On the determination of cyclic strain history

Th. Triantafyllidis, T. Wichtmann & A. NiemunisInstitute of Soil Mechanics and Foundation Engineering, Ruhr-University Bochum

ABSTRACT: The information about the void ratio and the stress is not sufficient to predict the deformationaccumulation rate under cyclic loading of definite amplitude. The rate of accumulation is significantly affectedby cyclic preloading. For an accurate prediction of the accumulation the determination of cyclic preloadinghistory is of high importance. In this paper we search for appropriate methods to determine the preloadinghistory in situ. It is demonstrated, that a cyclic preloading cannot be correlated with small strain stiffness. Acorrelation with material damping or the undrained cyclic material behaviour is somewhat more promising. Analternative approach for shallow layers using heavy vibration is proposed.

1 INTRODUCTION

The importance of cyclic preloading history concern-ing the accumulation rate is illustrated in Figure 1.The reduction of void ratio e in three cyclic triax-ial tests with identical average and cyclic loads butslightly different initial void ratios e0 is presented.Considering a state of identical void ratio (marked bythe solid horizontal line) the accumulation rate ė =∂e/∂N of the freshly pluviated specimen is higherin comparison with the rates of those specimens, thathad already been subject to several thousands of loadcycles. Thus, at the same average stress, amplitude ofcyclic loading and void ratio cyclic preloading signif-icantly affects the cumulative behaviour of a soil.

Niemunis et al. (2004) and Wichtmann etal. (2004a,b) present an explicit constitutive modelfor sand under cyclic loading and its experimental ev-idence. The formulas were derived from laboratorytests performed on freshly pluviated specimens. How-ever, an accumulation under cyclic loading in situ isslower due to a cyclic preloading history from seismicactivity, sedimentation and erosion processes (in thecase of natural deposits), construction site machinesor mechanical densification processes (especially inthe case of artificial deposits). Thus, using the consti-tutive model and starting the calculation with N = 0(corresponds to the freshly pluviated laboratory spec-imen without any cyclic preloading) could stronglyoverestimate the accumulation of deformation or thepore water pressure development. Therefore, it is ofcrucial importance to determine the cyclic preloadinghistory of a soil and to introduce it into the calcula-

tion.This paper presents our search for methods to de-

termine the cyclic preloading history. A correlation ofcyclic preloading history with the small strain stiff-ness (dynamic soil property, measurable in situ crosshole or down hole) was tested. The theoretical basisfor such correlation is given. Unfortunately, no exper-imental validation of the hypothesis could be obtainedfrom the results of resonant column tests and fromcyclic triaxial tests with measurement of small straincompression and shear wave velocities. It was foundthat the small strain stiffness is sparsely affected bycyclic preloading.

0.61

0.62

0.63

0.64

0.65

0

Number of cycles N [-]

Voi

d ra

tio e

[-]

identical void ratioidentical stresse1 e2 e3

1052 104 2 104 2 104 2 104

e1 > e2 > e3

Test No 1: e0 = 0.629Test No 2: e0 = 0.636Test No 3: e0 = 0.644

all tests: pav = 100 kPa, qav = 77 kPa, σ1 = 55 kPa

ampl

Figure 1. Influence of a cyclic preloading history on theaccumulation rate ė in drained cyclic triaxial tests

A correlation of cyclic preloading with materialdamping is outlined. Furthermore, it is demonstrated,that cyclic preloading significantly affects the cyclic

undrained material behaviour and thus, for saturatedsoils the cyclic preloading history could possibly bedetermined in situ via measurements of pore waterpressure development in cone penetration tests.

For shallow soil layers it is proposed to measurethe settlement and settlement accumulation of aheavy vibrator on the surface after a number ofexcitation cycles. The cyclic preloading history couldbe calculated back from the measured curve of thesettlement over time.

2 CORRELATION OF CYCLIC STRAIN HIS-TORY WITH DYNAMIC SOIL PROPERTIES

2.1 Motivation

Cyclic loading changes the fabric of an assemblage ofgrains, i.e. the number of particle contacts, their ori-entation and shape are varied. With increasing num-ber of cycles the soil fabric becomes more resistantagainst further accumulation. This is due to an in-crease of the coordination number (number of con-tacts per grain), smaller spatial fluctuation of grain-to-grain forces etc.

1

1

4

6

σ (log scale)σ *

hertzian contactsphere - sphere

contact conus - sphere

E (

log.

sca

le)

F'

S'

S

Fα

Figure 2. Comparison of stiffness of contact types ”sphere-sphere” and ”conus-sphere”

The stiffness E and the elastic energy W of an elas-tic contact of two ideal spheres with an equal radiusR loaded by an axial force F was derived by Hertz(1881):

E =3

2

[

2Ḡ

3(1− ν̄)

]2

3

σ1

3 (1)

W =4

8

3

5

[

3(1− ν̄)

8Ḡ

]2

3

R4

3 σ5

3 (2)

with Ḡ and ν̄ being the shear modulus and the Pois-son’s ratio of the sphere material. For a simple cubepacking σ = F/D2 is the stress in the axial direction.Goddard (1990) derived the corresponding modulus

and energy of a contact of a conus (asperity α) and asphere:

E =

(

Ḡ

1− ν̄

)1

2(

6

πα

)

1

2

σ1

2 (3)

W =4

3

2

3

[

3(1− ν̄)

8Ḡ

]1

2

(πα)1

2 R3 σ3

2 (4)

Goddard assumed Equation (3) to be valid in the caseof pressures σ below a transition pressure σ∗ whichcan be deduced from Equations (1) and (3):

σ∗ =1

96

Ḡ

1− ν̄π3 α3 (5)

For σ > σ∗ Equation (1) is valid. This is shownschematically in Figure 2. The curve F’-F is cal-culated from Equation (1) and S’-S corresponds toEquation (3). Goddard thought of Figure 2 as a kind of”thermodynamic” phase diagram with the more sta-ble phase being represented by the curve F’-F (con-tact sphere - sphere) and with S’-S (contact conus -sphere) being a metastable phase. The transition pres-sure σ∗ is strongly dependent on the asperity α, seeFigure 2. Goddard assumed that during vibration therelatively soft contacts of the type ”conus - sphere”are replaced by stiffer and more stable Hertzian con-tacts due to particle re-orientation and abrasion ef-fects. The possible increase of stiffness due to a tran-sition of the shape of a particle contact is indicated bythe vertical arrow in Figure 2.

Changes of stiffness and energy of an assemblageof ideal spheres caused by an abation of spatial stressfluctuation are demonstrated by the simple examplepresented in Figure 3. In case (a) the whole exter-nal force 2F is carried by one column and in case(b) it is distributed evenly. Assuming a Hertzian con-tact law the relations Eb/Ea = 2

2

3 (Et: stiffness forcase t corresponding to Fig. 3) and Wb/Wa = 2−

2

3

are valid. Since a particle assemblage should spon-taneously tend towards the state of the lowest possi-ble energy it is expected that vibration of dry particlesleads to a reduction of stress fluctuation and thus, to ahigher stiffness.

A more elaborate study concerning energies re-sulting from different stress fluctuations in particleassemblages using the ”q-model” of Coppersmith(1996) is documented by Triantafyllidis & Niemunis(2000).

These theoretical considerations were supported bytest results of Drnevich & Richart (1970) who foundthe small strain shear modulus G0 to increase stronglyunder a repeated torsional shearing in the resonantcolumn device (Fig. 4). However, the literature is not

2F

F F

2F

2F

a) b)

Stiffness: Eb = 1.58 EaEnergy: Wb = 0.63 Wa

Figure 3. Reduction of stress fluctuation as a cause of stiff-ness increase and energy decrease

unanimous concerning the influence of cyclic strainson small strain stiffness. While some authors (e.g.Drnevich & Richart 1970, Shen et al. 1985) found astiffness increase some others did not (e.g. Alarcon-Guzman et al. 1989, Lo Presti et al. 1993, Teachavo-rasinskun et al. 1994). A more detailed review is givenby Wichtmann & Triantafyllidis (2004a,b).

10-5 10-4 10-30

0.5

1.0

1.5

2.0

2.5

3.0

Shear strain amplitude γ [-]

G(γ

) / G

0(N

=0)

[-]

N = 22 106N = 10 106N = 1 106N = 6 103N = 0

p = 112 kPaID = 0.86

γprestrain = 6 10-4

Figure 4. Increase of small strain stiffness G0 due totorsional prestraining in Resonant Column tests afterDrnevich and Richart (1970)

2.2 Resonant Column (RC) Tests

The effect of a prestraining history on shear modulusand material damping was studied in resonant columntests. Details of the device and the specimen prepara-tion method (dry pluviation through air) were givenby Wichtmann et al. (2001). Some test data concern-ing the dependence of the dynamic properties of thesands under consideration on stress, void ratio andtime can be found in Wichtmann & Triantafyllidis(2004a,b).

One fine and two medium grained sands wereused in this test series documented in this paper.The grain distribution curves are given in Figure 5while the minimum and maximum densities as wellas some characteristic quantities of the grain distri-

0.002 0.006 0.02 0.06 0.2 0.6 2 6 20 600

20

40

60

80

100

Silt Sand Gravel

Diameter [mm]

Fin

er b

y w

eigh

t [%

]

Medium sand 1Medium sand 2

Fine sand

Fin

e

Med

ium

Cou

rse

Fin

e

Med

ium

Cou

rse

Fin

e

Med

ium

Cou

rse

Figure 5. Grain distribution curves of tested materials

Sand %s %d,min %d,max d50 U[

g/cm3] [

g/cm3] [

g/cm3]

[mm] [-]

Fine sand 2.65 1.453 1.753 0.11 1.7

Medium sand 1 2.65 1.437 1.715 0.51 2.0

Medium sand 2 2.65 1.414 1.680 0.55 1.8

Table 1. Minimum / maximum densities and grain sizecharacteristics of used sands

bution curves (mean diameter d50, uniformity indexU = d60/d10) are summarized in Table 1.

In a first test series a dynamic torsional prestrainingwas applied to full (not hollow) cylindrical specimensin the RC device. The tests were conducted on the finesand and the medium sand 1 with a medium density(0.62 ≤ ID0 ≤ 0.68). Two different prestraining am-plitudes (γprestrain = 5 · 10−5 and γprestrain = 1 · 10−4)and different maximum numbers of cycles Nmax werestudied.

After the application of the confining pressure theshear strain amplitude γ was increased up to γprestrainand a definite number of cycles was applied. Next theshear strain amplitude was reduced to the lowest pos-sible value in order to measure the shear modulus atsmall strains G0 = G(γ ≈ 10−6). Having reached thedesired maximum number of cycles Nmax the curvesG(γ) and D(γ) were measured over the whole rangeof applicable shear strains in order to observe changesof the shape of these curves due to cyclic torsionalloading.

Figure 6 presents typical curves of shear modulusand damping ratio versus shear strain amplitude. Af-ter the first package of cycles a slight reduction ofG0 in comparison with the value of a non-prestrainedspecimen was observed. During the subsequent tor-sional prestraining only a slight increase of G0 couldbe detected (see also Fig. 7). At N = Nmax the curves

0.07

0.06

0.05

0.04

0.03

0.02

0.01

Dam

ping

rat

io D

[-]

She

ar m

odul

us G

[MP

a]

10-6 10-5 10-4 10-3

Shear strain amplitude γ [-]

10-6 10-5 10-4 10-3

Shear strain amplitude γ [-]

200

180

160

140

120

increase of � over γprestrain

increase of � over γprestrain

developmentof plateau

first increase of �

first increase of �

re-increase of �

re-increase of �

reduction of �

reduction of �

developmentof plateau

prestraining amplitude

�prestrain

prestraining amplitude�prestrain

a)

b)

Figure 6. Development of plateaus in the curves a) G(γ)and b) D(γ) due to dynamic prestraining in the RC device,test on dry fine sand with Nmax = 3 · 106 cycles, confiningpressure p = 200 kPa and initial relative density ID,0 = 0.64

G(γ) and D(γ) exhibited a plateau around the pre-straining amplitude γprestrain. These plateaus were al-ready reported by Li et al. (1998a,b, 1999). They be-come more pronounced with increasing prestraining,i.e. with the number of cycles and the prestrainingamplitude (Wichtmann & Triantafyllidis 2004a). Evi-dently, sand memorizes not only the prestraining am-plitude but also the number of applied cycles. If pack-ages of cycles with different amplitudes γprestrain wereapplied, the sequence of these packages was of im-portance for the shape of the resulting curves G(γ)and D(γ) (Wichtmann & Triantafyllidis 2004a). Achange of stress after prestraining blurred the signa-ture of cyclic strain history (Wichtmann & Triantafyl-lidis 2004a). No significant differences in the materialbehaviour could be detected, if hollowed cyclindri-cal specimens (shear strains more evenly distributedover the cross section) were used (Wichtmann & Tri-antafyllidis 2004a).

The strain amplitudes during torsional prestrainingin the resonant column device were relatively small.In an additional test series hollow cylinder specimenswere cyclically (0.5 Hz) prestrained at γprestrain ≥ 5 ·10−3 in a special torsional shear device. They had dif-

a)

b) 1.2

1.1

1.0

0.8

0.7

0.9

1.2

1.1

1.0

0.8

0.7

0.9

104 105 106 107

104 105 106 107

Number of cycles N [-]

Number of cycles N [-]G

0(N

) / G

0(N

=0)

[-]

G0(

N)

/ G0(

N=

0) [-

]

p = 200 kPa, ID0 = 0.63 - 0.67

�prestrain = 1 10-4

all tests:

p = 200 kPa, ID0 = 0.62 - 0.68

�prestrain = 1 10-4

all tests:

Fine sand

Medium Sand

Figure 7. Evaluation of G0 with the number of cycles dur-ing dynamic torsional prestraining with γprestrain = 10−4 inthe RC device, tests with different maximum numbers ofcycles Nmax

ferent initial densities and the number of prestrainingcycles was varied. After prestraining the specimenswere placed into the RC device in order to measurethe small strain stiffness G0 as well as the curves G(γ)and D(γ). Figure 8 presents the shear moduli of spec-imens prestrained with γprestrain = 5 · 10−3 versus thevoid ratio e after prestraining. The test data of non-prestrained samples is shown as black filled circles.The shear moduli of the samples subject to 100 pre-straining cycles are all smaller than the mean value ofthe non-prestrained samples. In the case of Nprestrain =50,000 all values of G0 are larger than the ones of thenon-prestrained samples. However, the stiffness in-crease after 50,000 prestraining cycles did not exceed20 % of the stiffness of the non-preloaded specimensalthough a significant densification was observed (εaccvup to 6.3 %). Hardly any changes could be detectedconcerning the curves G(γ)/G0 and D(γ).

2.3 Cyclic triaxial tests with measurement of P- andS-wave velocity

In cyclic triaxial tests the development of the com-pression and the shear wave velocity vp and vs andthe corresponding stiffnesses at small strains Es0 andG0, respectively with the number of cycles was stud-ied. The test device is shown in Figure 9. The spec-imen end plates are instrumented with three kinds ofpiezoelectric elements, i.e. one pair of compressionelements (P-wave), one pair of bender elements (S-

Void ratio e [-]

She

ar m

odul

us G

0 [M

Pa]

140

120

100

80

60

400.54 0.58 0.62 0.66 0.70 0.74 0.78

non-preloadedsamples

Nprestrain1001,00010,00050,000

=

all tests: hollow cylindermedium sand 2, p = 80 kPa, γprestrain = 5 10-3

Figure 8. Influence of a cyclic torsional prestraining onsmall strain shear modulus G0

Wave) and one pair of shear plates (S-Wave). Wavevelocities can be measured in the axial direction ofthe specimen. Details of the test device can be foundin Wichtmann & Triantafyllidis (2004b). The termi-nology of the cyclic triaxial tests is given by Wicht-mann et al. (2004a). An illustration to the tests is pre-sented schematically in Figure 10. After the applica-tion of the average stress σav and consolidation (1 h)the wave velocities vp and vs were measured. Next thefirst package with 10 cycles was applied to the spec-imen. The wave velocities were measured again andthen the cyclic loading was continued. In this way vpand vs were measured at σav = const. after differentnumbers of cycles.

vsvhvpv

compressionelement (CE)

F + � F-

ball bearing

sealing ofload piston

load cell

plexiglascylinder

load cell

soilspecimen

shear plate (SP)bender element (BE)

water

air pressurepiezoelectricelements:

Figure 9. Triaxial device with piezoelectric elements

In the first test series the initial density (0.59 ≤ID0 ≤ 0.61) and the stress amplitude σ

ampl1 = 60 kPa

were held constant while the average stress σav wasvaried (100 kPa ≤ pav ≤ 200 kPa, 100 kPa ≤ qav ≤200 kPa). The dry medium sand 2 was loaded with100,000 cycles at 1 Hz. Figure 11 shows the develop-ment of the small strain stiffnesses Es0 und G0 withthe number of cycles. In order to separate the influ-

t,N

σ1

σ1

σ1

σ1....

Measurement of vs and vp

ampl

amplav

Figure 10. Proceeding of cyclic triaxial tests with measure-ment of compression and shear wave velocities

ence of fabric changes from the effect of void ra-tio reduction the stiffnesses were normalized by voidratio functions F (e) (Wichtmann & Triantafyllidis2004b). Although a significant accumulation of strainεacc = ‖εacc‖ (especially in the case of pav = qav = 100kPa, Fig. 12) was observed only moderate changesof the small strain stiffnesses could be detected. Inno test stiffness changes during cyclic loading with100,000 cycles exceeded 15 % the initial stiffness.

0.9

1.0

1.1

1.2

[

Es0

/ F

(e)]

/

[Es0

/ F

(e)(

N=

0)] [

-]

0.9

0.8

1.0

1.1

G0

/ F(e

)] /

[G

0 / F

(e)(

N=

0)] [

-]

Number of cycles N [-]100 101 102 103 104 105 106

pav = 200 kPa; qav = 100 kPapav = 200 kPa; qav = 200 kPapav = 200 kPa; qav = 150 kPapav = 100 kPa; qav = 100 kPapav = 150 kPa; qav = 100 kPa

all tests:

�

1 = 60 kPa, ID0 = 0.57 - 0.59

ampl

Figure 11. Development of Es0 and G0 with the numberof cycles in cyclic triaxial tests with varying average stressσ

av

Wichtmann & Triantafyllidis (2004b) presentedthree other test series. In the first series the averageand cyclic stresses were held constant (pav = 200 kPa,qav = 150 kPa, σampl1 = 60 kPa) while the initial densitywas varied (0.29 ≤ ID0 ≤ 1.05). In the second seriesthe average stress (pav = 200 kPa, qav = 150 kPa) andthe initial density (0.58 ≤ ID0 ≤ 0.61) were held con-stant and different stress amplitude (30 kPa ≤ σampl1 ≤= 90 kPa) were tested. In a last series of test, addi-tional tests on fine sand were conducted. In neither of

p = 200 kPa, q = 100 kPap = 200 kPa, q = 200 kPap = 200 kPa, q = 150 kPap = 100 kPa, q = 100 kPap = 150 kPa, q = 100 kPa

0

1

2

3

4

Number of cycles N [-]100 101 102 103 104 105 106

ε

[%

]ac

c

all tests: σ1 = 60 kPa, ID,0 = 0.57 - 0.59

ampl

Figure 12. Accumulation of strain εacc with the number ofcycles N in cyclic triaxial tests with varying average stressσ

av

these series the stiffness increased after 100,000 cy-cles more than 10 % of the initial value.

The resonant column tests on cyclically prestrainedspecimens as well as the cyclic triaxial tests withmeasurement of compression and shear wave veloc-ities did not show any correlation between cyclicpreloading and small strain stiffnesses. Thus, adetermination of cyclic preloading history in situ viasmall strain stiffnesses does not seem to be feasible.

3 CORRELATION OF CYCLIC STRAIN HIS-TORY WITH MATERIAL DAMPING

In the cyclic triaxial tests documented in subsection2.3 loss of intensity of the received signals of benderelements and shear plates with the number of cycleswas observed while the intensity of the compressionelement signal stayed almost constant (Wichtmann &Triantafyllidis 2004b). The amplitude of the emittedsignal (a single sinus impulse) was not changed dur-ing the test. The reduction of the received signal en-ergy with the number of cycles could be caused byan increased material damping. However, a change ofthe bedding of the elements could also be a possiblecause of theses observations.

Four additional cyclic triaxial tests were performedon medium sand 2 studying these effects. The ex-periments concerned bender element measurementsas such they produced the clearest signals at the re-ceiver. While the average stress (pav = 200 kPa, ηav =qav/pav = 0.75) was held constant the initial density(0.59 ≤ ID0 ≤ 0.90) and the stress amplitude (30 kPa≤ σampl1 ≤ 90 kPa) were varied.

Figure 13 presents the received signals in a test withσampl1 = 60 kPa and ID0 = 0.59 after different numbersof cycles N . Although the source signal was a singlesinus impulse the received signal contained severalcrossings of the zero axis due to multiple reflections

at the sample boundaries and probably also due to in-ternal inhomogeneities. The signal intensity grows upto N = 5,000 and afterwards a strong abation was ob-served. The double amplitude was defined as the spanbetween the total minimum and the total maximumof the signal as demonstrated for N = 0 in Figure 13(keeping in mind, that this measure is influenced bythe boundary reflections).

Figure 14 presents the receiver signal amplitudeversus N in the four tests. In the tests with smallstrain accumulation (εacc ≈ 0.1% for N = 100,000)only slight changes of signal intensity could be ob-served due to a high initial density (Test 3) or a lowamplitude (Test No. 4). In tests 1 and 2 with higherresidual deformation (εacc ≈ 0.8% for N = 100,000)an increase of signal intensity up to N = 2000–5000(reduction of material damping?) and a subsequentdecrease (increase of material damping or changes ofthe bender element bedding?) could be detected.

101 102 103 104 105

Number of cycles N [-]

0

0.5

1.0

1.5

2.0A

mpl

itude

of s

igna

l [V

]

Test 1: ID0 = 0.59, σ1 = 60 kPa Test 2: ID0 = 0.59, σ1 = 60 kPa Test 3: ID0 = 0.90, σ1 = 60 kPaTest 4: ID0 = 0.65, σ1 = 30 kPa

ampl

ampl

ampl

ampl

Figure 14. Development of signal amplitude with the num-ber of cycles

The question arises if aging effects (see Afifi etal. 1971, 1973, Baxter 1999, Wichtmann & Tri-antafyllidis 2004a) could be responsible for thechanges in signal intensity. Therefore a test with con-stant stress pav = 200 kPa, ηav = 0.75 and an initial den-sity ID0 = 0.70 was performed without cyclic loading.Over a period of one week the signals were recordedin appropriate time distances but their intensity didnot significantly change. Thus, the observations in thecyclic triaxial tests could not be described as aging ef-fects.

In yet another test the receiver signals were ac-quired over a range of frequencies of the source im-pulse fTRM (5 kHz ≤ fTRM ≤ 60 kHz) after every ap-plied package of cycles . Figure 15 shows the receivedsignal amplitude against fTRM for different numbersof cycles. It can be seen that the frequency of thebest response, i.e. the location of the peak does notchange due to cyclic loading while its height becomessmaller. Evidently there is no shift of the amplitude

N = 0

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

0

1.0

-1.0

0.5

-0.5

1.5

-1.5

Vol

tage

[V]

Time [µs]0 400 800 1200 1600

N = 100

N = 1,000 N = 10,000

N = 50,000 N = 100,000

doublesignalamplitude

Figure 13. Reduction of the receiver element signal intensity with the number of cycles in a cyclic triaxial test with p av =200 kPa, qav = 150 kPa, σampl

1= 60 kPa and ID0 = 0.59

- frequency spectrum which could explain the aba-tion of the receiver signals (it is well known thathigher frequencies undergo a stronger damping thanthe lower ones).

The observed effects are somewhat amazing. Res-onant column tests on preloaded specimens (cyclicaxial preloading in a load press, starting from anisotropic pressure pav = 80 kPa applied via vacuum,superimposing 2σampl1 = 96 kPa) show a slight but con-tinuous decrease of material damping with the num-ber of preloading cycles Nprestrain (Fig. 16). Furtherclarification on this field is necessary.

4 CORRELATION OF CYCLIC PRESTRAIN-ING HISTORY WITH LIQUEFACTION PO-TENTIAL

4.1 Motivation

The strain accumulation under drained cyclic load-ing and the pore water pressure increase during cyclicundrained loading are two occurrences of the samephenomenon (just like creep and relaxation). There-fore one could expect that liquefaction is governed bysimilar structural effects as the accumulation of strain.

Several publications (Finn et al. 1970, Ladd 1974,Mulilis et al. 1975, 1977, Seed et al. 1977, Seed et

0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60 70

frequency of transmitter impulse fTRM[Hz]

Sig

nal a

mpl

itude

[V]

N = 0N = 1N = 100N = 1,000N = 10,000N = 100,000

Figure 15. Receiver signal amplitude for varying frequen-cies of the transmitter signal fTRM after different numbersof cycles N

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10-6 10-5 10-4 10-3

Shear strain amplitude γ [-]

Dam

ping

rat

io D

[-]

Nprestrain = 0 10 100 1,000 10,000100,000

all tests: ID = 0.56 - 0.66

Figure 16. Curves of damping ratio D(γ) after different

al. 1988, Teachavorasinskun et al. 1994) documentedan impact of cyclic preloading on undrained cyclicstrength.

In the study presented in the following cyclicundrained triaxial tests were performed (see alsoPoblete 2004).

4.2 Cyclic undrained tests without drained preload-ing

The triaxial device used in the tests was similar to theapparatus presented by of Wichtmann et al. (2004a)with the only difference, that the vertical force wasapplied by the upper traverse of a free programmableload press driven by an electric servo motor. The ac-tual vertical force was measured by a force transducerinside the pressure cell below the lower end plate. Anadditional force transducer was mounted below thetraverse of the load press. A displacement transduceroutside the pressure cell was used to monitor the ver-tical deformation. Volume changes during consolida-tion and in the drained test phases were measured bythe pore water squeezed out from the sample. Cellpressure and back pressure were monitored by means

of pressure transducers.The cyclic loads were applied at a constant dis-

placement velocity recording the actual vertical force.This was more convenient than force-controlled test-ing since the specimen stiffness (which usually variesduring a test) affects the control process. Some varia-tions in frequency during the test resulted due to dis-placement control especially around the initiation ofliquefaction. However, the frequencies were generallyless than 0.01 Hz and these variations were assumedto be of no importance.

All tests were performed on medium sand 2. Thesand specimens were prepared by air-dry pluviation,and saturated first by CO2 and subsequently by water.A back pressure of 300 kPa was applied. Skempton’sB-value was larger than 0.95 in all tests.

In the tests without drained preloading mediumdense specimens (ID = 0.60–0.65) were consolidatedunder isotropic conditions (pc = 100 kPa, qc = 0). Thenthe specimen drainage was closed and the vertical to-tal stress σ1 was cyclically varied by σ

ampl1 while the

lateral total stress σ3 was kept constant.Figure 17 presents the increase of pore pressure u

and the accompanying decrease of the effective verti-cal and lateral stresses σ1′ and σ3′ during undrainedcyclic loading in a typical test. If the pore waterpressure reached the total lateral stress σ3 the speci-men had undergone ”initial liquefaction” (here afterapproximately 1,700 seconds) followed by the wellknown cyclic mobility phenomenon (e.g. Seed & Lee1966).

0 500 1,000 1,500 2,000 2,500

Time [s]

0

100

200

300

400

500

Str

esse

s σ 3

, u, σ

3', σ

1' [k

Pa]

u

σ3'σ1'

σ3 = const.

ID0 = 0.65

�

1 = 40 kPaampl

�

1 ampl~~

Figure 17. Development of pore water pressure and effec-tive lateral and vertical stresses during undrained cyclicloading

While the vertical strain amplitude εampl1 was verysmall during the first cycles it increased strongly dur-ing the cycle that lead to initial liquefaction (Fig. 18)and grew with each subsequent cycle. The generatedstrain amplitudes were nearly symmetric in extensionand compression. Figure 19 presents the stress-strainhysteresis and the stress path is shown in Figure 20.

0 500 1,000 1,500 2,000 2,500

Time [s]

ID0 = 0.65

�

1 = 40 kPaampl

Ver

tical

str

ain

ε 1 [%

]

-10

-5

0

5

10

14

1617

13N =

15

Figure 18. Development of axial strain during undrainedcyclic loading

-60

-40

-20

0

20

40

60

-10 -8 -6 -4 -2 0 2 4 6 8 10Vertical strain ε1 [%]

Dev

iato

ric s

tres

s q

[kP

a]

13 14

14

15

15

16

16

17

13

N =

N =

1-12

1-12

ID0 = 0.65

�

1 = 40 kPaampl

Figure 19. Stress-strain hysteresis for different numbers ofcycles during undrained cyclic loading

0

0

20

20

40

40

60

-20

-40

-60

60 80 100p [kPa]

q [k

Pa]

ID0 = 0.65

�

1 = 40 kPaamplPT

PT

FL

FL

Figure 20. Stress path in the p-q-plane, PT: phase transfor-mation line, FL: failure line (from static tests)

Seven tests with identical consolidation stress (pc =100 kPa, qc = 0), similar densities (0.60≤ ID0 ≤ 0.65)but different stress amplitudes σampl1 were performed.Figure 21 presents the stress ratios σampl1 /pc and thecorresponding cycle numbers that were necessary togenerate 2, 5 or 10 % double amplitude vertical strain2εampl1 . The sample was defined as fully liquified at2εampl1 = 10 %. It is obvious that higher stress ampli-tudes caused earlier liquefaction. Only a short timewas needed between the initial and full liquefaction.

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0 10 20 30 40 50 60 70

Number of cycles N [-]

Str

ess

ratio

� 1

/ p

c [-

]am

pl

all tests: ID0 = 0.60 - 0.65 pc = 100 kPa

double amplitudes:2 %5 %

10 %

Figure 21. Relationship between stress ratio σampl1

/pc andthe number of cycles to cause 2,5 or 10 % double amplitudeof vertical strain

5 Cyclic undrained tests with drained preloading

The impact of drained cyclic preloading on theundrained cyclic behaviour was studied in several tri-axial tests. All specimens were prepared with densi-ties in the range 0.63 ≤ ID0 ≤ 0.68 and isotropicallyconsolidated under pc = 100 kPa. Thereafter, a drainedcyclic preloading with σampl

1,preload and Npreload was ap-plied (stress-controlled). The accumulation of strainwas measured. After preloading the drainage wasclosed and the undrained cyclic phase was started.

Three different preloading histories were tried out,Table 2. The stress amplitudes σampl1 = 30 kPa andσampl1 = 50 kPa lead to shear strain amplitudes of ap-proximately γampl = 3.3 · 10−4 and γampl = 7.2 · 10−4,respectively during preloading. Several amplitudesσampl1 during the undrained phase were applied foreach preloading history in order to evaluate curvesσampl1 /pc(N) analogously to those in Figure 21.

Some of the volumetric accumulation curves areshown in Figure 22. For each preloading history threecurves εaccv (N) are presented. As could be expectedthe cyclic preloading with an amplitude of 30 kPacauses less accumulation than with 50 kPa. The devi-atoric accumulation εaccq was negligible since the av-erage state of stress was isotropic.

Preloading σampl1,preload Npreload

history [kPa] [-]1 30 102 50 103 50 100

Table 2. Tested preloading histories

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

Number of cycles N [-]

ε vol

[%

]ac

c

σ1,preload = 50 kPaampl

σ1,preload = 50 kPaampl

σ1,preload = 30 kPaampl

all tests: ID0 = 0.63 - 0.68

Figure 22. Volumetric strain accumulation curves due todrained cyclic preloading with different stress amplitudes

The development of the pore pressure in four testswith different preloading histories but identical am-plitude σampl1 = 45 kPa in the undrained phase can befound in Figure 23. An increase of the intensity ofdrained cyclic preloading (in the amplitude and/or inthe number of cycles) reduces the rate of pore pres-sure accumulation u̇ = ∂u/∂N and it takes more cy-cles to reach initial and full liquefaction. The non-preloaded specimens needed about 5 cycles to initialliquefaction whereas 8 cycles were necessary in thecase of the specimen preloaded with σampl

1,preload = 30

kPa and Npreload = 10. The preloading with σampl1,preload

= 50 kPa and Npreload = 10 caused the initial of lique-faction to occur after 43 cycles and 205 cycles wereneeded for the specimen preloaded with σampl

1,preload = 50

kPa and Npreload = 100. The stress ratios σampl1 /pc and

the number of cycles necessary to cause 10 % doubleamplitude of vertical strain 2εampl1 are shown in Figure24. It is therefore obvious that preloading may con-siderably delay liquefaction.

The undrained cyclic strength is defined here as thestress ratio σampl1 /pc necessary to cause 10 % doubleamplitude vertical strain in 25 cycles. Non-preloadedspecimens exhibited an undrained cyclic strength ofabout 0.35. Preloading with σampl

1,preload = 30 kPa andNpreload = 10 increased the undrained cyclic strengthto 0.39, preloading with σampl

1,preload = 50 kPa and Npreload= 10 lead to 0.48 and a preloading with σampl

1,preload= 50 kPa and Npreload = 100 caused the undrained

cyclic strength to increase to 0.55. If the amplitudesin the undrained phase were much higher than thepreloading amplitudes the curves σampl1 /pc(N) of thepreloaded specimens fell together with the curves ofthe non-preloaded ones, i.e. cyclic preloading doesnot affect subsequent pore pressure accumulation in-duced by substantial larger cycles.

0 100 150 200

Number of cycles N [-]

all tests: ID0 = 0.63 - 0.68

�

1,preload = 30 kPa,Npreload = 10

ampl

�

1,preload = 50 kPa,Npreload = 100

ampl

�

1,preload = 50 kPa,Npreload = 10

ampl

no preloading0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Str

ess

ratio

� 1

/ p

c [-

]am

pl cyclicundrained strength 50

Figure 24. Relationship between the stress ratio σampl1

/pcand the number of cycles to cause 2,5 or 10 % double am-plitude of vertical strain for varying preloading histories

It was demonstrated that a drained preloading sig-nificantly decreases the rate of pore water pressurerise under undrained conditions and thus, increasesthe cyclic undrained strength of a soil. A clear cor-relation between the intensity of the cyclic preload-ing and the undrained cyclic behaviour exists. Nextthe transfer of this result on the in situ determinationof preloading history (e.g. via cone penetration testswith measurement of pore water pressure rates) hasto be considered. Note that this method is restricted tofully saturated soils. For partly saturated deposits thesituation is more complex.

6 BACK ANALYSIS

As an alternative way to assess cyclic preloading his-tory (at least for soil layers near the surface) one mayconsider the application of a cyclic test in situ usingheavy vibration on the soil surface (Fig. 25). The set-tlement as a function of time or cycles s(t) or s(N)can be recorded. The corresponding boundary valueproblem can be computed (Niemunis et al. 2004) andthe constitutive parameters discussed by Wichtmannet al. (2004a,b) can be used (measured from disturbedsamples). The only problem left is the determinationof the preloading parameter (accumulated structuralstrain εacc A, see Wichtmann et al. 2004 and Niemu-nis et al. 2004). This can be done by varying εacc A

until the curve s(N) obtained from the heavy vibra-tion is reproduced. Having the preloading parameter

0

100

200

300

400

500

Str

esse

s σ 3

, u, σ

3' [k

Pa]

σ3 = const.

ID0 = 0.64

�1,preload = 50 kPa

Npreload = 100ampl

σ1 = 45 kPaampl

u

σ3'

0 10,000 15,000 20,0005,000

Time [s]

0 1,000 2,000 3,000 4,000 5,000 6,000

Time [s]

0

100

200

300

400

500

Str

esse

s σ 3

, u, σ

3' [k

Pa]

u

σ3'

σ3 = const.

ID0 = 0.66

�1,preload = 50 kPa

Npreload = 10ampl

σ1 = 45 kPaampl

0 1,000 2,000

Time [s]

0

100

200

300

400

500

Str

esse

s σ 3

, u, σ

3' [k

Pa]

σ3 = const.

u

σ3'

ID0 = 0.65

�1,preload = 30 kPa

Npreload = 10ampl

σ1 = 45 kPaampl

0 1,000 2,000

Time [s]

ID0 = 0.63

σ1 = 45 kPaampl

no preloading

0

100

200

300

400

500

Str

esse

s σ 3

, u, σ

3' [k

Pa]

σ3 = const.

u

σ3'

Figure 23. Development of pore water pressure u and effective lateral stress σ3′ during undrained cyclic loading in fourtests with varying preloading history (all tests: σampl

1= 45 kPa in the undrained phase)

from in-situ cyclic loading the boundary value prob-lems of interest can be calculated.

s

s = f( � acc A)

Vibrator

6 m~~s

s

N1

Figure 25. Test loading with a heavy surface vibration insitu and back analysis of measured settlement curves s(t)

7 SUMMARY AND CONCLUSIONS

Void ratio, average stress and amplitude of cyclicloading are insufficient for the prediction of accumu-lation rates in situ. Cyclic preloading significantly af-fects the cumulative behaviour and has to be deter-

mined. Several approaches were studied in this paper.Cyclic preloading could not be correlated with the

small strain stiffness. Neither resonant column testson cyclically preloaded specimens nor cyclic triax-ial tests with measurements of compression and shearwave velocities after definite numbers of cycles ex-hibited significant changes of these stiffnesses due tocyclic loading.

In the cyclic triaxial tests a strong change of theintensity of the received signal was observed. Sinceresonant column tests show only slight changes (i.e. amoderate decrease) of damping ratio D due to cyclicloading further investigation is necessary.

A cyclic preloading significantly decreases therate of pore water pressure rise and thus, increasesthe undrained cyclic strength. A possible conclusionabout the preloading in situ needs further correlations,e.g. with CPT measurements.

An alternative could be the back analysis of settle-ment versus time obtained from a heavy vibration insitu.

8 ACKNOWLEDGEMENTS

The authors are grateful to DFG (German Research

Council) for the financial support. Parts of the studiespresented in this paper were done in the frameworkof the project A8 ”Influence of the fabric changes insoil on the lifetime of structures” of SFB 398 ”Life-time oriented design concepts”. Other parts were sup-ported by DAAD (German Academic Exchange Pro-gram) in the framework of the project ”Analytical andNumerical Studies on Microstructure Effects of theResponse of Elastic Solids and Structures” as a partof the IKYDA program (Prof. Beskos, Prof. Polyzos,University of Patras). This support is also gratefullyappreciated.

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