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

0

F4

CcI0

Cd

Z 0ChCh

0 lbCd

M 0 ChrJ

IOf4

le

'll

I

0Vl

1W

4g

I IJ

0 0lFl IPl

I ILl

0

0 CcI0 I4

% QCl Z

CLI W <Z % Cd

O Cd lgO C; Z

tll h3Cd WC>H 0CLC W A

0 0C4 4 tL

0 Cd

0 O g5

Ill LPGA 0 QN

I ~ I I

4J

0rkg0CL

800

0 ~Ch Qlh Q ~ %

ChI I NIg

IJ

I I I

gh

V

C4

08Wqj

O IIZ

H 0Cd Cj C4

CCI

'U

pC hlCh

0 00 4

0

0 IV% CD

Cd W

W CDCh

III IM'W IN

0 Cdg 04 0

223

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

Tide70. 2

TABLE 6-3

Large ~iax head lossTide +1/2 Max wave load, 111.9

hiax Nave 108. 6+1/2 Max head loss

15 ~ 925.3

36.548.761.870. 2

8.012.718.324.530.9

35.1

24.6

39.156.575.3

95.6108.6

15 ~ 925.336.548.761.870,2

284

187

90

31

9 2

284

187

90

31

9 2

284

187

90

31

9 2

284

187

90

31

9 2

224

ea

CL0

rt! a

n 0 Qo

0 O

W Cl

I 0 e 4I0

W 0

0IA

+ 0I g4

0U'

CV

+

+0

+ I

o

0 a!

gOgW

0ml

TABLE 6.4

225

0 0

0 00

~ Q rIPI

. IA~ + ~

~ NA CV

FrlA

'0 Cl«I

0 C40

hr

0 oD

0 W 0 N h'ul ~ 0! O

g ~ Q r ~0 N N

IA

A A M

A

8H Ch

Ch0 hr0

0Crl Kl

0

CaT«h

WCh

0

Cir

g! Q

Q 0 tho o ~

N0 Ill

O % N

0 e 0

r Q ~

0o ~

~ Q ~ ~N N

Irl 0 Vl nfl«h

N

e

~ «h

Ch ~ 0~ ~ Ce

L«l

0hC A

K CcrCa <

aa

w 0 o «h~ Q Q ChCh m 0 WCh Ccl <

A F4 oR C4

o AH 4 ChQLl~A AÃ%Ca Q<O+gH«h

pC~O«hch

Ch CCC 00<o0Ch AW Calpc oQ 'V

Ch

0 >o

226

ICI

LI

an

I

A ccI ICW ~

CCI Ã

LI CIIA

hl

Q ROW

O ChtcI ch

5

4 cclW

W ACh 'gg

Ll cd4

0 cC014

0

a O Oa

'0 CINl

0 C40

LI

tel

&4

O W CCO gl

a al~ rt4JIb

O

Ihc

A

K

fal ClINg

O ga

& coK mLlX <Ll OCJW ea

4 a

f4

A H 4' a

Ch W M O

a

Ch C! ChO4

CV

gl O 4

cO pC& DC

Ll XXLl X0 aQ CC

gayC3 Ã

cn 4 < OW R a Ul4 < CC rn

+ A Dc 4< LI Q C7H AROOSLl

g r 4hZ 'R AW H Q Llg4 W Was ~ogg M M ch

4 W OC/ca A OC IEIa IQ cc: R

Oucnev a0+AOCII W g ~w DOCW XI4 OX NIa I4O

227

0 0 O 0O 0O ~ Cl

LPI O chpg

O &

O N

CA

o '8

Ig O

0 04CP

a

lA

228

Cd OXmCd C»U w

4 O

C» Cd

a ~

0

04 acd a

Ã

a K Cd

0 DCoC»

C4C»t4a

00 54PC

Ieel

C4

Ie

ee

lA

~ n IO 00

tV ee

O 04

O H hl

O

e aIN4

44tllQ S

44 08

CtIt

. 44l5CI

It4t

II CICI c»eU Wqj

444

ctt

CC CJ

'CP

O 4»

~ e&

+ CACdgg

O alOC Cd 44

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! .