investigating effects of end platens on stresses in soil during consolidation

T E C H N I C A L A R T I C L E

Investigating Effects of End Platens on Stresses in Soil DuringConsolidationY.-S. Bae1 and J.A. Bay2

1 Environmental and Safety Research Department, Seoul Development Institute, Seocho-dong, Seocho-gu, Seoul, South Korea

2 Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322

KeywordsShear Modulus at Small Strains (Gmax),

End Effects, Resonant Column Test,

Finite Element Analysis, Effective Stresses,

Consolidation

CorrespondenceY.-S. Bae,

Environmental and Safety Research

Department, Seoul Development Institute,

Seocho-dong, Seocho-gu,

Seoul, South Korea

Email: [email protected]

Received: April 15, 2011; accepted:

March 12, 2012

doi:10.1111/j.1747-1567.2012.00828.x

Abstract

The objective of this study is to investigate the effects of rigid end platens oneffective stresses in soil mass during consolidation. The friction between theteeth of top cap/base pedestal and the specimen during consolidation decreasesthe radial and tangential effective stresses in resonant column (RC) speci-mens. However, it is unpractical to measure the effective stresses in the soilspecimen. Two approaches were used to evaluate the state of stress in RC spec-imens during consolidation. First, careful measurements were made of smallstrain shear modulus, Gmax in specimens with carefully controlled void ratiosand stress histories, to infer the state of stress. And second, a finite elementanalysis was performed to analytically evaluate the effect of various soil param-eters on the state of stress in RC specimens during consolidation. By combiningthese experimental and analytical results, an example was performed to predictthe average state of stress in RC specimens during consolidation.

Introduction

In the resonant column (RC) device, surfaces of thetop cap and base pedestal are purposely roughenedto ensure that there is no slippage at the interfacesbetween the specimen and end platens during shear.However, the friction between the teeth and the spec-imen affects the effective stresses in the soil specimenduring consolidation.

The effect of end restraints during compression testshas been studied by several investigators.1 Saada andTownsend2 recommended that the following criteriabe used for hollow cylindrical specimens to minimizedend effects:

L ≥ 5.44∗(r2o − r2

i )0.5 (1)

and

ri ≥ 0.65∗ro (2)

where ri, inside diameter of specimen; ro, outsidediameter of specimen; and L, length of specimen.

These criteria were developed for research type testconditions (e.g., different inner and outer pressures,

axial forces, large strains). For routine type testing onsolid specimens a length-to-diameter ratio of 2.0 isusually considered adequately.

Little work has been done regarding the influenceof end effect on resonant column test results duringconsolidation. This end effect is more pronouncedin short specimens, therefore it is important toproperly account for it when test short specimens.Unfortunately, it is practically impossible to makea direct measurement of normal stresses in a soilspecimen.

The deformation characteristics of soils are inde-pendent of strain amplitude at small strains below10−5%. The shear modulus at small strains of soilsis dependent of only effective stresses if identical soilmaterials under same conditions (i.e., same void ratio,same duration of confinement, and same appliedconfining pressure) are tested. By testing otherwiseidentical soil specimens with different aspect ratios todetermine shear modulus at small strain, Gmax value,a comparison of relative stress conditions after con-solidation can be made experimentally. To determinethe influence of end effect using a longer sample than

Experimental Techniques (2012) © 2012, Society for Experimental Mechanics 1

The Effects of End Platens on Soil Y.-S. Bae and J.A. Bay

the conventional RC specimen is important issue;however making specimen longer than the length-to-diameter ratio of 2.0 is not practical because ofthe difficulty in maintaining the specimen geometryduring soil setup.

Bancroft3 studied the influence of specimen geom-etry on the frequency equation of one-dimensionalwave propagation. He concluded that the one-dimensional theory of rods can be used, withoutsignificant error, if the wavelength of the propagatingwaves is significantly larger than the diameter of therod. McNeil4 also states if the wavelength to diameterratio is greater than 2.5, the one-dimensional theorycan be validated. The standard resonant column spec-imen is constructed to meet this criterion. Since allspecimens with aspect ratios of 2:1, 1:1, and 1:2 meetthe criterion, they can be used without significantlyviolating the frequency equation.

Analytically, stresses in specimens can be assessed.The effective stresses generated in soil specimenswere investigated using the finite element program(PLAXIS). Soil parameters were varied to study therelationship between shear modulus at small strains,Gmax and various soil parameters. A method is pre-sented for predicting average normal stress in shortspecimens. This method combines the experimentaland analytical analyses.

Sample Preparation

Test materials

The basic soil properties are summarized in Table 1.

Sample preparation

It has been shown that specimen preparation effectsthe specimen structure.5 They evaluated the effects ofspecimen compaction technique (kneading, staticloading, and impact) on resonant column test resultsand concluded that the compaction technique is animportant variable in specimen preparation for anylaboratory tests.

In this study, 3.56 cm diameter specimens wereprepared using the raining method. A speciallydesigned mold and rod shown in Fig. 1 were usedto make specimens with different aspect ratios. Oven

Table 1 Soil properties index of tested sand specimen

USGS

Curvature

coefficient, Cc

Uniformity

coefficient, Cu PI

Median particle

size, D50, mm

SP 1.423 3.1 NP 0.57

15.2 cm

6.4 cm

φ 3.56 cm

φ 3.56 cm

φ 1.5 cm

φ 1.5 cm

φ 3.56 cm

5.3 cm

0.95 cm × 12

4.4 cm

1.9 cm

Figure 1 Mold and rod.

dry sand was poured uniformly into the mold througha small hole with the diameter of 10 mm from a con-stant height of 1 cm above the sand surface. Afterthat, the rod with scale was used to maintain a pre-cise lift thickness of 0.95 cm. To make void ratio anddensity as uniform as possible along the whole heightof the specimen, the soil was divided into portionswhich had the same weight for each lift. A specimenwith an aspect ratio of 2:1 was made with 8 lifts.Similarly, a 1:1 aspect ratio required four lifts, and a1:2 aspect ratio required two lifts. The soil of bottomin the mold can be denser than the soil of top in themold because of the weight of soil itself. But this effectcan be negligible so it was not considered. The initialvoid ratios for specimens with aspect ratios of 2:1, 1:1,and 1:2 are 0.81, 0.84, and 0.83, respectively.

Next the soil specimens were inundated with waterand frozen. Frozen soil samples were used for ease inmeasuring and setting-up tests. An important aspectof testing saturated soils is the capillary effect on theshear modulus. Wu et al.6 showed that there was lit-tle difference in value of Gmax found in the range of70–100% Sr and the value of Gmax was the same forSr = 0 and 100%. If water is poured on top of thesoil, air bubble can be trapped between particles. Toprevent this, water was slowly poured on top of the

2 Experimental Techniques (2012) © 2012, Society for Experimental Mechanics

Y.-S. Bae and J.A. Bay The Effects of End Platens on Soil

tube connected to the bottom of mold so that trappedair could escape upward through voids.

Pressure measure equipments

The accurate measurement of applied confining pres-sure is critical in studying the effect of confiningpressure on shear modulus at low strain, Gmax. Preci-sion regulators were used to regulate the air pressureconfining the soil specimens. The calibration for thetwo omega pressure transducers was carefully per-formed using a dead weight tester, which is generallyregarded as the most accurate method in calibrationof pressure sensors. This test was performed by read-ing the transducer pressure while applying pressurewith dead weight.

Frozen soil sample set up

The soil samples with the mold were placed in afreezer (−20◦C) for 24 h. The samples were allowedonly to expand in vertical direction during freezingprocess. The volume change during freezing was con-sidered to calculate void ratio and degree of saturationof the sample. Bae7 presented the details of procedurein calculating the void ratio during consolidation. Thedimensions and weight of frozen soil were measuredcarefully using calipers in the freezer.

After measurement, a small circular piece of filterpaper was placed on the drain part of the resonantcolumn base pedestal. The filter paper was used toprevent soil grains from clogging the drainage lineof base pedestal. The top cap was then placed onthe sample and a membrane was placed around thesample. Two membranes with vacuum silicon greasewere used to minimize leakage of vapor through thespecimen during the test. O-rings were used to sealthe membranes to the top cap and base pedestal. Theinner support cylinder with the electro-magnetic driv-ing system was then placed around the sample. Next,the LVDT, accelerometer, and proximity probes withproximity target were placed in position. This proce-dure requires about 40 min. In order to maintain thefrozen state of the soil, this specimen preparation wasdone in the freezer.

After the sample setup, the base pedestal with soilsample and drive system was placed on the isolationtable, and the lowest confining pressure was applied.The low strain shear modulus of the soil samples wasevaluated at three confining pressure: 25.3, 101.3,and 202.7 kPa. Each pressure was sustained for 24 hbefore incrementing the confining pressure.

After the applied confining pressure was applied,the resonant frequency was measured at preselected

time intervals. The first readings at each pressure weremeasured after the soil samples thawed completelyduring confinement (it takes about 2 h). Sampledrainage was permitted during the entire testingsequence. A burette was used to maintain specimensaturation. The burette was connected to the twodrainage line on the base plate. The level of waterof the burette maintained at the middle of specimen.The hydrostatic pressure below water level can benegligible since negative pore pressure occurs abovethe free water surface. This assumption is reasonableperforming a Stokoe-type RC/TS device.

After all measurement were completed, the con-fining pressure was reduced to zero, the device wasdissembled, and final sample dimensions and weightwere measured to determine final void ratio and finalwater content of the soil. The final void ratios forspecimens with aspect ratios of 2:1, 1:1, and 1:2 are0.80, 0.81, and 0.80, respectively and the final watercontent (%) for specimens with aspect ratios of 2:1,1:1, and 1:2 are 25 (%), 22 (%), and 24.9 (%),respectively.

Low Strain Test Results

The low strain shear modulus, Gmax varies withconsolidation time. In the laboratory, the time-dependent response for shear modulus can beseparated into a period of primary behavior, which isanalogous to primary consolidation, and a period ofsecondary behavior.

The primary consolidation can be neglected in thesand soil and only secondary behavior is considered.The secondary time effect cannot be accounted formerely by the change in void ratio and the rateof secondary increase in shear modulus or shearwave velocity is a function of logarithm of time.The magnitude of secondary increase, as defined bychange in modulus per logarithmic cycle of time, wasfound to vary with soil type.

It has been known that Gmax is a function of voidratio.8 Even though the specimens were constructedas carefully as possible, there were variations invoid ratio. To remove only void ratios effect, Gmax

is corrected by:

Gmaxcorrected = Gmaxmeasured· F(e0)

F(eact), (3)

where F(e) = 1/(0.3 + 0.7e2), e0 = corrected voidratio, and eact = measured void ratio.

Table 2 shows the ratio (�G)per log cycle/(G)1000 min

for this study. The low strain shear modulus, Gmax

was corrected for a void ratio of 0.8. The average



Table 2 Ratio of (�G)per log cycle to (G)1000 min

Aspect ratio 25.3 kPa (%) 101.3 kPa (%) 202.7 kPa (%)

2:1 3.44 2.57 2.311:1 3.10 2.98 2.221:2 3.25 2.38 2.64

140 × 103

120

100

80

60

40

20

0

Low

Str

ain

She

ar M

odul

us, k

Pa

1 10 100 1000

Time, minute

2 to 1 ratio, 202.7 kPa 1 to 1 ratio, 202.7 kPa 0.5 to 1 ratio, 202.7 kPa 2 to 1 ratio, 101.3 kPa 1 to 1 ratio, 101.3 kPa 0.5 to 1 ratio, 101.3 kPa 2 to 1 ratio, 25.3 kPa 1 to 1 ratio, 25.3 kPa 0.5 to 1 ratio, 25.3 kPa

Figure 2 Effect of time of confinement on all soil specimens.

ratio of (�G)per log cycle/(G)1000 min is 2.77 %, which ismuch higher than the ratio investigated by Afifi andRichart.9

Figure 2 shows that the low strain shear modulus,Gmax versus time for soil specimens with aspect ratioof 2:1, 1:1, and 1:2, respectively. The fitted equationsfor the time effect are indicated in Table 3.

Effect of End Restraint

In RC test, the top and bottom of the specimenare rigidly fixed to the base pedestal and top cap,respectively, which each have roughened contact sur-faces to enhance fixidity. The specimen adheres to the

Figure 3 Hyperbolic stress-strain relation of Hardening Soil model.11

top cap on the driving system and the base pedestal,which prevents radial deformation at the top andbottom of the specimen. During consolidation, thefrictional restraint between the end platens andthe specimen also creates nonhomogeneous stressesthroughout the specimen.

PLAXIS hardening model

To investigate the effective stresses in the soil, a com-mercial finite element program, PLAXIS was used.PLAXIS incorporates advanced soil models. One ofthese models is the hardening soil model. The PLAXIShardening soil model is an advanced soil model whichsimulates the nonlinear (hyperbolic) soil stress–strainbehavior and different loading and unloading behav-ior. The hardening soil model accounts for the stressdependency of soil moduli and the dilatency behaviorand uses a Mohr–Coulomb failure model.

The hardening soil model accounts for the stressdependency of soil moduli using a variation of Janbu’smethod.10 The relationship shown in Fig. 3 is:

E50 = E50ref

(c cot φ − σ3

′

c cot φ + pref

)m

(4)

where E50, Young’s modulus of 50% strength; E50ref,

Young’s modulus of 50% strength at referencepressure; c′, cohesion; σ ′

3, confining pressure; φ′,friction angle; pref, reference pressure; and m, powerfor stress-level dependency of stiffness.

E50ref, Eoed

ref, Eurref, effective friction angle (φ’),

dilatancy angle (ψ), and stress dependency power

Table 3 Fit equation for time effect

2:1 aspect ratio 1:1 aspect ratio 1:2 aspect ratio

25.3 kPa 26.15log(t) + 694.48 22.6log(t) + 683.54 263.19log(t) + 659.99101.3 kPa 41.32log(t) + 1564.8 46.8log(t) + 1483.1 36.24log(t) + 1503.3202.7 kPa 54.58log(t) + 2338.5 50.48log(t) + 2267.2 59.33log(t) + 2176.6



(m) are the major parameters in the hardening soilmodel. A value of 0.5 was used for the power forstress-level dependency of stiffness, m, for sand andthe parameter, Eref

50 introduced in the PLAXIS manualis based on Janbu’s test results.10

The hardening soil model accounts for the stressdependency of soil moduli using a variation of Janbu’smethod.10 E50

ref (Young’s modulus of 50% strengthat reference pressure) is a major parameter in thehardening soil model.

In this study, E50ref was calculated based on Hooke’s

law using:

E50ref = 2(1 + υ)G50

ref (5)

where G50ref = 0.5 Gmax

ref (half of Gmax at 101.3 kPa)and υ = 1/3. Gmax at 101.3 kPa was obtained usingresonant column test device.

Consolidated, drained (CD) triaxial compressiontest was performed to measure the friction angle ofthe sand soil (φ′). Four specimens were prepared andconsolidated isotropically to confining pressures of25.3, 101.3, and 202.7 kPa. Ice cubes were placed inthe triaxial cell to slow the thawing of the specimens.

The procedures of sample set up are as follows:

(1) The ice cubes were placed in the cell afterthe frozen soil sample is placed in the triaxialchamber.

(2) The tube between water tank and triaxialchamber was connected and water filled the cell.

(3) The hydrostatic confining pressure was applied,and the soil was allowed to thaw and consolidatesfor 24 h.

(4) A deviator stress was applied to the soil specimenuntil the soil specimen failed in shear.

The internal friction angle (φ′) is then obtained by:(

σ ′1

σ ′3

)f

= tan2

(45 + φ′

2

)(6)

The failure is defined as the maximum σ ′1/σ ′

3. Theprinciple stress ratio (σ ′

1/σ ′3) versus axial strain (%)

and failure points identified with arrows are plotted inFigs. 4 and 5, respectively. Table 4 shows the internalfriction angles (φ′) for each consolidated pressures.

PLAXIS analysis results

To investigate the effect of soil parameters on theaverage effective stresses generated in soil mass,the following components of stress were investigated:σ ′

r : effective radial stress; σ ′θ : effective vertical stress;

σ ′z: effective tangential stress.

6

5

4

3

2

1

0

stre

ss r

atio

(σ 1′/σ

3′)

302520151050

strain (%)

confining pressure =25.3 kPa

Figure 4 Principle stress ratio (σ ′1/σ ′

3) vs axial strain (%) for 25.3 kPa cell

pressure.

5

4

3

2

1

0302520151050

strain (%)

confining pressure=202.7 kPast

ress

rat

io (

σ 1′/σ3′)

Figure 5 Principle stress ratio (σ ′1/σ ′

3) vs axial strain (%) for 202.7 kPa cell

pressure.

Table 4 Internal friction angles (φ′) for each consolidated pressure

Consolidated pressures (kPa) 25.3 101.3 202.7

Friction angles at failure, φ’ (◦) 43 40.1 39.2Average friction angle (◦) 40.8

Stress points are used to calculate the averageeffective stresses in the soil mass. The average effectivestresses are then calculated by averaging the stresses ateach stress point. Details of procedures in calculatingthe effective stresses in soil are introduced by Bae.7

Effect of interface strength (Rinter) on average effective stresses

The soil contacts both top cap and base pedestal. Tomodel the interaction between the soil and metal,interfaces are used in PLAXIS. Interface strength(Rinter) is a strength reduction factor for the shearstrength along the soil–metal interface. The interfaceproperties are calculated from soil properties asfollows:

cinter = Rinter · csoil (7)

tan φinter = Rinter · tan φsoil (8)

In general, for real soil–structure interactionthe interface is weaker and more flexible than the



Table 5 Material properties of the sand, base plate, and top cap

Parameter Sand Top cap Base plate

Material model Hardening soil Linear elastic Linear elastic

Unit weight 1.45 g/cm3 N/A N/A

Stiffness Eref50 = 1.16E + 05 kPa Eref = 3.02E + 07 kPa Eref = 2.0E + 08 kPa

Poisson’s ratio 0.33 0.33 0.285

associated soil layer, which means that the value ofRinter is less than 1. Suitable values for Rinter for thecase of the interaction between various types of soiland structures in the soil can be found in PLAXISmanual, Version 7:11

Interaction sand/steel = Rinter ≈ 2/3;Interaction clay/steel = Rinter ≈ 0.5;Interaction sand/concrete = Rinter ≈ 1.0–0.8;Interaction clay/concrete = Rinter ≈ 1.0–0.7;Interaction soil/geogrid = Rinter ≈ 1.0;Interaction soil/geotextile = Rinter ≈ 0.9–0.5 (foil,textile).

In the absence of detailed information, it maybe assumed that Rinter is of the order or 2/3 for asand–steel contact and of the order of 1/2 forclay–steel contact, whereas the interaction withrough concrete usually gives a somewhat highervalue. The same material properties of the sand, basepedestal, and top cap presented in Table 5 were usedin these analyses.

Figures 6 and 7 present the effect of Rinter on aver-age effective stresses in the soil mass at 202.7 kPafor an aspect ratio of 2:1 and 1:2. The average effec-tive stresses were also normalized by the confiningpressures at each component (radial, vertical, andtangential). It was seen that the radial and transverseaverage effective stresses decrease with increasingvalues of Rinter, and this effect is pronounced usingshorter specimens. The tangential average effectivestresses are most effected by the values of Rinter and

1.0

0.8

0.6

0.4

0.2

0.0

Nor

mal

ized

avg

. effe

ctiv

e st

ress

es

1.00.80.60.40.20.0

vertical stresses component radial stresses component tangential stresses component

Rinter

Figure 6 Effect of Rinter on normalized effective stresses for a specimen

with an aspect ratio of 2:1 using hardening soil Model at 202.7 kPa cell

pressure.

1.0

0.8

0.6

0.4

0.2

0.0Nor

mal

ized

avg

. effe

ctiv

e st

ress

es

1.00.80.60.40.20.0

vertial stresses component radial stresses component tangential stresses component

Rinter

Figure 7 Effect of Rinter on normalized effective stresses for a specimen

with an aspect ratio of 1:2 using hardening soil Model at 202.7 kPa cell

pressure.

the vertical average effective stresses are less effectedby the value of Rinter. Figure 6 also shows that theaspect ratio based on ASTM standard D4015-92 doesnot have large effect on average effective stresses.

Effect of internal friction angle (φ′) on average effective stresses

Effects of friction angle (φ′) on average normalizedeffective stresses in the soil mass for specimens withan aspect ratio of 2:1 and 1:2 are presented in Figs. 8and 9. It was seen that the vertical average effectivestresses are not affected by the friction angle for allaspect ratios using the hardening soil model. Thetangential average effective stresses are most affectedby friction angle and this effect is pronounced usingshorter specimens. As shown in Fig. 8, the effect offriction angle on average effective stresses is negligiblewith a specimen with an aspect ratio of 2:1.

1.0

0.8

0.6

0.4

0.2

0.0

Nor

mal

ized

avg

. effe

ctiv

e st

ress

es

50403020

axial normalized stressesradial normalized stressestangential normalized stresses

φ' (degree)

Figure 8 Effect of φ′ on normalized average effective stresses for a

specimen with an aspect ratio of 2:1 using hardening soil model.

1.0

0.8

0.6

0.4

0.2

0.0

Nor

mal

ized

avg

. effe

ctiv

e st

ress

es

50403020

axial normalized stresses radial normalized stresses tangential normalized stresses

φ′ (degree)

Figure 9 Effect of φ′ on normalized average effective stresses for a

specimen with an aspect ratio of 1:2 using hardening soil model.



Verification of soil parameters

An empirical relationship between Gmax and void ratioand effective stress (σ ′) is presented in Eq. (9).

Gmax = A · F(e) · (σ ′)n (9)

It was shown that the shear modulus at smallstrains is dependent of only mean effective stresses(σ ′

0) if identical soil types (same void ratios and timeconfinement) are tested and Gmax versus σ ′

0 plotsshould be collapsing to one line on the log-log scaleirrespective of the ratio of radius to sample height.

As presented in previous sections, PLAXIS programwas employed to investigate input soil parametersaffecting the effective stresses in soil mass. Hence,varying soil parameters, curve-fitting techniques wereperformed to satisfy Eq. (9). The best values ofthe soil parameters are the ones which minimize thevalue of chi-square values. The chi-square values aredefined as:

chi-square =∑(

y − yi

σi

)2

(10)

where y, fitted value for a given point; yi, original datavalue for the point; and σi, standard deviation of eachdata value.

As described in section Effect of Interface Strength(Rinter) on Average Effective Stresses, interface strength(Rinter) influences the effective stresses in soil mass.PLAXIS Version711 suggests to use 2/3 for interactionbetween sand and steel. The interface strength valuesto minimize the value of chi-square are comparedwith the suggestion values.

The internal friction angles (φ′) are measured usingtriaxial compression test. The internal friction anglesto minimize the value of chi-square are also comparedwith the measured ones.

Another comparison is done by using the average ofvertical and transverse effective stresses (σ ′

avg) insteadof mean effective stress (σ ′

0) to calculate low strainshear modulus, Gmax.12 The average of vertical andtransverse effective stresses is given by:

σ ′avg = (σ ′

θ + σ ′z)/2 (11)

where σ ′θ , effective stress in vertical direction and σ ′

z,effective stress in tangential direction.

Table 6 presents the chi-square values varyinginternal friction angles (φ′) and interface strength(Rinter). The minimum chi-square value is representedby bold character. The lowest chi-square value wasobtained using interface strength (Rinter = 0.6), andinternal friction angle (φ′ = 40◦). And the relationshipbetween Gmax versus effective stresses is well

Table 6 Chi-square values varying internal friction angles (φ′) and

interface strength (Rinter)

φ′ = 30◦

σ ′0 = (σ ′∗

r + σ ′v† + σ ′

t ‡)/3 σ ′avg = (σ ′

v + σ ′t )/2

Rinter = 0.5 6.503e-2 7.822e-1Rinter = 0.6 4.387e-2 5.767e-2Rinter = 0.7 4.486e-2 5.529e-1

φ′ = 35◦

σ ′0 = (σ ′

r + σ ′v + σ ′

t )/3 σ ′avg = (σ ′

v + σ ′t )/2


φ′ = 40◦

σ ′0 = (σ ′

r + σ ′v + σ ′

t )/3 σ ′avg = (σ ′

v + σ ′t )/2


∗Effective stress in radial direction.

†Effective stress in vertical direction.

‡Effective stress in tangential direction.

4

5

6

7

8

9

105

Gm

ax (k

Pa)

3 4 5 6 7 8 9 100 2

mean effective stresses (kPa)

Gmax = 407.1*effctive stresses^0.563

2 to 1 ratio1 to 1 ratio1 to 2 ratio

Figure 10 Relationship between Gmax and mean effective stresses,

σ ′0 = (σ ′

r + σ ′v + σ ′

t )/3.

represented by using mean effective stress (σ ′0) rather

than the average of vertical and transverse stresses(σ ′

avg). The result verifies the measured internalfriction value (φ′ = 40.8◦) and the suggested interfacestrength (Rinter ≈ 0.67) for the interaction betweensand and steel. Given that strength parameters, themeasured Gmax values versus predicted mean effectivestresses (σ ′

0) were shown in Fig. 10.

Procedure to estimate mean effective stresses in soilmass

According to Perloff and Pombo,1 the frictionalrestraint between the end platens and the spec-imen, stresses are not homogeneous throughoutthe specimen. They showed nonhomogeneous stress



Figure 11 Friction on soil during consolidation due to rigid platens.

contours throughout specimen at 1% axial strain. Theschematic friction effect on soil due to rigid platenswas shown in Fig. 11.

To estimate the average effective stress in thesoil mass, the experimental and analytical analysesare combined. Several soil strength parameters fromexperimental test are needed to perform the analyticalanalyses. The internal friction angle (φ′) and cohesion(c′) can be measured using CD triaxial compressiontest and Young’s modulus of 50% strength atreference pressure (Eref

50 ) is calculated from the shearmodulus at low strains (Gmax) using resonant column(RC) test. The values of interface strength (Rinter) areavailable in PLAXIS manual.

Using these strength parameters, the effectivestresses in the soil specimens are then assessed ana-lytically using the finite element program (PLAXIS).Finally, the mean effective stresses (σ ′

0) are obtainedby averaging the vertical, tangential, and radial com-ponent stresses.

Conclusion

The effects of rigid end platens on effective stressesin soil during consolidation were investigated. Exper-imental and analytical analyses were performed toevaluate the state of stress in RC specimens. Carefulmeasurements were made of small strain shearmodulus, Gmax in specimens with carefully controlled

void ratios and stress histories, to infer the state ofstress. And a finite element analyses was performed toanalytically evaluate the effect of various soil parame-ters on the state of stress in RC specimens during con-solidation. The results showed that the soil strengthparameter (φ′) and interface strength (Rinter) have alarge effect on the effective stresses. And these effectsare more pronounced on the shorter specimens. Bycombining these experimental and analytical results,an example was performed to predict the averagestate of stress in RC specimens during consolidation.

References

1. Perloff, W.H., and Pombo, L.E., ‘‘End RestraintEffects in the Triaxial Test,’’ Proceeding of the 7thInternational Conference on Soil Mechanics andFoundation 327–333 (1969).

2. Saada, A.S., and Townsend, F.C., Young, R.N., andTownsend, F.C., (eds), State of the Art: LaboratoryStrength Testing of Soils, Laboratory Shear Strength ofSoil, ASTM 740, ASTM. pp. 7–77 (1981).

3. Bancroft, D., ‘‘The Velocity of Longitudinal Waves inCylindrical Bars,’’ Physics Review 59: 588–593 (1941).

4. McNeil, R.L. A Study of the Propagation of Stress Wavesin Sand. Air Force Weapon Laboratory. TechnicalReport No. AFWL-TR-65-180 (1966).

5. Anderson, D.G., and Woods, R.D., ‘‘Time-dependentIncrease in Shear Modulus of Clay,’’ Journal of theGeotechnical Engineering Division ASCE 102 (5):525–537 (1976).

6. Wu, S., Gray, D.H., and Richart, F.E., ‘‘CapillaryEffects on Dynamic Modulus of Sands and Silts,’’Journal of the Geotechnical Engineering Division ASCE110 (9):1427–1447 (1984).

7. Bae, Y.S., Modeling Soil Behavior in Large StrainResonant Column and Torsional Shear Tests, Utah StateUniversity, Logan, Utah (2007).

8. Hardin, B.O., ‘‘The Nature of Stress-Strain Behaviorfor Soils,’’ Proceedings, Earthquake Engineering and SoilDynamics, Pasadena, California ASCE 3–90 (1978).

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investigating effects of end platens on stresses in soil during consolidation

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