALCULATED PERMANENT DISPLACEMENTS, LOAD COMBINATION SERIES
FIGURE 6-22
251
DESIGN CASE
0 0: O'' ~ O. ~ o'o > -'. o '~..o
NO
2 cms
LARGE TIDE
LARGE WAIE
+ DEHNFKD HOLOCKhK
SAIN
o UNOENSIFKO HOLOCEtK
SAIC
CALCULATED PERMANENT DISPLACEMENTS. DENS IFIED VERSUS UNDENSI-
F IED HOLOCENE SAND
FIGURE 6-23
252 V!lh4J
4JC9
I-
LLI
O K O COX O M R hJO D 4J4
FIGURE 6-24
253 lhI�
LLII-
UJ 9
U RhJC3
O Z LLIX
FZGIJRR 6-25
254
Torei newer o< eye>cs, s
CENTRIFUGE LOADING PROGRAM
Rowe and Crai.g, 1978!
FIGURE 6-26
255
CHAPTER VII
S~ary and Conclusions
7 ~ 1
A.n analytical method was developed to predict the
permanent displacement of soils that res ults f rom cyclic
loading under drained conditions.
The method is bas ed on a 1 inear v iscoelast i c
formulation of the problem in which "time" is replaced with
the number of cycles of loading experienced by the soil
el ement .
The advantage of the viscoelastic model lies in its
capacity to give a general set of stress-strain-number of
cycles relationship which are adaptable to a variety of
boundary and loading conditions us ing parameters determined
f rom a f ew, relatively simple cyclic tests. The assumption
of isotropic, homogenous behavior allows two parameters, a
bulk compliance and a shear compliance, to completely define
the general ized three-dimensional linear viscoelast.ic
stress- strain- nunber o f cycles relat ions . Th ese two
compliances are a function of stress path parameters and are
determined f rom cyclic laboratory tests.
256
- For Oosterschelde sand A, the two compliances are
derived from drained cyclic compression triaxial tests ~
These compl iances show a strong ef f ect of stress path
parameters, soil porosity and cyclic shear stresses.
The model is evaluated under one-dimensional strain
boundary conditions Mch provides a stress system dif f erent
from triaxial conditions. A series of cylic oedometer tests
were performed in which measurements of both accanulation of
vertical deformation, and lateral stress were monitored.
Cumulative strain predictions made by the model match
the test results in approximate magnitude and trend. The
model is able to predict the shallower slope of the
log-strain-log cycle relationship produced in the cyclic
However, the model overpredicts theoedometer tests.
stresses by as much as 7G% ~
settlements and horizontal displacements than the Marr and
Christian approach.
Th ree s eries of two d imens ional analyses were
performed ~ The f irst series assessed the relative
importance of stress path parameters on the magnitude and
pattern of permanent deformations of a cyclically loaded
f 1 ex ibl e f o undat ion. Th e s econd s e r i es i nv es t ig at ed the
acccumulat ion of displacements for the Oosterschelde
closure, and compared the results with the permanent
displacements predicted by Narr and Christian �981! . The
linear viscoelastic method predicts larger vertical
257
In the last series, results obtained with the model
were compared to those obtained using the centrif uge, for
the Oosterchelde barrier. In all cases, the p red icted
horizontal and vertical displacements were three to f ive
times smaller than the displacements measured in the
centrifuge. Despite the difference in magnitude between the
predicted and measured values, the linear viscoel*stic
approach was able to predict the same trend of accumulation
of permanent displacements as the centrif uge for the four
cases analyzed.
The small permanent displacement predictions occur in
large part because the permanent displacement compliances
parameters come f rom laboratory tests on one Oosterschelde
sand with a relatively high resistance to cyclic loading.
7. 2 Conclusions
a. � A linear viscoelastic formulation with parameters
from stress path type tests promises a rational means of
solving the dif f icult problem of predicting permanent
displacements from cyclic loading with a suf f icient degree
of accuracy for most practical purposes.
b. � In evaluating the formulation for one dimensional
strain conditions, strains predicted f rom the model with
parameters from triaxial tests agree with measured strains
both in magnitude and rate. Predicted versus measured
horizontal stress does not agree.
258
c. � In evaluating the formulation for two-dimensional
conditions the patter~ and mechanism of displacement appear
reasonable bas ed on three s eries o f analys es: parametric
studies of a cyc1 ical ly loaded fl ex ibl e f oundat ion,
comparison with Marr and Christian' s approach and comparison
with centrifuge results.
d. � The two dimensional series of analyses showed
that the initial stresses, cyclic shear stresses and
porosity significantly affect the magnitude and pattern of
permanent displacements, and that they interact in a very
complicated way that can not easily be inferred.
7 ~ 3 Future Research
Two areas for f uther research are identif ied:
l . � Exper imental analys is:
Cyclic laboratory tests. These cyclic laboratory tests
would include the ef f ect of other stress systems on the
response of soils to drained cyclic behavior. Anisotropic
cyclic s imple shear tests and drained extension triaxial
tests will provide the model with parameters from other
stress systems. With these parameters approximate effects
o f stress anisotropy can be included into the linear
viscoelastic approach.
259
2. � Extension of the method:
The following should be studied in more detail:
a. � For one dimensional strain conditions:
comparison of the viscoelastic model predictions
with those from a different model such as that of
Prevost �977,1978!, Mroz et al �978.1979! i etc.
check the one 0 imens ional horizontal stress
measurements with a different laboratory test, such
as the centrif uge.
b. � Further comparison of predictions f rom the
viscoelastic model with measured performance, particularly
in model and f ield tests.
c ~ � Expand th e f o rmul at ion to cons ider undrained
cond it ions ~
d. � Modifications of the computer program to include:
overstressed elements
irregular cyclic loading or parcels of cycles of
dif f erent amplitude.
e. � Extension o f the method to include other stress
paths tests, particularly cycling such that the mean shear
stress becomes zero cyclic shear stress reversal!, the
effects of the inclination of the cyclic stress path and
260
cycling with a negative mean shear stress extension
conditons! .
f . - parametric stud ies using diff erent footing
rigidities.
g. � extens ion of the analysis for other foundation
systems i. e.piles! .