EVALUATION OF MODULUS AND POISSON'S

RATIO FROM TRIAXIAL TESTS

M. Hassa¡ Farzin, Stone and Webster Engineering Corporation, Boston; and

Raymond J. Krizek and Ross B. Corotis, Technological Institute'Northwestern UniversitY

A method is suggested to determine piecewiSe linear, stress-dependentrelationships for the modulus and Poissonrs ratio of soils. The method isbased on linear elasticity and conditions associated with conventional tri-axial tests at different states of stress. The tangent modulus at a givenstress level is shown to be the slope of the axial stress-axial strain curveat that stress level, and the value of Poissonts ratio is evaluated by use oftheoretical considerations and a simple graphical construction. Thismethod of interpretation is applied to e4perimental data from two naturalsoils used in an actual full-scale field installation of buried concrete pipe,and the results are shown to be in reasonable agreement with those deter-mined by more sophisticated analyses and more extensive experimentalmeasurements. It is also demonstrated that other analytical methods forinterpreting these test data may yield significantly different values for themechanical properties of soils, and this must be taken into account whensuch results are incorporated into mathematical models for the responseof soil-structure sYstems.

oONE of the major difficulties associated with currently availabte techniques of analy-sis for problems in soil-structure interaction is the specification of soil properties.Although high*speed digital computers and the finite element method have provided theopporfunity to hãndle material properties in a more realistic manner, the determinationoi ttrese properties has been the subject of much controversy. Several recent studies(4, S,6) frave tretpeO irnmensely in the understanding of soil behavior under complicateds-taftlof istress, but the tests discussed involve the use of sophisticated equipment thatis generally not available in most laboratories. Therefore, there is still a pressingneed for a procedure that can be used with data from a standard laboratory test to de-duce the mâterial properties required in the formulation of a problem involving soilresponse. Accordingly, this study wilt describe and evaLuate e4perimentally a methodthaf interprets the results of a conventional triaxial test to obtain piecewise linearvalues for the modulus and Poissonrs ratio of soil.

STRESS-STRAIN PARAMETERS

The theoretical formuiation of any problem based on the theories of piecewise linearelasticity requires two parameters, the modulus of elasticity E and Poisson's ratio u,

to characterize an isotropic material. By definition, E is the slope of the axial stress-axial strain curve in a uniaxial stress test, and y is the ratio of the lateral strain to thetongitudinal strain for a specimen that is uniaxially stressed in the iongitudinal direc-tion. The state of stress in both cases is assumed to be homogeneous. Since it is

Publication of this paper sponsored by Comm¡ttee on Subsurface Soil-Structure Interaction and Commitïee

on Soil and Rock Properties.

69

70

generally not practical to test soil under homogeneous states of stress, three differentand somewhat less ideal laboratory tests (uniaxial strain test, triaxial test, and planestrain test) are usually used; however, the results of these tests are gene"á[y i¡iter-preted on the basis that the soil specimen is an inJinitesimal element with an associatedhomogeneous state of stress. This assumption is not true, and serious errors may beintroduced because of the various boundary conditions. Accordingly, unless otherwiseshown, none of the conventional soil tests can provide directly the values of E and yfor soils.

Even if the notion of an inÍinitesimal element and homogeneous state of stress is ac-cepted, the usual test data (which do not include lateral deformation or lateral stressmeasurements) and method of interpretation allow the evaluation of only one parameter,especially for partially saturated soils. In such cases, v is often assumed, and Hooke'slaw is used to determine E. For example, the results of a uniaxial strain test and anassumption lor u may be used in conjunction with

n=o'( â

to determine E. However, the test is usually performed to determine the values ofboth parameters.

SCOPE OF STUDY

The theoretical part of this sfudy considers the boundary value problem associated witha triaxial compression test on a cylindrical soil specimen, and the effect of the assump-tion of a homogeneous state of stress is evaluated. Results of the theoretical findingswere verified experimentaily by conducting triaxial tests with radial deformation mea-surements on two soils from an actual full-scale field installation of buried concretepipe, and comparison was made with other methods for determining these parameters.Throughout this investigation, we assumed the soil to be isotropic, time-independent,and piecewise linear elastic and the loading to be monotonic.

THEORE TICAL CONSIDERATIONS

For the boundary conditions usually associated with a conventional triaxial test withrough end platens, the radial and axial displacement components, u and w, of a linearelastic cylinder subjected to the axially symmetric loading conditions shown schematic-ally in Figure 1 can be closely approximated by

u = uor * f lrrrt * å u2r5 + f,urrr' *! unr"z'* | o, rza + tJ (1)

Q)1w=woz+fwtz'+

where the periodic parts

Ðn=1

!*rru *L*"r', *Lwn*'zt * f *, fz +w

U and W are given by

- çrI,(kr)]u=F * t,to"l cos kz (3)

77

Figure '1. Test specimen and boundary conditions.

Looding ûnd Coordìnole Syslôm

w = Ë [* r, to'l + f;rr'(r<r)] sin kz (4)

in which Io (kr) and I, (kr) are Bessel functions of the zero and first order respectively,and kr is the imaginary argument. The derivation of equations 1,,^2,, 3r. and 4 and the

associated coeffiõients arJgiven by Farzin, Corotis, and Krizek (Ð. An evaluation of

the vertical displacement w-at ttre ends (z = +c) of the specimen yields

- 1 --- -g 1 --- ^s (b)w = woc + t wrc- + 5

wzc-

The vertical displacement w can be shown to vary with z in essentially a linearmanner; thus, the âxial strain 6" at any point in the middte three-quarters of the speci-

-u., *iy be áefined as essentiallyw/c, ãnd equation 5 maybe reasonablywritten as

€a =wo*f "'*r*f,"n*,(o)

(?)

(a)

(e)

where

*"=|g;o(+_)

*, =H {"t*.#t2]- fr} ,' *,) (r - z,{

*, ="fu"#r( å')

in which

2(r + v) [î,t - ,) - ,lD=m (ro)

72

n=u(rl;l*å"-å*ffi"

,' = #W.'-'] - å". åHi# - å(--+)]'-

r,, = - fs* (*") - +( à. å.' (#-_ )

. - å ,n lrr+ 1)a + [te, *ri r - (Bv + 4) totu -

u?=tæ

1

'-z(t-v)

4-kâ.

S =9a

'fffc

where Io and L are Bessel functions of the zero oriler respectively,argunent, and p" is the average distributed axial load given by

Ft .2nn^=#l I ozrdrdo

to to

deu _ dwo - 1 ^zdwr -

1.+ dwzdp. -dp. '3" dp. ' 5" dp"

Differentiation of equations ?, 8, and g gives

(1 1)

(rz)

(13)

(14)

(15)

(18)

e is the imaginary

(ro)

(17)

(rg)

The rate of change of the axial strain (" for the rlverage distributed a¡nal load p. can bedetermined by the differentiation of equation 6 with respect to p.. The result is

(zr)

# = + {f *J*i' t,"m;+xhr'="J}

Since equations 21, 22, and,23 ate independent of pu, it follows that the slope given byequation 20 is also independent of p". When equations 21, 22, ænd 23 are substitutedinto equation 20, a long and tedious computation yields

?3

Qz)

(23)

(2 b)

(26)

Q+)d("41dp"=E=E

where Er is the slope of the axial stress versus axial strain measured in a conventionaltriaxial test, A is â function of Poisson's ratio u, and the slenderness ratio S of the

specimen is

A=(1 - rrluL

$t - z,t

When equation 24 is rewritten asin which ô is given bY equation 14.

n = a@1= en,d€u

it canbe seen on Figure 2atltat,A approaches unity, or E approaches ET, as S increases.¡r particular, for uulous of 0.3 or 0.4 tor Poissonts ratio, the error is on the order of4 tä 6 percent at a slenderness ratio of 2 (commonly used for triaxial test specimens)'Howevèr, since the actual boundary conditions at the ends of the specimen may.be some-what lesé constrained than the idealized rough boundary conditions assumed in this so-lution, the actual error may be slightly less. For small values of S, the_value oj $from êquation 25 is such that Er approaches the constrained modulus M (Figure 2b).

The preceding results can also be shown by direct algebraic computation. Fo_r ex-ample, ìt U"ge lalues (S > Z) are assumed for the slenderness ratio S, 4 is small, and

the ratio of I¡/h can be approximated by

ro/r,=å('-'i-#. ) Q7)

Retention of terms up to the fourth order and direct substitution of equation 27 intoequation 14 give

(n{s'

74

ô=e6 Ën=1 å{'

d4 fzg + 24(1 - ,)lì- so L--zn-¡-J1 (ze)

(zs)

(¡ r)

which, on expansion and retention of only the fourth order terms, becomes

o=ffisn+m 23 +24(1 - u)

2Í+u)

where ra is approximated by 90, and m is the largest number of terms taken in equation28 for which the approximation given by equation 2? (small Q, where c = mn /S) is ac-ceptable. Substitution of the values for ô given by equation 29 into equation 24 yields

+ 24(r- u)l m - #tt - 2ù (t + v)(30)

-f,tzs +24(L- y)l m - Sl - rl

Since, for small values of { the first terms in both the numerator and the denominatorof the right side of equation 30 are large relative to the other terms, they dominate themagnitude of equation 30. Thus, for a long specimen (S = Z), equation 30 can be ap-proximated by

d€" t{r=ul

-f,rzz

n, =ff-n

Implicit in the conclusion that E approaches dp./de" as S increases is the suggestionthat the radial deformations u are approximately uniform over most of the length of thespecimen. As shown in Figure 3 for three typical cases, this is indeed true from atheoretical point of view. For purposes of emphasis, the scale of the deformations isdi-fferent from that used for the dimensions of the specimen. In the middie case, theapplied pressures and the value of Poissonts ratio are such that the radial deforma-tions are almost zero.

Letting

pa = mpr (sz)

and

<" = Ép' (33)

equation 6 can be plotted conveniently, as in Figure 4, in terms of generalized pa-rameters po/p,, Fp", and Poissonfs ratio u. As previously explained, the influence ofS is negligible if S > 2. It is observed that the ordinate of the point where the plot in-tersects the axial pressure axis is directiy proportional to the value of Poissonrs ratio.This characteristic can be used advantageously to determine the value of Poissonts

?5

Figure 2. Variation of moduli versus slenderness ratio.

Figure 3, Radial deformations along

specimen,

"l .5d!ßa&

=9"

cõi<l

.9

E^

ilni.=o.o,r=opd fr 'o.t'v=o.zs fr =o., ,=ou

Figure 4, ldealized relationships among

presure, strain, and elastic propert¡es.

Figure 5, Stepwise determination of linear elastic properties from experimental data.

.5dl"

.tE¿E

E€5&

À'i.

.9 I

Èlc oto

Axiol Stroin, €a

76

ratio from the results of a conventional triaxial test in which only the axial displace-ment of the specimen is measured. This is accomplished by setting €a eeual to zeroand by combining equations 6, 7, 8, and g to obtain

p_A-(1 .r)e)^(.+)

(34)

For S > 2, A.o 1 and equation 34 reduces to

Hence, if p"/p, is known to be e. = 0r

(3 5)

v may be readily determined from equation 35.

EVALUATION OF MATERIAL PROPERTIES

Modulus

As shown by equation 31, the value of E at a given stress level can be evaluated by tak-ing the slope of the axial stress-axial strain curve, which is obtained from a conven-tional triaxial test with a constant cell pressure. Such a curve is shown in Figure 5a.These values of moduli are termed tangent moduli since they characterize a change inloading in terms of a series of incremental loads.

Poisson's Ratio

Equation 35 shows that, if the value of the axial pressure p'" is given at €' = 0, theinitiai value of Poisson's ratio can be readily calculated. As shown in Figure 5b, thevalue of pao cân be determined by measuring the initial axial strain €,r of the soil spec-imen due to the hydrostatic cell pressure and by extending the e:çerimental stress-strain curve to the stress axis. This latter extension, if a straight-line segment isused, corresponds to the assumption of linear soil properties under the action of thecell pressure alone. Although radial displacements were not measured directly as thecell pressure was applied because of changes in the cell diameter (and thereby the ref-erenèe datum for thè radial displacement measurements), the initial radial strain €"rdue to the hydrostatic cell pressure is assumed to be equal to (.r. Therefore, plots ofboth axial strain and ra.dial strain versus axial pressure begin at the point (€.r - €,r, p"o).

The tangent value of Poissonts ratio at any other point A along an acfual stress-straincurve can be determined in a manner similar to that previously described by establish-ing a new coordinate system parallel to the original one. To position this new coordi-nate system, one may conveniently assume that point A has the coordinates (<"r, 1) inthe new system. In effect, this implies that the bulk modulus of the soil does not varywith the state of stress. Although this is not strictly correct, the stress levels ofinterest are relatively low. IThe maximum axial stress is onthe order of 50 psi (345kPa) and thereby tends to minimize the anticipated stiffening of a soil as stress in-creases. The relationship K = E/t3(1 - 2u)J qualitatively favors such an assumptionbecause the modulus decreases and Poissonts ratio increases as the axial stress is in-creased in a triaxial test. This causes both the numerator and denominator to decreasesimultaneously, but not necessarily proportionately.J Within the limitati.on of this

pr

þ^ =zvpr

assumption, the interceptonthe new verticalaxis of a straight line drawntangent r" J:,A will yielá twice the tangent value of Poissonls ratio at point A, as shown in Figure 5c.

EXPERIMENTAL INVES TIGATION

To evaluate the foregoing theory, we conducted triaxial tests on two soils with radialand axial displacement measurements at several boundary points. Soii EB-1 is a mix-fure of coarsè to very fine sand and gravel. Of material passing a No. I sieve, about95 percent was sand and about 5 percent was silt. Soil EC-l is a mixture of sand, silt,and clay. Of material passing a No. 8 sieve, about 40 percent was sand, 35 percentwas silt, and 25 percent was clay. One specimen of each soil was tested under a con-

stant confining pressure of 10 psi (09 tpa), and typical data from two tests are givenin Figure 6. Measured radial displacements near the ends of the specimens con-firmedthe assumption of essentially zero radial e4pansion at the soil-platen interface.

The values of E and u shown in Figure 7 were determined by the method described,which uses onty axial displacement r¡reasurements, and by the inverse method (3), whichuses both axial and radial displacement measurerrents. AIso included are values ofpoissonts ratio calculated as a simple ratio of the increment of radial strain, measuredfrom €¡rr to the increment of axiat strain, measured from ("t: and there is no adjust-ment for the fact that the test involves a triaxial, rather than a uniaxial, state of stress'For small-strain, monotonic-loading, time-independent, and drained conditions, theagreement between the first two methods and the ratio of radial to axial strain taken aspãisson's ratio is good. It appears reasonable to use the theory of elasticity to char-acterize soil behavior in a piècewise linear manner. Although a comparison of thesey values with values deterniined by some other method (2) would have been desirable,this test program was not sufficiently comprehensive toTacilitate this comparison.Values for E, however, were determined by the method described by Duncan and Chang( 1) anO are shown in Figure ?. The method used to interpret data lrom a laboratorytest can make significant differences in the resulting mechanical properties, and thisfact must be fulty appreciated when such results are incorporated into mathematicalmodels. Since the vãtues of Poisson's ratio are greaily affected by the initial axialstrain due to the cell pressure, extreme care must be exercised to ensure that the ini-tiat displacement has been measured accurately and does not include seating errors orerrors due to the e4pansion of the cell itself when pressured to p".

DISCUSSION OF RESULTS

The foregoing method describes a relatively straightforward means for interpretingdata from a Conventional triaxial compression test and for determining E and z valuesthat are consistent with the use of a piecewise linear elastic relation for formulatingproblems in soil-structure interaction. Modulus values are often calculated by use ofieveral oversimplified procedures in which the soil specimen is treated as an infini-tesimal elementìubjected to a given state of stress and the actual boundary conditionsof the test specimen are not incorporated into the analysis. For the simplest of these

approaches the tangent modulus is defined as

o.-ap.A(^

(36)

Although strictly incorrect because it represents the ratio of shear stress to normalstrain, an analogous modulus sometimes used in mathematical models i.s defined as

78

Figure 6. Loaddeformation data from triaxial m

têst.

Figure 7. Modulus and Poisson's ratio versus 60

axial pressure for soils EB-1 and EC-1,50

40

30

zô

.9 ro

odt:oõæoù-50'v

40

20

to

,r'

/

{.,

So¡l EC-l

/d = lo8.l9cf

r = 18.4%pr = lO Ps¡

qtrI €or '€¡

o Ax¡ol slroino Rodiol slroin ot mid'height of specimenltr

o oæ4 0.m oot? ootS 0.020

Sl¡o¡n, .o or 6r

o'ê

oto3oooIL

i

Soil Eg-lI

I

I

Sôil EC-l

So E8-

I

soit Ec-I

\

\

oo.20-40,60246S

Þ Írtàod þrcÞo¡ld hù.|¡a Érlàod DroÞ!.!d br omcoi a¡d cno¡.llglo)

/.P-'

I ,ÁSoil EB-l

7à 'l32 8Pcr

w .9.7%pr' lOps¡

þf//I €o¡' €,r

Poisson's Rol¡o. y ModuluÊ. E {tsil

In equations 36 and 37, p" and p" are usually termed the principal stresses 91 ând o3

respectively. For the often encountered case in which p" is maintained constant duringthe test, equations 36 and 37 are identical, and both are essentially the same as equa-tion 31. Hõwever, there is a fundamental difference between equations 36 and 37 andequation 31 because a homogeneous state of stress has not been assumed to deriveequation 31. Rather, the solution of the boundary value problem used and the condi-tions under which equation 31 is reasonably comect are evaluated. Similar reasoningapplies to the determination of Poisson's ratio. For an idealized infinitesimal elementof a linear elastic material subjected to a state of stress given by or ) gz = 03, one maywrite

- gL 2UO"tr = -^EE

which, on setting (r = 0, becomes

?9

(3?)

(38)

=2v (¡s)

Equation 39 is the same âs equation 35, except that the latter was deduced by consider-ing tire actual test specimen and its associated boundary conditions. However, theg"apirical method shown in Figure 5 to determine stress-dependent values of Poisson'sratio has not generally been recognized heretofore.

CONCLUSIONS

Data from a standard triaxial compression test with no volume change measurementsand no radial strain measurements can be used to determine piecewise linear valuesof the modulus and Poisson's ratio for a soil. The tangent modulus at a given stresslevel can be simply taken as the slope of the axial stress-axial strain curve at thatstress level, and the value for Poissonts ratio can be evaluated by use of conceptsfrom the theory of elasticity and some simple graphicat constructions. Based on testdata from two different soils, results deduced by this method are shown to be in rea-sonable agreement with results determined by more sophisticated analyses and by moreextensive experimental measurements. Other methods for interpreting these data mayyield signifiôantly different values for the mechanical properties of the soil, and thismust be taken into account when such results are incorporated into mathematical modelsfor the response of soil-structure systems.

ACKNOWLEDGMENT

This work was performed in connection with a project supported by the American Con-crete Pipe Assõciation to investigate the soil-structure interaction of buried concretepipe.

o1

O3

80

REFERENCES

1. J. M. Duncan and C. Y. Chang. Nonlinear Analysis of Stress and Strain in Soils.Journal, Soil Mechanics and Foundations Division, American Society of Civil En-gineers, Vol. 96, No. SM5, 1970, pp. 1629-1653.

2. F. H. Kuthawy and J. M. Duncan. Stresses and Movements in Oroville Dam.Journal, Soil Mechanics and Foundations Division, American Society of Civil En-gineers, Vol. 98, No. SM?, 1972, pp. 653-665.

3. M. H. Farzin, R. B. Corotis, and R. J. Krizek. Inverse Method for DeterminingApBroximate Stress-Strain Behavior of Soils. Journal of Testing and Evaluation'America¡r Society for Testing and Materials' 1974.

4. H. H. Roscoe. The Influence of Strains in SoiI Mechanics. Geotechnique, Vol. 20,

No. 2, 1970, pp. 129-L70.5. H. Y. Ko and R. F. Scott. A New Soil Testing Apparatus. Geotechnique, Vo]. 17,

1967, pp. 40-57.6. R. Ñ. yong and E. McKyes. Yield and Failure of CIay Under Triaxial Stresses.

Journal, Soil Mechanics and Foundations Division, American Society of Civii En-gineers, Vol. 97, No. SM1, 1971, pp. 159-176.