Journal of Transportation Engineering, ASCE, Vol. 135, No. 7, pp. … · 2014. 10. 28. · Journal of Transportation Engineering, ASCE, Vol. 135, No. 7, pp. 467-478, 2009 Comparison

Journal of Transportation Engineering, ASCE, Vol. 135, No. 7, pp. 467-478, 2009

Comparison of cyclic triaxial behavior of unbound granular

material under constant and variable confining pressure

H. A. Rondoni), T. Wichtmannii), Th. Triantafyllidisiii), A. Lizcanoiv)

Abstract: Cyclic stresses due to passing wheels impose an accumulation of permanent strains in layers of unboundgranular materials (UGMs) of flexible pavements. The hollow cylinder triaxial test would be the most appropriate testto simulate the in situ stress conditions but it is difficult to perform on UGMs due to their large maximum grain size.The simpler axi-symmetric cyclic triaxial test does not consider the shear stress components. It can be performed witha constant (CCP) or a variable confining pressure (VCP). CCP and VCP tests are commonly assumed to deliver similarresidual and resilient strains as long as the average stress is the same. Thus, the simpler CCP test is mostly used inpavement engineering. However, this assumption is based on limited test data in the literature and may not be on the safeside. The present paper documents a comparative study of CCP and VCP tests on UGM. The study is mainly dedicatedto the permanent deformations. The results show that only for some special stress paths both types of test deliver similarpermanent axial or volumetric strains. For some other stress paths the CCP test may underestimate the permanent axialstrain in comparison to the corresponding VCP test.

CE Database subject headings: Flexible pavements; Unbound granular material (UGM); Cyclic loading; CCP triaxialtests; VCP triaxial tests

1 Introduction

Many design procedures for pavements (e.g. [1,18,20,32,37,43]) suppose that permanent deformations occur mainly inthe subgrade. The behavior of base and subbase layersmade of unbound granular materials (UGMs) is assumedlinear or non-linear elastic. However, due to the large cyclicstresses occuring in the UGM layers of flexible pavementsan inelastic portion of deformation accumulates with eachcycle (e.g. [8,15,19,45]). The settlements may become sig-nificant after a large number of cycles.

Starting in the 1960s significant research has been doneon the resilient characteristics of UGM materials. Mathe-matical models for the prediction of the resilient modulusfor different cyclic stress paths have been developed. Themodels assume isotropically (e.g. [9, 13, 27]) or anisotrop-ically non-linear elastic (e.g. [3, 26, 52, 58]) or hyperelastic(e.g. [28, 51]) material behavior.

Permanent deformations of UGMs were less frequentlystudied than the resilient properties (e.g. [7, 15, 17, 22, 24,25,31,38,39,50,57,59–61,63]). Empirical or semi-empiricalmodels for the prediction of permanent strains in depen-dence of stress (usually described by the confining pressureand the amplitude of the cyclic load) and the number ofload cycles have been developed. A discussion of factorsaffecting the UGM behavior and a summary of the mostimportant models can be consulted in [38] and [48]. How-ever, in spite of a great amount of research, the complexbehavior of UGMs is not totally understood yet. While a

i)Research Assistant, Department of Civil and Environmental En-gineering. Los Andes University, Bogota D. C. (Colombia)

ii)Research Assistant, Institute of Soil Mechanics and Rock Me-chanics, University of Karlsruhe, Germany (corresponding author).Email: [email protected]

iii)Professor and Director of the Institute of Soil Mechanics and RockMechanics, University of Karlsruhe, Germany

iv)Professor, Department of Civil and Environmental Engineering.Los Andes University, Bogota D. C. (Colombia)

satisfactory progress has been made in the prediction of theresilient behavior, the advances in the prediction of perma-nent strain accumulation were less significant [19].

One limitation is the lack of appropriate test devices tosimulate the in situ stress paths in a pavement (Fig. 1).Ideally, in a laboratory element test the axial, the two lat-eral and the shear stress components should be controlledindependently of each other. The device should be capa-ble to simulate rotations of the principal stress directions.The most appropriate device to apply this loading wouldbe the hollow cylinder apparatus [19, 44, 55]). However, itsusage for tests on UGM is limited due to the large maxi-mum grain size of such materials in comparison to the wallthickness of the hollow cylinder specimen. In true triaxialdevices the three principal stresses can be controlled in-dependently but their directions do not change. Most ofthe research on UGMs is made by means of axi-symmetriccyclic triaxial tests. Those tests are either performed witha constant (CCP) or with a variable (VCP) confining pres-sure. CCP tests simulate only the vertical component of thecyclic stresses. VCP tests are somewhat better since theysimultaneously simulate vertical and lateral cyclic stresses.However, both types of tests cannot reproduce the shearstress component in addition to the normal stresses, whichmeans that the rotation of the principal stress directions isnot considered.

In comparison to CCP tests only a few studies with VCPtests have been performed due to different reasons. Testdevices with the possibility to vary the confining pressureare less widespread than those for an application of ax-ial cyclic loads. The loading frequencies in VCP tests areusually smaller than in CCP tests due to technical limi-tations. Thus, it takes longer time to reach a sufficientlylarge number of cycles. A variation of the confining pres-sure implicates the problem of membrane penetration [42],which means that complicated measurements of lateral de-

1

Wichtmann & Triantafyllidis J. Transp. Eng., ASCE, Vol. 135, No. 7, pp. 467-478, 2009

σ11

σ22

σ11

σ22

σ12

wheel

pavement

σ11

σ22

σ12

stress

time

Stress loop:

Time history:

σ12

Fig. 1: Stress variation beneath a rolling wheel load [38]

formations directly on the specimen are indispensable (atleast for studies of the resilient behaviour). Most of the em-pirical equations available for the prediction of permanentstrain in an UGM were developed based on CCP triaxialtest results (e.g. [7, 30, 31, 50, 57, 60]).

In standard codes or design guidelines it is often assumedthat CCP and VCP tests deliver similar results as long asthe average stress is the same. Thus, for engineering appli-cations it is easier to use the simpler CCP tests to quantifythe permanent deformations. However, this assumption isbased on a limited number of tests and thus it is question-able if it is on the safe side. Some comparative studies withCCP and VCP triaxial tests have been documented in theliterature (e.g. [4, 12, 41, 67]) but most of them have beenconcentrated on the resilient characteristics. Only somewere dedicated to the permanent strains:

• Chan & Brown [16] compared CCP triaxial tests andhollow cylinder tests on a sand with a maximum grainsize of 5 mm. In both types of tests the confining pres-sure was kept constant and the vertical stress was var-ied cyclically. The additional oscillation of shear stressin the hollow cylinder test lead to a larger rate of shearstrain accumulation. Based on shaking table tests,multidimensional simple shear tests or VCP triaxialtests on sand also other researchers [34,46,65] demon-strated an increase of the accumulation rate with in-creasing number of oscillating stress or strain compo-nents (i.e. with increasing dimensionality of the cyclicstress or strain path). For example, Wichtmann etal. [65] observed approximately twice larger accumu-lation rates for a circular cyclic strain path comparedto a one-dimensional strain path with an amplitudeidentical to the radius of the circles.

• Only a few comparative studies were performed onUGM. Although no attempts were made to quantifythe permanent strains, Allen & Thompson [4] reportedon larger permanent strains in CCP tests than in VCPtests. Allen & Thompson [4] compared the stress pathspresented in Fig. 2a. The maximum stress (pmax,qmax) with the mean pressure p = (σ1 + 2σ3)/3 andwith the deviatoric stress q = σ1 − σ3 was identicalfor the CCP and the VCP tests. Thus, both stresspaths have the same amplitude of deviatoric stressqampl = (qmax − qmin)/2 but different amplitudes of

mean pressure pampl and also different average meanpressures pav = (pmin + pmax)/2.

p

q

VCP

qmax

2qam

pl

2pampl

qmaxqampl

qav

pmax

CCP

σavσav

13

p

q

VCP

qmax

qav

pav

CCP

σav

q

t

a) b)

2pampl

2qam

pl

Fig. 2: Scheme of stress paths used a) by Allen & Thompson [4]and b) recommended by Brown & Hyde [12]

0 1 2 3 4 50

1

2

3

4

5

CCP tests (Brown, 1974) CCP tests VCP tests

qmax / σav [-]3

Perm

anen

t axi

al s

trai

n ε 1 [

%]

p

CC

P2C

CP3

CC

P4

VC

P1

VC

P3

0 100 200 300 400 5000

50

100

150

200

250

q [k

Pa]

p [kPa]

VC

P1V

CP2

VC

P3C

CP2

CC

P3

CC

P4

VC

P4V

CP5

VCP6

CC

P1

VC

P4,5

,6 /

CC

P1

VC

P2

Fig. 3: Tests of Brown & Hyde [12]: Permanent axial strainversus the ratio of the maximum deviatoric stress qmax and theaverage confining pressure σav

3

• Brown & Hyde [12] argued that a comparison of cyclicstress paths with an identical average stress, (i.e. iden-tical values of pav and qav, as shown in Fig. 2b) is moremeaningful. In such a case the CCP and VCP stresspaths are the same except for the amplitude pampl.Brown & Hyde tested an UGM under the stress pathsshown in Fig. 3. The test data was supplemented bydata from an older CCP test series [11]. In Fig. 3the residual axial strain εp

1 is plotted versus the ratioqmax/σav

3 . Since for same values of qmax/σav3 the data

of CCP and VCP tests did not differ much, Brown &Hyde concluded that the accumulation of permanentstrain is similar for the two types of tests. However,the scatter of data for qmax/σav

3 = 4 is significant inFig. 3 and the CCP tests seem to deliver larger strainaccumulation rates. Furthermore, the tests of Brown& Hyde were restricted to a single value of qmax = 200kPa.

If CCP and VCP triaxial tests are equivalent with re-spect to the accumulation of permanent strain, as long as

2


Index property ValueMaximum grain size, dmax [mm] 16Mean grain size, d50 [mm] 6.3Coefficient of uniformity, Cu = d60/d10 100Specific weight, %s [g/cm3] 2.65Maximum dry density, %d,max [g/cm3] 2.16Minimum dry density, %d,min [g/cm3] 1.84Maximum void ratio, emax 0.44Minimum void ratio, emin 0.23Maximum proctor density, %Pr [g/cm3] 2.30Optimum water content, wopt [%] 5.2Critical friction angle, ϕc [◦] 38.0Grain shape Subangular

Table 1: Index properties of the tested UGM

the average confining stress σav3 and the amplitude of de-

viatoric stress qampl are the same in both types of tests,seems not fully clearified yet. This paper intends to makea contribution to this open question. It presents a studycomparing the results of CCP and VCP tests, similar tothe one performed by Brown & Hyde [12], but with a widerrange of deviatoric stress amplitudes qampl. It shows thatonly for some special stress paths both types of test deliversimilar permanent axial or volumetric strains.

2 Tested material

The grain size distribution curve (Fig. 4) used in the testsis in accordance with the Colombian Specification [20] forbase layer construction in flexible pavements except for themaximum grain size. It was reduced to dmax = 16 mmin order not to fall below a ratio b/dmax of 5 with b ×

b being the dimensions of the specimen cross section inthe triaxial tests. This grain size distribution curve wasmixed from different gradations of a natural quartz sandwith subangular grain shape. For the fine particles a quartzmeal was used. The index properties of the tested materialare summarized in Table 1. A Proctor test with modifiedenergy (E = 2700 kNm/m3) resulting in a maximum drydensity %Pr = 2.30 g/cm3 and an optimum water contentwopt = 5.2 % has been presented by Rondon et al. [49].The critical friction angle ϕc = 38.0◦ has been determinedas the inclination of a pluviated cone of the dry material.

0.10.05 0.2 0.5 1 2 5 10 200

10

20

30

40

50

60

70

80

90

100

tested grain size distribution curve upper limit according to INVIAS, 2002 lower limit according to INVIAS, 2002

Fine

r by

wei

ght [

%]

Particle size [mm]

Fig. 4: Grain size distribution curve of the tested UGM com-pared to the limits of Colombian Specification (INVIAS [20])

3 Test device and specimen preparation

A scheme of the used triaxial device is given in Fig. 5.The cyclic axial load was applied with a pneumatic loadingsystem and also the confining pressure could be cyclicallyvaried by means of a pneumatic valve.

load cell

displ. transducer (axial deform.)

pressure transducers (cell- and back press.)

differential pressure transducer

back pressure

soil specimen (8.7 x 8.7 x 18 cm)

drainage

cell pressure σ3

ball bearing

Volume measuring unit:

reference pipette

plexiglas cylinder

water in the cell

ball bearing

load piston

inner rods

outer rods

membrane

measuring pipette

axial load F

Fig. 5: Scheme of the used triaxial device

The prismatic specimens had dimensions 8.7 × 8.7 × 18cm. Specimens with a square cross section were used in or-der to measure lateral (and also axial) deformations locallyby means of local displacement transducers (LDTs). Localmeasurements of lateral deformations are indispensable inVCP tests in order to get an information about the resilientbehavior which is free from membrane penetration effects(Nicholson et al. [42]). In contrast, the CCP test resultsand also the permanent deformations in the VCP tests arenot falsified by membrane penetration effects.

The LDTs are strips of phosphor bronze applied withstrain gauges (see also Goto et al. [23], Hoque et al. [29]).In order to mount an LDT two hinges are glued to themembrane of the specimen. The LDT is placed betweenthese hinges in an initially bended condition (Fig. 6g,h).The lateral deformation of the specimen between the twofixing points of the LDT is measured as a change of thebending of the LDT. Unfortunately, the LDTs available forthe present study could not be used in combination withwater in the cell. However, for the long-term cyclic tri-axial tests the water in the cell is indispensable. Thus, atwo-step testing program was chosen. The cyclic CCP andVCP triaxial tests were performed without LDT measure-ments. Afterwards, two additional short-term tests withLDT measurements were performed in which all CCP andVCP stress paths were applied in succession. The testingprogram is explained in detail in a following section.

The axial load was measured inside the pressure cell witha load cell located below the bottom end plate. The axialdeformation was obtained from a displacement transducerattached to the load piston (Fig. 5). The system com-pliance was determined in preliminary tests on a dummyspecimen. It was subtracted from the measured displace-ment. Volume changes were measured via the pore waterof the fully saturated specimens using a pipette system anda differential pressure transducer. Deviations between thecorrected total measurements and the local LDT measure-

3


ments of deformations are discussed later in this paper.

Fig. 6: Preparation of an UGM specimen for triaxial tests: a)Steel mould fixed to the bottom end plate of the triaxial cell(preparation outside the device), b) Moist tamping of the spec-imen with a miniature proctor hammer, c) Specimen after re-moval of one side of the mould, d) Specimen after removal of allsides of the mould, e) Placement of the membrane with a specialstretcher, f) Specimen in the mounted triaxial cell prior to test-ing, g),h) Specimen equipped with local displacement transduc-ers (LDTs, eight for the lateral and two for the axial direction)

For the specimen preparation a steel mould consistingof four plates was fixed to the bottom end plate of thetriaxial cell (Fig. 6a). The specimen preparation was per-formed outside the triaxial cell in order to preserve theload cell. Specimens were prepared by tamping in n = 6layers each with a thickness of 3 cm. The material wasin the moist condition (water content w = wopt = 5.2 %).A miniature proctor hammer (Fig. 6b) was used. Its fallweight (m = 1 kg, i.e. G = 10 N) was dropped from aheight of H = 20 cm and N = 250 blows were appliedto each layer. An energy per volume (total volume of aspecimen V = 1362 cm3) in the order of magnitude of

E = N · n · G · h/V ≈ 2200 kNm/m3 was induced intoa specimen. It was chosen lower than the energy used inthe modified Proctor test in order to reach densities slightlylower than the modified Proctor density (93 - 96 % of %Pr),which are typical for UGM layers in situ. After the tamp-ing procedure the bottom end plate with the mould and thespecimen was placed into the triaxial cell and the mouldwas removed (Figures 6c,d). The specimen stands due to

capillary pressure. Afterwards the membrane was placedusing a stretcher with square cross section (Fig. 6e). Thespecimen end plates have a special shape at the transitionfrom the square to the round cross section. The roundcross section is necessary to enable a proper sealing of themembrane by O-rings. Fig. 6f presents a specimen afterthe top plate was placed, the membrane was sealed, thetriaxial cell was mounted and filled with water and the cellpressure was applied. Finally, the specimens were satu-rated with de-aired water. A back pressure of 200 kPa wasused in all tests in order to improve saturation. The satu-ration was controlled by Skempton’s B-Value. In all testsB > 0.95 was achieved.

Fig. 6g,h presents a specimen equipped with LDTs. Ontwo adjacent sides of the specimen four LDTs were mountedhorizontally for local measurements of the lateral deforma-tion (Fig. 6g). The other two sides were equipped withLDTs mounted vertically for local measurements of the ax-ial deformation (Fig. 6h).

Despite its extensive use (mainly in Japanese soil me-chanics laboratories) specimens with a square cross sectionare sometimes set into question because an inhomogeneousdeformation is expected. Surprisingly, experimental studiescomparing a circular and a square cross section can hardlybe found in the literature. Thus, prior to the tests on theUGM material, results of prismatic and cylindrical spec-imens have been compared for similar test conditions inmonotonic [49] and cyclic triaxial tests on a medium coarsequartz sand. In both types of tests the material behaviordid not depend on the geometry of the specimen. Thus,the use of prismatic specimens does not imply any disad-vantages in comparison to cylindrical ones.

4 Monotonic triaxial tests

For the determination of shear strength parameters, threemonotonic triaxial tests on specimens with large initial den-sities (relative density index ID0 = (emax−e0)/(emax−emin)= 1.06 - 1.13, dry density > 95% of %Pr) were performed.The effective lateral stresses were σ3 = 50, 100 and 200kPa. The curves q(ε1) and εv(ε1) are given in Fig. 7. Apeak friction angle ϕPeak ≈ 54◦ was determined (Rondonet al. [49]).

5 Testing program of the cyclic triaxial tests

Table 2 and Fig. 8 present the stress paths applied in thecyclic tests. Five pairs of VCP and CCP stress paths withidentical pav, qav and qampl were tested. All VCP and CCPstress paths start from qmin = 0. The VCP stress pathshave a low minimum mean pressure pmin= 20 kPa in orderto simulate realistic in situ stress conditions for pavements.Three different inclinations ηampl = qampl/pampl = 0.75,1.125 and 1.5 of the VCP stress paths in the p-q-plane weretested.

The critical state line (CSL) in Fig. 8 was plotted withan inclination Mc = 6 sinϕc/(3 − sinϕc) = 1.55 usingϕc = 38.0◦ (Table 1) determined for the dry material de-posited by pluviation. The final stage of the monotonictests (Fig. 7) may indicate that the critical stress ratio ofthe pre-compacted UGM is larger. For example, the stressratio at the end of the test with σ3 = 200 kPa was η ≈

1.9. In that case even the stress path of test CCP5 wouldcompletely lay below the CSL.

The loading frequency f was 0.05 Hz in the VCP tests

4


0 1 2 3 4 5 60

200

400

600

800

1000

1200

1400

1600

1800D

evia

tori

c st

ress

q [

kPa]

Axial strain ε1 [%]

σ3 = 200 kPa ID0 = 1.10

σ3 = 100 kPa ID0 = 1.13

σ3 = 50 kPa ID0 = 1.06

1

0

-1

-2

-3

-4

Vol

umet

ric

stra

in ε

v [%

]

0 1 2 3 4 5 6


σ3 = 200 kPa ID0 = 1.10

σ3 = 100 kPa ID0 = 1.13

σ3 = 50 kPa ID0 = 1.06

dilatancycompaction

a)

b)

Fig. 7: Results of monotonic triaxial tests with different confin-ing pressures: a) q vs. ε1 and b) εv vs. ε1

0 50 100 150 200 2500

50

100

150

200

250

300

350

q [k

Pa]

p [kPa]

CSL

peak

stre

ngth

(mon

. tes

ts)

1

VCP1

VCP2

VCP3

VCP5

CC

P1

CC

P2C

CP3

CC

P4

CC

P5

VCP4

1.5

0.75

1.1251

1

Fig. 8: VCP and CCP stress paths used for the present study

and 1 Hz in the CCP tests. The low frequency in the VCPtests was chosen due to technical limitations. Numerousresearchers have reported on a negligible influence of theloading frequency on the resilient characteristics and onthe permanent strain accumulation in sand (e.g. [64, 66])or UGM (e.g. [4, 10, 11, 35, 47, 53, 54]). In order to provethe frequency-independence and in order to check the re-peatability of the tests, four preliminary CCP tests wereperformed on the UGM (tests CCP2a to CCP2d, see Table2). Either 3 · 104 cycles with f = 0.05 Hz or 2 · 105 cycleswith f = 1 Hz were applied. Fig. 9 presents the curves ofpermanent axial strain εP

1 versus the number of cycles N .

It shows no significant influence of the loading frequencyand a satisfying reproducibility of the tests.

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Test CCP2a, f = 1 Hz Test CCP2b, f = 1 Hz Test CCP2c, f = 0.05 Hz Test CCP2d, f = 0.05 Hz

100

101

102

103

104

105

106

N [-]

ε 1 [%

]p

Fig. 9: Curves εp

1(N) in preliminary tests: No influence of theloading frequency f and satisfying repeatability of the tests

Two additional tests were conducted with LDT mea-surements. These tests were performed without water inthe cell. The lateral stress was applied via air pressure. Inorder to keep the specimens saturated, water was flushedthrough the specimen from time to time. After a largecyclic preloading (as recommended in [2] for studies of theresilient response); 20,000 cycles with the stress path oftest CCP5, see Fig. 8, were applied) all stress paths of theCCP and VCP triaxial tests were applied in succession.The two tested specimens had similar initial densities asthe specimens in the long-term cyclic tests. The same testwas performed twice in order to check the repeatability.

6 CCP test results

Fig. 10a,b presents the curves of the axial permanent strainεp1 and of the volumetric permanent strain εp

v versus thenumber of cycles N in the CCP tests. While for small ratiosqmax/σav

3 (tests CPP1 and CPP2) the residual strains growalmost proportional to ln(N), the curves show a steeperrise for larger stress ratios (tests CPP3 - CPP5). However,the inclination of the curves εp

1(N) and εpv(N) in the semi-

logarithmic scale decreases again for large numbers of cycles(N > 105).

Fig. 10c,d shows that εp1 and εp

v increase with the ra-tio qmax/σav

3 . The wide-spread empirical function εp1 =

c1(qmax/σav

3 )c2 can be fitted to the test data (solid curvesin Fig. 10c,d), wherein the parameters c1 and c2 could beformulated as functions of N . However, such empiricalformulas are not the scope of the present study. Similarrelationships εp

1 ∼ qmax/σav3 have been reported by several

other researchers, e.g. [5–7,11,14,21,33,36,39,40,56,61–63].The permanent deviatoric strain εp

q = 2/3(εp1 − εp

3)is plotted versus the permanent volumetric strain εp

v inFig. 10e. The inclination of the εp

q-εpv-strain paths (i.e. the

ratio εpq/εp

v) increases with increasing average stress ratioηav = qav/pav of a test. These findings are concordant withtest results on sand specimens [64]. However, while forsand a pure volumetric accumulation (i.e. εp

q = 0 with the

rate defined as t = ∂ t/∂N) was observed for stress cycleswith ηav = 0, in the present tests on UGM the condition

5


100 101 102 103 104 105 1060

1

2

3

4

5

6

7

N [-]101 102 103 104 105 106

N [-]

0

1

2

3

4

5a) b)

c) d)

e) f)

g) h)

Perm

anen

t axi

al s

trai

n ε 1

[%

]p

Perm

anen

t vol

umet

ric

stra

in ε

v [%

]p

0

1

2

3

4

5

6

0 1 2 3 4 5

Stress ratio qmax/σav [-]3

Perm

anen

t vol

umet

ric

stra

in ε

v [%

]p

CC

P 1

CC

P 2

CC

P 3

CC

P 4

CC

P 5

0 1 2 3 4 50

1

2

3

4

5

6

7

N = 200,000 100,000 50,000 10,000 1,000

N = 200,000 100,000 50,000 10,000 1,000


Perm

anen

t axi

al s

trai

n ε 1

[%

]p

CC

P 1

CC

P 2

CC

P 3 CC

P 4

CC

P 5

0 1 2 3 4 5-1

0

1

2

3

4

5

Perm

anen

t dev

iato

ric

stra

in ε

q [%

]p

Permanent volumetric strain εv [%]p

Test / ηav = qav/pav =

CCP1 / 0.50 CCP2 / 0.61 CCP3 / 0.92 CCP4 / 1.00 CCP5 / 1.23

0 50 100 150 200 2500

100

200

300

q [k

Pa]

p [kPa]

CSL

peak

stre

ngth

1CC

P1

CC

P2C

CP3

CC

P5

CC

P4

1.23

0.921.00

0.500.61

Permanent axial strain ε1 [%]p

Rat

e of

per

man

ent a

xial

str

ain

ε 1 [

-]p

CCP1, qmax/σav = 1.20 CCP2a, qmax/σav = 1.54 CCP3, qmax/σav = 2.66 CCP4, qmax/σav = 3.00 CCP5, qmax/σav = 4.153

3

3

3

3 CCP1, qmax/σav = 1.20 CCP2a, qmax/σav = 1.54 CCP3, qmax/σav = 2.66 CCP4, qmax/σav = 3.00 CCP5, qmax/σav = 4.153

3

3

3

3

0 1 2 3 4 50

4

8

12

16

20


Res

ilien

t str

ain

[10-4

]

0 1 2 3 4 5 6 7

CCP1, qmax/σav = 1.20 CCP2a, qmax/σav = 1.54 CCP3, qmax/σav = 2.66 CCP4, qmax/σav = 3.00 CCP5, qmax/σav = 4.153

3

3

3

3

10-5

10-6

10-7

10-8

10-9

10-10

0

50

100

150

200

Dev

iato

ric

stre

ss q

[kP

a]

0 0.1 0.2 0.3 0.4 0.5


Test CCP3

50 -54

100 -104

200 -204

500 -504

1000 -1004

N = 1 -25

ε1, long-term CCP testsr

ε1, local LDT meas.r



Fig. 10: Results of the CCP tests: a) εp

1 versus N , b) εpv(N) versus N , c) εp

1 versus qmax/σav3 , d) εp

v versus qmax/σav3 , e) εp

q versus εpv,

f) εp

1 versus εp

1 g) q-ε1-hystereses, h) resilient strain εr1 versus qmax/σav

3

6


Test pmin pmax pav pampl qmin qmax qav qampl ηampl σav3

qmax/σav3

L f ID0 %%Pr[kPa] [kPa] [kPa] [kPa] [kPa] [kPa] [kPa] [kPa] [-] [kPa] [-] [kPa] [Hz] [-] [%]

VCP 1 20.0 100.0 60 40 0 60.0 30.0 30.0 0.75 50 1.2 100.0 0.05 1.03 95VCP 2 20.0 200.0 110 90 0 135.0 67.5 67.5 0.75 87.5 1.5 225.0 0.05 1.05 95VCP 3 20.0 200.0 110 90 0 202.5 101.25 101.25 1.125 76.25 2.7 270.9 0.05 0.92 93VCP 4 20.0 100.0 60 40 0 120.0 60.0 60.0 1.50 40 3.0 144.2 0.05 0.99 94VCP 5 20.0 200.0 110 90 0 270.0 135.0 135.0 1.50 65 4.2 324.5 0.05 0.98 94CCP 1 50.0 70.0 60 10 0 60.0 30.0 30.0 3.0 50 1.2 63.2 1.00 1.04 95CCP 2a 87.5 132.5 110 22.5 0 135.0 67.5 67.5 3.0 87.5 1.5 142.3 1.00 1.06 95CCP 2b 87.5 132.5 110 22.5 0 135.0 67.5 67.5 3.0 87.5 1.5 142.3 1.00 1.07 95CCP 2c 87.5 132.5 110 22.5 0 135.0 67.5 67.5 3.0 87.5 1.5 142.3 0.05 1.07 95CCP 2d 87.5 132.5 110 22.5 0 135.0 67.5 67.5 3.0 87.5 1.5 142.3 0.05 1.05 95CCP 3 76.25 143.8 110 33.8 0 202.5 101.25 101.25 3.0 76.25 2.7 213.5 1.00 1.11 96CCP 4 40.0 80.0 60 20 0 120.0 60.0 60.0 3.0 40 3.0 126.5 1.00 1.00 94CCP 5 65.0 155.0 110 45 0 270.0 135.0 135.0 3.0 65 4.2 94.9 1.00 1.12 96

Table 2: Program of the cyclic triaxial tests (pmin, pmax, pav, pampl = minimum, maximum, average or amplitude values of mean

pressure, qmin, qmax, qav, qampl = minimum, maximum, average or amplitude values of deviator stress, ηampl = qampl/pampl =

inclination of the cyclic stress path, σav3 = average confining stress, L =

√

(2pampl)2 + (2qampl)2 = length of the cyclic stress pathin the p-q-plane, f = frequency, ID0 = initial relative density index)

εpq = 0 was fulfilled at considerable larger average stress ra-

tios (ηav ≈ 0.5). Thus, the specimen preparation (tampingof UGM compared to pluviation of sand) seems to influencethe direction of accumulation εp

q/εpv (so-called ”cyclic flow

rule” [64]). Probably the preparation method induces ananisotropy and the ”isotropic” axis is shifted in the stressspace. Plotting the strain paths in a εp

1-εp3-diagram shows

that stress cycles with an average stress close to the p-axiscause a reduction of the sample diameter (positive valuesof εp

3) while the cross section increases (negative values ofεp3) with N for larger average stress ratios (e.g. ηav = 1.23

in test CCP5). The direction of accumulation εpq/εp

v for anUGM under cyclic loading will be the subject of furtherexperimental studies in future.

Fig. 10f shows that the decrease of the rate of perma-nent vertical strain εp

1 with the permanent strain εp1 itself is

faster for low values of the ratio qmax/σav3 . The data points

in Fig. 10f refer to N ≥ 1000. For low stress levels qmax/σav3

= 1.20 and 1.54 the granular material reaches a state called”Shakedown” where the rate of permanent strain accumula-tion is very small. According to [2] a shakedown is reachedif the increment ∆εp

1 between N = 3000 and N = 5000 issmaller than 4.5 ·10−5. For the present tests with qmax/σav

3= 1.20 and 1.54 the criterium is fulfilled.

The banana-shaped q-ε1-hystereses are given exemplaryfor test CCP3 in Fig. 10g. The early stage of the test(N ≤ 1, 000) is shown. The shape of the hystereses doesnot significantly change with the number of cycles. Thehystereses observed in the CCP and in the VCP tests willbe compared later.

The resilient axial strain εr1 = 2εampl

1 and the resilient

lateral strain εr3 = 2εampl

3 are plotted versus the stress ratioqmax/σav

3 in Fig. 10h. The filled symbols correspond to themeasurements in the long-term CCP tests, that means themeasurements of axial deformation with the displacementtransducer (corrected by system compliance) and of volu-metric deformation using the pipette system. The strainsshown as void symbols were derived from the LDT mea-surements in the two additional tests. The εr

1-data is av-eraged from the values of the two vertical LDTs and theεr3-data is presented as a mean value of the measurements

with the eight horizontal LDTs. The data from both (cor-rected) total and local measurements coincide quite well.The under-linear curve εr

1(qmax/σav

3 ) in Fig. 10h is due tothe stress-dependence of the resilient modulus Er which is

defined as

Er =σmax

1 − σmin1

εmax1 − εmin

1

=2qampl

εr1

(1)

It can be approximated by Er ∼ (θav)n with n = 0.75 andθav = 3pav being the sum of the average principal stresses.

Beside an influence of the average stress, the increaseof the accumulation rate with an increasing ratio qmax/σav

3(Fig. 10c,d) is mainly due to the increase of the strain am-plitude (Fig. 10h).

7 VCP test results

Fig. 11a presents the increase of the permanent axial strainwith the number of cycles N in the VCP tests. In orderto describe the dependence of εp

1 on the VCP stress path,Brown & Hyde [12] used the ratio qmax/σav

3 . From thediagram in Fig. 11c it becomes clear that the data pointscannot be described by a unique function εp

1(qmax/σav

3 ). Asan alternative the longitude of the stress path

L =√

(qmax − qmin)2 + (pmax − pmin)2

=√

(2qampl)2 + (2pampl)2 (2)

may be used, although it does not supply an informa-tion about the stress path inclination ηampl = qampl/pampl.Fig. 11d shows that εP

1 correlates better with L than withqmax/σav

3 . However, the permanent strain seems to in-crease with the stress path inclination ηampl. An increase ofthe accumulation rate with increasing stress path inclina-tion qampl/pampl was also reported by Gidel et al. [22] andHabiballah & Chazallon [24]. Thus, both quantities, L andηampl, are necessary as influencing parameters in empiricalequations for εP

1 .The permanent volumetric strains, given in Figures 11b

as a function of N , can be better understood by lookingat the εp

q-εpv-diagram in Fig. 11e. For the tests VCP1,

VCP3 and VCP5 with large amplitudes (pmax = 200 kPa)the ratio εp

q/εpv increases with increasing stress path in-

clination ηampl (i.e. with increasing average stress ratioηav = qav/pav). The tendency is similar as in the CCPtests. For ηav = 0.5 the accumulation is almost pure vol-umetric (εp

q = 0). Small ηampl-values lead to a decrease of

the sample diameter (positive values of εp3) while large ones

cause an increase. The test VCP2 with a small amplitude(pmax = 100 kPa) at ηampl = 0.5 shows the same direction

7


0 1 2 3 4 5

10-5

10-6

10-7

10-8

a) b)

0

0.5

1.0

1.5

2.0

0

1

2

3

4 VCP1, L = 100 VCP2, L = 225 VCP3, L = 271 VCP4 ,L = 144 VCP5, L = 325

VCP1, L = 100 VCP2, L = 225 VCP3, L = 271 VCP4 ,L = 144 VCP5, L = 325

100 101 102 103 104 105

N [-]100 101 102 103 104 105

N [-]

Perm

anen

t axi

al s

trai

n ε 1

[%

]p

Perm

anen

t vol

umet

ric

stra

in ε

v [%

]p

0 1 2 3 4 50

1

2

3

4

5

N = 30,000 20,000 10,000 5,000 1,000


Perm

anen

t axi

al s

trai

n ε 1

[%

]p

c)

VC

P 1

VC

P 2

VC

P 3

VC

P 4

VC

P 5

0 100 200 300 400

Stress path length L [kPa]

0

1

2

3

4

5

Perm

anen

t axi

al s

trai

n ε 1

[%

]p

d)

VC

P 1

VC

P 2

VC

P 3

VC

P 4

VC

P 5

N = 30,000 20,000 10,000 5,000 1,000

0 0.5 1.0 1.5 2.0-1

0

1

2

3

4

Perm

anen

t dev

iato

ric

stra

in ε

q [%

]p

Permanent volumetric strain εv [%]p

e) f)

g) h)

Test / ηav = qav/pav = VCP1 / 0.50 VCP2 / 0.61 VCP3 / 0.92 VCP4 / 1.00 VCP5 / 1.23

0 50 100 150 200 2500

100

200

300

q [k

Pa]

p [kPa]

CSL

peak

stre

ngth

1

1.23

0.921.00

0.500.61

VCP3VCP4/5

VCP1/2

Permanent axial strain ε1 [%]p

Rat

e of

per

man

ent a

xial

str

ain

ε 1 [

-]p

VCP1, L = 100 VCP2, L = 225 VCP3, L = 271 VCP4 ,L = 144 VCP5, L = 325

0 100 200 300 4000

4

8

12

16

20

Stress path length L [kPa]

Res

ilien

t axi

al s

trai

n [1

0-4]

0

50

100

150

200

Dev

iato

ric

stre

ss q

[kP

a]

0 0.1 0.2 0.3 0.4 0.5


Test VCP3

50 -54

100 -104

200 -204

N = 1 -25




pav = 110 kPa

pav = 60 kPa

Fig. 11: Results of VCP tests: a) εp

1 versus N , b) εpv versus N , c) εp

1 as a function of stress ratio qmax/σav3 , d) εp

1 as a function of the

length L of the stress path (empty symbols: stress path inclination ηampl = qampl/pampl = 0.75, symbols with dot: ηampl = 1.125,

filled symbols: ηampl = 1.5), e) εpq versus εp

v, f) εp

1 versus εp

1, g) q-ε1-hystereses, h) resilient strain εr1 versus qmax/σav

3

8


of accumulation as the test VCP1 with the same ηampl-valuebut with a larger amplitude. However, the small cycles atηampl = 1.5 (test VCP4) also cause an almost volumet-ric accumulation (εp

q ≈ 0), although a much larger ratioεp

q/εpv was observed for the large amplitudes with the same

ηampl-value (test VCP5). These effects may be due to ananisotropy of the sample induced by the preparation pro-cedure. Further studies on the direction of accumulationεp

q/εpv of pre-compacted UGM materials are necessary.

Looking at Fig. 11b, there are two effects that affectthe rate of volumetric strain accumulation. First, the plas-tic volumetric strain increases with the amplitude (longi-tude) of the stress path (εp

v in tests Nos. VCP2 and VCP5with large amplitudes is larger than in tests Nos. VCP1and VCP4 with same ηampl but with smaller amplitudes).Second, the ratio of the rates of deviatoric and volumetricstrain accumulation εp

q/εpv increases with increasing average

stress ratio ηav (or with increasing stress path inclinationηampl, respectively). Both effects are counteracting.

In a similar way to the CCP tests, the rate of permanentaxial strain decreases with an increase of the permanentstrain itself (Fig. 11f). The data points in Fig. 11f referto N ≥ 1, 000. The decrease of the rate is faster for thesmaller amplitudes. The criteria for a shakedown [2] arefulfilled for none of the VCP tests.

Fig. 11g shows the q-ε1-hystereses in the test VCP3 forN ≤ 100. Similar to the hystereses in the CCP tests(Fig. 10g) they have a banana-like shape. The scale of theε1-axis is the same as in Fig. 10g in order to compare thearea encompassed by the hystereses in the q-ε1-diagram. Itis larger for VCP stress paths than for CCP ones.

Fig. 11h presents the increase of the resilient strain εr1

with stress path length L. The values obtained from thelong-term tests coincide well with the additional LDT mea-surements. The increase of the accumulation rate with L(Fig. 11c) is mainly due to the accompanying increase ofthe strain amplitude (Fig. 11h). Due to membrane pene-tration the values εr

3 = 0.5(εrv − εr

1) in the VCP long-termtests are not reliable. The LDT measurements show thatfor a constant pav the lateral resilient strain decreases ap-proximately linear with increasing stress path inclinationηampl = qampl/pampl. This is due to the decreasing dis-tance to the critical state line. For ηampl = constant, εr

3increases with increasing amplitude of the cycles.

8 Comparison of CCP and VCP test results

A comparison of the residual axial strains after N = 20,000cycles in CCP and VCP tests is presented in Fig. 12a (whichis similar to Fig. 3 of Brown & Hyde [12]). Except for thetests VCP4 and CCP4, the absolute differences in the εp

1-values between the CCP and the VCP tests seem to besmall. Fig. 12b contains a plot of εp

v versus qmax/σav3 . For

small stress ratios qmax/σav3 ≤ 1.5 the VCP tests deliver

significantly larger volumetric strains than the correspond-ing CCP tests. The behaviour is opposite for the largertested stress ratios qmax/σav

3 ≥ 2.7.The differences, especially for the smaller amplitudes,

become clearer from the left column of diagrams in Fig. 13showing εp

1 versus N . Comparing VCP tests Nos. 1 and 2with ηampl = 0.75 with the respective CCP tests (Figures13a,d) the permanent axial strain was larger in the VCPtests. After N = 20,000 cycles the permanent axial strainεp1 in the test VCP1 was 1.8 times larger than that in the

test CCP1. This factor was 1.6 for tests CCP2 and VCP2.In the test VCP3 with ηampl = 1.125 (Fig. 13g) the residualaxial strain was similar as in the corresponding CCP test(factor 1.1 between εp

1 in the VCP and in the CCP test atN = 20,000). When a stress path with ηampl = 1.50 wasapplied in the VCP tests the behavior depended on the am-plitude. For a small amplitude (tests No. 4, Fig. 13j) theUGM experienced a larger permanent strain in the CCPtest and the differences significantly increased with N (fac-tor 4.1 between εp

1 in the CCP and in the VCP test at N =20,000). There is no significant difference in the εp

1-valuesin the case of a large amplitude (tests No. 5, Fig. 13m, fac-tor 1.2 between εp

1 in the CCP and in the VCP test at N =20,000). Thus, CCP and VCP tests seem to deliver similarpermanent axial strains εp

1 for stress paths with intermedi-ate and large stress path inclinations 1.125 ≤ ηampl ≤ 1.5as long for the latter case the amplitudes are large. Theassumption of Brown & Hyde [12] that the permanent ax-ial strain is not affected by the type of test is disproved forsome of the tested stress paths. Especially for low stresspath inclinations ηampl = 0.75 it is not on the safe side.

0 1 2 3 4 50

1

2

3

4

5

CCP tests VCP tests

0 1 2 3 4 50

1

2

3

4

Per

man

ent v

olum

etri

c st

rain

εv

[%]

p



Per

man

ent a

xial

str

ain

ε 1 [

%]

p

Test

s 1

Test

s 2

Test

s 3 Te

sts

4

Test

s 5

Test

s 1

Test

s 2

Test

s 3

Test

s 4

Test

s 5

a)

b)

CCP tests VCP tests

Fig. 12: a) Permanent axial strain and b) permanent volumetricstrain after N = 2 · 104 cycles versus stress ratio qmax/σav

3

The permanent volumetric strains are compared in themiddle column of diagrams in Fig. 13. While VCP stresspaths with small inclinations ηampl = 0.75 delivered largerrates of volume changes than the corresponding CCP tests(Fig. 13b,e), the behavior was opposite for intermediateand large stress path inclinations (Fig. 13h,k,n), especiallyfor large numbers of cycles. The distance of the maximumstress to the CSL may play a role.

The εpq-ε

pv-strain paths on the right side of Fig. 13 show

9


100 101 102 103 104 105

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

Number of cycles N [-]

100 101 102 103 104 105



100 101 102 103 104 105



100 101 102 103 104 105



100 101 102 103 104 105



ε1

[%]

p

εv

[%]

p

ε1

[%]

p

εv

[%]

p

ε1

[%]

p

εv

[%]

p

ε1

[%]

p

εv

[%]

p

ε1

[%]

p

εv

[%]

p

0

0.5

1.0

1.5

2.0

2.5

0

0.1

0.2

0.3

0.4

CCP1 VCP1

0

0.1

0.2

0.3

0.4

0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

3

4

5

6

7

0 50 100 150 200 2500

100

200

300 CSL

1

1

1

1.5

1.125

0.75

0 50 100 150 200 2500

100

200

300

q [k

Pa]

p [kPa]

CSL

1

1

1

1.5

1.125

0.75

0 50 100 150 200 2500

100

200

300 CSL

1

1

1

1.5

1.125

0.75

0 50 100 150 200 2500

100

200

300 CSL

1

1

1

1.5

1.125

0.75

0 50 100 150 200 2500

100

200

300 CSL

1

1

1

1.5

1.125

0.75

a) b)

d) e)

g) h)

j) k)

m) n)

0

0.2

0.4

0.6

0.8

1.0

0

0.5

1.0

1.5

2.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

1

2

3

4

0

1

2

3

4

5

CCP1 VCP1

CCP2 VCP2

CCP2 VCP2

CCP3 VCP3

CCP3 VCP3

CCP4 VCP4

CCP4 VCP4

CCP5 VCP5

CCP5 VCP5

ηampl = 0.75

ηampl = 0.75

ηampl = 1.125

ηampl = 1.5

ηampl = 1.5

q [k

Pa]

p [kPa]

q [k

Pa]

p [kPa]

q [k

Pa]

p [kPa]

q [k

Pa]

p [kPa]

VCP1

VCP2

VCP3

VCP4

VCP5

CC

P1

CC

P2C

CP3

CC

P4C

CP5

εq

[%]

p

εv [%]p

εq

[%]

p

εv [%]p

c)

f)

i)

l)

o)

0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.4 0.8 1.2 1.6 2.0-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.8 1.6 2.4 3.20

0.4

0.8

1.2

1.6

0 0.8 1.6 2.4 3.2-0.4

0

0.4

0.8

1.2

1.6

2.0

0 1 2 3 4 50

1

2

3

4

5

6

CCP1 VCP1

CCP2 VCP2

CCP3 VCP3

CCP4 VCP4

CCP5 VCP5

εq

[%]

p

εv [%]p

εq

[%]

p

εv [%]p

εq

[%]

p

εv [%]p

Fig. 13: Permanent axial and volumetric strain versus number of load cycles, permanent deviatoric versus permanent volumetricstrain: comparison of CCP and VCP tests

10


a quite similar direction of accumulation εpq/εp

v in CCPand VCP tests for small VCP stress path inclinations(Fig. 13c,f). For intermediate and large ηampl-values com-bined with large stress amplitudes (Fig. 13i,o), the devia-toric portion of the direction of accumulation was signif-icantly larger in the VCP tests than in the correspond-ing CCP tests (although the average stress ratios ηav wereequal).

For small and intermediate values 0.75 ≤ ηampl ≤ 1.125,the VCP stress paths caused larger resilient axial strainsεr1 than the CCP ones. For large VCP stress path inclina-

tions ηampl = 1.5, the amplitudes of axial strain were quitesimilar in both types of tests.

Summary and conclusions

Several pairs of cyclic triaxial tests with (VCP tests) andwithout (CCP tests) a variation of the confining pressurehave been performed on an unbound granular material inorder to compare the residual deformations in both typesof tests. The present study demonstrates, that for mostof the tested stress paths CCP and VCP tests with equalaverage stress σ

av and equal amplitude of deviatoric stressqampl deliver different permanent and resilient strains. Forsome special stress paths, some of the components of strainmay be equal in CCP and VCP tests (e.g. εp

1 for interme-diate or large VCP stress path inclinations ηampl and largeamplitudes).

For the design of pavements, the permanent axial strainis the most important strain component. For small valuesof the VCP stress path inclination ηampl = qampl/pampl =0.75 the residual axial strain in the VCP tests was largerthan in the corresponding CCP test, independently of theapplied amplitude. If VCP cycles were applied along anintermediate inclination ηampl = 1.125 the residual strainsin VCP and CCP tests were similar. The same applies tolarge VCP stress path inclinations ηampl = 1.5 in combi-nation with large amplitudes. For ηampl = 1.5 and smalleramplitudes the residual strain in the CCP test exceededthe one of the corresponding VCP test. Thus, the presentwork disproves the assumption that VCP and CCP testsgenerally deliver similar results.

Since small values of ηampl are realistic for pavements,CCP tests seem to underestimate the residual axial strainin comparison to VCP tests. Thus, VCP tests should bepreferred to CCP ones. However, it has to be kept in mindthat both, CCP and VCP tests cannot simulate the shearstress component during the cycles and thus both may un-derestimate the accumulation rates observed in situ.

9 Acknowledgments

The experimental work has been done at the Instituteof Soil Mechanics and Foundation Engineering at Ruhr-University Bochum, Germany. The stay of H. Rondon inBochum was financed by scholarships of Colciencias andDAAD which is gratefully acknowledged herewith.

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