Semi-empirical model for the prediction of modulus of elasticity for unsaturated soils

Semi-empirical model for the prediction ofmodulus of elasticity for unsaturated soils

Won Taek Oh, Sai K. Vanapalli, and Anand J. Puppala

Abstract: A semi-empirical model is proposed in this paper to predict the variation of modulus of elasticity with respectto matric suction for unsaturated sandy soils using the soil-water characteristic curve (SWCC) and the modulus of elastic-ity under saturated conditions. Using this model, comparisons are provided between the predicted and measured moduli ofelasticity and elastic settlements from model footing test results on three different sandy soils. The results of this study areencouraging as there is good agreement between the predicted and measured moduli of elasticity and settlements.

Key words: modulus of elasticity, soil-water characteristic curve, elastic settlement, matric suction, model footing tests.

Resume : Dans cet article, un modele semi-empirique est propose afin de predire la variation du module d’elasticite enfonction de la succion matricielle dan des sols sableux non-satures, en utilisant la courbe de retention d’eau (CRE) et lemodule d’elasticite a saturation. Des comparaisons sont faites entre le modules d’elasticite predits et mesures ainsi que letassement elastique en utilisant les resultats d’essais sur une semelle avec trois differents types de sable, et le modele pro-pose. Les resultats de cette etude sont encourageants puisqu’il existe une bonne concordance entre les modules d’elasticitepredits et mesures, ainsi qu’avec les valeurs de tassement.

Mots-cles : module d’elasticite, courbe de retention d’eau, tassement, succion matricielle, essais sur semelles modeles.

IntroductionBearing capacity and settlement are two key parameters

that have a significant influence on the design of founda-tions. In several scenarios it is the settlement behavior thattypically governs the design of a foundation as opposed tothe bearing capacity. This is particularly true for coarse-grained soils such as sands in which foundation settlementsare immediate in nature. In sandy soils, there are two mainreasons why settlement must be estimated or predicted reli-ably. Firstly, the differential settlements in sandy soils arepredominant in comparison with clayey soils because sanddeposits are typically heterogeneous in nature. Secondly,the settlements in sandy soils occur quickly and may causesignificant damage to the superstructure immediately afterconstruction (Maugeri et al. 1998).

Foundations are conventionally designed assuming thatthe soil in which they are placed is in a saturated condition.The concepts of conventional soil mechanics may not bevalid in the estimation of elastic settlement of foundationsin unsaturated soils. Some investigations have been per-formed to study the contribution of matric suction towardsbearing capacity of unsaturated sandy soils (Steensen-Bach

et al. 1987; Mohamed and Vanapalli 2006). However, thereis limited information in published literature with respect tothe estimation or prediction of the elastic settlement behav-ior of foundations in unsaturated sandy soils.

The modulus of elasticity is used as a key parameter inthe estimation of elastic settlement of foundations oncoarse-grained soils such as sands and gravels. This value istypically assumed to be constant both below and above thegroundwater table in homogeneous soil deposits. In otherwords, the influence of capillary or matric suction (i.e., un-saturated conditions) is not taken into account. A close ex-amination of the experimental results of stress versusdisplacement relationships for model footing tests conductedon soils that are in an unsaturated condition show that themodulus of elasticity is significantly influenced by matricsuction (Vanapalli and Mohamed 2007).

In this paper, stress versus displacement relationshipsfrom model footing tests performed on three different sandsunder unsaturated conditions are analyzed. The variation ofmodulus of elasticity with respect to matric suction is de-rived from the above results and plotted along with their re-spective soil-water characteristic curve (SWCC) behavior.These plots show that there is a relationship between theSWCC and the elasticity behavior similar to the relationshipbetween the SWCC and the shear strength – bearing ca-pacity of unsaturated soils. A semi-empirical model is pro-posed in this paper for predicting the variation of modulusof elasticity with respect to matric suction using the SWCCand the modulus of elasticity under saturated conditions, ex-tending techniques that were followed for the prediction ofthe shear strength (Fredlund et al. 1996; Vanapalli et al.1996) and bearing capacity (Vanapalli and Mohamed 2007)of unsaturated soils. Comparisons are provided between themeasured modulus of elasticity and elastic settlements from

Received 12 October 2007. Accepted 5 March 2009. Publishedon the NRC Research Press Web site at cgj.nrc.ca on 24 July2009.

W.T. Oh and S.K. Vanapalli.1 Department of CivilEngineering, University of Ottawa, 161 Louis Pasteur Street,Ottawa, ON K1N 6N5, Canada.A.J. Puppala. Department of Civil and EnvironmentalEngineering, The University of Texas at Arlington, Box 19308,Arlington, TX 76019, USA.

1Corresponding author (e-mail: [email protected]).

903

Can. Geotech. J. 46: 903–914 (2009) doi:10.1139/T09-030 Published by NRC Research Press

model footing test results and those predicted using the pro-posed method. The results show a good comparison betweenthe measured and predicted values of both moduli of elastic-ity and elastic settlements for the three different sandsstudied in this paper.

Background

Foundation settlementThe total settlement of foundations, d, consists of three

components

½1� d ¼ di þ dc þ ds

where di is the elastic settlement, dc is the consolidation set-tlement, and ds is the secondary settlement or creep.

Elastic settlement occurs under drained loading conditionsin coarse-grained soils. The total settlement in sands is asso-ciated only with elastic settlement as there will be no pri-mary or secondary consolidation settlement.

Foundation settlement in sands is conventionally esti-mated based on the theory of elasticity using two soil pa-rameters: modulus of elasticity, E, and Poisson’s ratio, n.Lade and Nelson (1987), Lade (1988), and Lancelotta(1995) studies show that the elastic settlement is signifi-cantly influenced by the stress–strain modulus, Es, whilePoisson’s ratio, n, does not play an important role.

The modulus of elasticity, E, can be estimated both fromlaboratory and field tests. In general, the modulus of elastic-ity from conventional triaxial tests can be underestimated dueto sample disturbance caused by stress relief and other me-chanical disturbances. To overcome this disadvantage, Davisand Poulos (1968) suggested the use of K0-consolidation tri-axial test results to derive the modulus of elasticity. Accord-ing to the test results by Simons and Som (1970), themodulus of elasticity from K0-consolidation triaxial tests aresignificantly higher than those determined from conventionalundrained triaxial tests.

Plate load tests, cone penetration tests, pressuremeter testsor geophysical methods (i.e., seismic methods) are usuallyused to estimate the in situ modulus of elasticity. In thecase of plate load tests (or model footing tests), the modulusof elasticity, E, can be calculated using eq. [2] (Timoshenkoand Goodier 1951)

½2� E ¼ ð1� n2ÞDd=Dqp

BpIw

where Dd/Dqp is the slope of the settlement versus platepressure, Bp is the width or diameter of the plate, and Iw isthe influence factor (i.e., 0.79 for a circular plate and 0.88for a square plate). A value of 0.3 was used for n in thisstudy, assuming drained loading conditions for the testedsands.

This value of E determined using eq. [2] is representativeof soil within a depth zone which is approximately1.5Bp*2.0Bp (Poulos and Davis 1974). Agarwal and Rana(1987) performed model footing tests in sands to study theinfluence of the groundwater table on settlement. The resultsof the study show that the settlement behavior of relativelydry sand is similar to that of a sand with a groundwater ta-

ble at a depth of 1.5B below the model footing (Fig. 1),where B is the width of the footing. These results indirectlysupport the concept that the increment of stress due to theload applied on the model footing is predominant in therange of 0 to 1.5B below the model footing. These observa-tions are also consistent with the results of modeling studiesby Oh and Vanapalli (2008).

The procedure used in the calculation of elastic settlementusing eq. [2] is illustrated in Fig. 2.

Relationship between SWCC and shear strength –bearing capacity of unsaturated soils

Several investigators have proposed empirical or semi-empirical procedures for predicting the shear strength of un-saturated soils. In many of these procedures, the SWCC isused as a tool in the prediction of the shear strength of anunsaturated soil (Fredlund et al. 1996; Vanapalli et al. 1996;Oberg and Sallfours 1997; Bao et al. 1998; Khalili andKhabbaz 1998; Xu and Sun 2002; Tekinsoy et al. 2004; Xu2004).

The semi-empirical nonlinear function proposed by Vana-palli et al. (1996) and Fredlund et al. (1996) to predict thevariation of the shear strength with respect to matric suctionusing SWCC is

½3� tunsat ¼ c0 þ ðsn � uaÞ tanf0 þ ðua � uwÞSk tanf0

where tunsat is shear strength of the unsaturated soil; c’ is co-hesion under saturated conditions; (sn – ua) is net normalstress, where sn is normal stress and ua is pore-air pressure;f0 is angle of internal friction under saturated conditions;(ua – uw) is matric suction, where uw is pore-water pressure;S is degree of saturation; and k is a fitting parameter. Vana-palli and Fredlund (2000) provided a relationship betweenthe fitting parameter, k, and plasticity index, Ip, using fivedata sets of shear strength as shown in eq. [4].

½4� k ¼ �0:0008ðI2pÞ þ 0:0801ðIpÞ þ 1

Garven and Vanapalli (2006) proposed a new relationship(eq. [5]) for eq. [4] using a large database (Fig. 3).

Fig. 1. Relationship between water table correction factor (Cw) anddepth of water table below base of footing (Dw) (Agarwal and Rana1987).

904 Can. Geotech. J. Vol. 46, 2009

Published by NRC Research Press

½5� k ¼ �0:0016ðI2pÞ þ 0:0975ðIpÞ þ 1

From eqs. [4] and [5], it can be seen that a fitting parameterk = 1 is required for predicting the shear strength of unsatu-rated soils with Ip = 0 (i.e., sandy soils).

Vanapalli et al. (1996), Fredlund et al. (1996), and Vana-palli et al. (1998) also provided mathematical relationshipstowards explaining the nonlinear variation of shear strengthwith respect to matric suction by differentiating the thirdterm (ua – uw)Sk tanf0 of eq. [3] (see eq. [6]). It is of interestto note that at matric suction values close to the residualstate conditions, the net contribution of matric suctioncauses a reduction in the shear strength of sandy soils be-cause S is small and the value of d(Sk)/d(ua – uw) is negative(see Fig. 4b).

½6�

tanfb ¼ dtunsat

dðua � uwÞ

¼ ðSkÞ þ ðua � uwÞdðSkÞ

dðua � uwÞ

24

35 tanf0

where fb is the rate of increase in shear strength relative tothe suction.

Extending similar concepts, Vanapalli and Mohamed(2007) suggested an equation for predicting the bearing ca-pacity of surface footings on unsaturated soils as follows:

Fig. 2. Estimation of elastic settlement from plate load test.

Fig. 3. Relationship between k and Ip (Garven and Vanapalli 2006).

Fig. 4. (a) SWCC and the variation of (b) shear strength, t, (c) ul-timate bearing capacity, qult, (d ) modulus of elasticity, E, and(e) elastic settlement, d, with respect to matric suction in sandysoils (subscript ‘‘sat’’ represents the values under saturated condi-tions).

Oh et al. 905


½7� qultðunsatÞ ¼ c0 þ ðua � uwÞSJ tanf0� �

Ncxc þ 0:5BgNgxg

where qult(unsat) is ultimate bearing capacity of the unsatu-rated soil, c’ is effective cohesion, f0 is effective angle ofinternal friction, J is a fitting parameter, g is soil unitweight, Nc is the bearing capacity factor from Vesic (1973),Ng is the bearing capacity factor from Kumbhojkar (1993),and xc and xg are bearing capacity factors from Vesic(1973). This equation was found to be suitable for interpret-ing the bearing capacity of unsaturated soils using the rela-tionship between Ip and J (Fig. 5). A fitting parameter J =1 was required for predicting the bearing capacity of threesands with Ip = 0.

Estimation of modulus of elasticity in unsaturated sandysoils

A simple equation is proposed in this paper for predictingthe variation of modulus of elasticity of unsaturated sandysoils using the SWCC and the modulus of elasticity undersaturated conditions, extending similar concepts describedearlier. In this equation, two fitting parameters, a and b areused

½8�

Eunsat ¼ Esat þ Esataðua � uwÞðPa=100Þ ðS

bÞ

¼ Esat 1þ aðua � uwÞðPa=100Þ ðS

bÞ

24

35

where Eunsat is modulus of elasticity under unsaturated con-ditions, Esat is modulus of elasticity under saturated condi-tions, a and b are fitting parameters, and Pa is atmosphericpressure (i.e., 100 kPa). In eq. [8], the terms Sb and a con-trol the nonlinear variation of the modulus of elasticity. Theterm (Pa/100) was used for maintaining consistency with re-spect to the dimensions and units on both sides of the equa-tion. A value of b equal to 1 was used for providingcomparisons between the measured and predicted modulusof elasticity of unsaturated sandy soils (i.e., Ip = 0) follow-ing the earlier concepts discussed for shear strength andbearing capacity behavior. The differential form of eq. [8]shown in eq. [9] can be used for providing mathematical ex-planations with respect to the nonlinear behavior of modulusof elasticity under unsaturated conditions.

½9� dEunsat

dðua � uwÞ¼ Esata

ðPa=100Þ ðSbÞ þ ðua � uwÞ

dðSbÞdðua � uwÞ

� �

It can be seen that eq. [9] is similar in form to eq. [6]. Inother words, the net contribution of matric suction towardsthe increase in modulus of elasticity starts decreasing as ma-tric suction approaches the residual suction value in coarse-grained soils, such as sands and gravels, which is similar tothe shear strength behavior (Fig. 4d).

Based on the discussion presented in this section, the me-chanical behavior of unsaturated sandy soils (i.e., the varia-tion of shear strength, bearing capacity, and modulus ofelasticity with respect to matric suction) could be predictedusing the SWCC and the saturated soil properties. A closeexamination of Fig. 4 shows that shear strength, bearing ca-pacity, and modulus of elasticity linearly increase up to the

air-entry value of the soil. Beyond the air-entry value thereis a nonlinear increase of these properties up to the residualsuction value. The contribution of matric suction towardsshear strength, bearing capacity, and modulus of elasticitystarts to decrease beyond the residual suction value forcoarse-grained soils. More details of the modulus of elastic-ity and its influence on the elastic settlement with respect tomatric suction are offered in later sections of the paper.

Test results

Summary of properties of the three sands studiedThe properties of the sandy soils and the model footing

sizes used for the present study are summarized in Table 1.The grain-size distribution curves and SWCCs for the threesands described in Table 1 are shown in Figs. 6 and 7, re-spectively. The coefficients for the equation shown in Fig. 7to plot the SWCCs and the resultant root squared (R-squared) values are provided in Table 2. The SWCC forcoarse-grained sand was established by Mohamed and Vana-palli (2006) and the SWCCs for Sollerod and Lund sandwere measured by Steensen–Bach et al. (1987).

From the SWCCs shown in Fig. 7, it can be seen that atthe same degree of saturation, Sollerod sand shows the high-est suction value followed by coarse-grained sand and Lundsand. Such a behavior can be attributed to Lund sand, whichoffers less resistance to desaturation due to a relatively lowpercentage of fines. In other words, the pore spaces in Lundsand are relatively larger than the other two sands.

Model footing tests on coarse-grained sandMohamed and Vanapalli (2006) carried out model footing

tests using two different footing sizes (i.e., 100 mm �100 mm and 150 mm � 150 mm) in a specially designedbearing capacity tank (900 mm � 900 mm � 750 mm) thathas provisions to simulate fully saturated and unsaturatedconditions (Fig. 8). The test results for the 100 mm �100 mm and 150 mm � 150 mm footings with different ma-tric suction values are shown in Figs. 9 and 10, respectively.The matric suction value at the center of gravity of the ma-

Fig. 5. Relationship between J and Ip for natural, statically com-pacted soils (Vanapalli and Mohamed 2007).



tric suction distribution diagram from 0 to the 1.5Bp depthregion was considered as the average value of matric suctionin the analysis of the results (Vanapalli and Mohamed2007). This is the zone of depth in which the stresses dueto loading are predominant (Poulos and Davis 1974). Fig-ure 11 and Table 3 show the variation of measured matricsuction with depth and typical data from the test tank for anaverage matric suction value of 6 kPa.

Model footing tests on Sollerod and Lund sandSteensen–Bach et al. (1987) performed model footing

(22 mm � 22 mm � 20 mm) tests using a circular steeltest pit (200 mm diameter � 120 mm height). The modelfooting tests results for Sollerod and Lund sand are shownin Figs. 12 and 13, respectively.

Analysis of the tests resultsFigures 14–16 show the SWCCs, the variation of modulus

of elasticity, and elastic settlement with respect to matricsuction for three model footing tests conducted on coarse-grained sand, Sollerod sand, and Lund sand, respectively.The moduli of elasticity were calculated from the linear por-tion of the stress versus settlement relationship using eq. [2].For all three sands studied in this paper, comparisons areprovided between the measured and predicted values of elas-tic settlements for an applied stress of 40 kPa. At this ap-plied stress value of 40 kPa, all sands studied in this paperexhibited elastic behavior.

Fitting parameter a

The fitting parameter a (see eq. [8]), used for the threedifferent sands in this research program are summarized inTable 4. The value of a is equal to 2.5 for coarse-grainedsand (tested with 150 mm � 150 mm model footing) and1.5 for Sollerod sand (tested with 22 mm � 22 mm modelfooting). The value of a was, however, equal to 0.5 forboth the 100 mm � 100 mm footing on coarse-grained sandand the 22 mm � 22 mm footing on Lund sand. These re-sults show no defined trend or relationship between a andthe size of the footing for the three different sands analyzed.Such a behavior can be attributed to the model footing testscarried out on Sollerod and Lund sands with a relativelysmall-size model footing in a relatively low suction range.More explanations are offered with respect to this behaviorin later sections.

Additional investigations were carried out to better under-stand the influence of the model footing size on the fittingparameter, a, using a new series of test results performedby Li (2008) on coarse-grained sand with a different sizefooting (i.e., B � L = 37.5 mm � 37.5 mm). Figures 17and 18 show the model footing test results and provide com-parisons between the measured and predicted moduli ofelasticity for a 37.5 mm � 37.5 mm model footing, respec-tively. From Fig. 18, it can be observed that there is a goodcomparison between the measured and predicted modulus ofelasticity using a equal to 0.5.

Figure 19 shows a relationship between the fitting param-eter a and three different model footing tests (i.e., 37.5, 100,and 150 mm) conducted on coarse-grained sand. These re-sults suggest that a decreases nonlinearly with decreasingmodel footing size. The more defined trend may be ex-

Fig. 6. Grain-size distribution curves of the three sands studied.

Fig. 7. Soil-water characteristic curves of the three sands studied(coefficients a, b, x0, and y0 are summarized in Table 2).

Table 2. Coefficients for the equation shown in Fig.7 to plot theSWCCs and the resultant R-squared values.

Soil a b x0 y0 R2

Lund 85.98 –0.2940 1.635 14.460 1.000Sollerod 78.91 –0.6631 7.212 20.680 0.998Coarse-grained 100.43 –0.9420 5.888 –0.517 0.994

Table 1. Summary of the data of the three sands studied.

Vanapalli andMohamed (2007)

Steensen–Bach etal. (1987)

ParameterCoarse-grainedsand

Sollerodsand

Lundsand

Shear failure type General General GeneralB (mm) � L (mm) 100� 100, 22� 22 22� 22

50� 150c’ (kPa) 0.6 0.8 0.6f0 (8) 35.3 35.8 44.0(ua – uw)b (kPa)a 4.2 5.7 1.1

Note: B and L, width and length of footing, respectively.aAir-entry value.

Oh et al. 907


plained using the ratio of the model footing size to the sizeof soil particles (i.e., particle-size scale effect) (Fig. 20). Thesize of the soil particles in Figs. 20a and 20b are shown tobe constant as the grain size distribution curve for all thesands tested fall in a narrow band and are approximatelyuniform in nature. However, the footing sizes used in thetesting program are different.

When a footing size is relatively large compared with thesoil particle sizes, the resistance in a soil is typically devel-oped along a well-defined failure plane (Fig. 20a). However,when a footing size is relatively small, the load applied onthe model footing is mostly carried by the individual soilparticles and is not due to the frictional resistance arisingbetween the soil particles (Fig. 20b). In this scenario, the

Fig. 8. University of Ottawa bearing capacity test equipment.

Fig. 9. Relationship between the applied stress versus settlement in coarse-grained sand (B � L = 100 mm � 100 mm) (Mohamed andVanapalli 2006).



contribution of matric suction towards modulus of elasticityis also less as the soil particles are relatively large in com-parison to the size of the model footing (i.e., Fig. 20b). Forthis reason, parameter a for the 150 mm � 150 mm footingis greater than that of the 37.5 mm � 37.5 mm footing. Thisbehavior can also be extended to explain the result of the100 mm � 100 mm footing behavior, which is in betweenthe behavior shown in Figs. 20a and 20b.

The above observations can also be extended for the ex-perimental results on Lund sand and Sollerod sand. The par-ticle sizes of the Lund sand are larger than that of Sollerodsand for the whole range of percent finer. Therefore, thecontribution of matric suction towards modulus of elasticityis less for Lund sand when the model footing tests are per-

Fig. 10. Relationship between the applied stress versus settlementin coarse-grained sand (B � L = 150 mm � 150 mm) (Mohamedand Vanapalli 2006).

Fig. 11. Variation of measured matric suction with depth alongwith hydrostatic distribution for an average matric suction of 6 kPain the stress bulb zone.

Table 3. Typical data from the test tank for an average suctionvalue of 6 kPa in the stress bulb zone (from Mohamed and Va-napalli 2006).

D(mm) gt (kN/m3) gd (kN/m3) e

w(%)

S(%)

AVR(kPa)

10 18.17 15.94 0.63 14.0 58 6150 18.75 15.85 0.64 18.3 76 4300 19.27 16.07 0.62 20.0 86 2500 19.40 15.77 0.64 23.0 94 1600 19.74 15.95 0.63 23.8 100 0

Note: D, depth from the surface of compacted sand; gt, moist unitweight; gd, dry unit weight; e, void ratio; w, water content; S, degree ofsaturation; AVR, average value of measured matric suction (ua – uw).

Fig. 12. Relationship between the applied stress versus settlementin Sollerod sand (Steensen–Bach et al. 1987).

Fig. 13. Relationship between the applied stresses versus settlementin Lund sand (Steensen–Bach et al. 1987).

Oh et al. 909


formed with the same model footing size (i.e., 0.5 for Lundsand and 2.5 for Sollerod sand).

Figures 21 and 22 show the variation of modulus of elas-ticity and elastic settlement with respect to the parameter afor the 150 mm � 150 mm footing in coarse-grained sand,respectively. The results show that the use of the parametera with a value between 1.5 and 2 provides conservativemoduli of elasticity values for sandy soils. The elastic settle-ments associated with this modulus of elasticity value alsoprovide conservative estimates.

Variation of modulus of elasticity with respect to matricsuction

From Figs. 14, 15, 16, and 18, it can be seen that themoduli of elasticity behavior is different in the three stages

of desaturation: the boundary effect zone, transition zone,and residual zone (Vanapalli et al. 1999).

(1) Boundary effect zone: The modulus of elasticity linearlyincreases up to the air-entry value.

(2) Transition zone: The modulus of elasticity increases non-linearly up to a certain matric suction value then gradu-ally decreases.

(3) Residual zone: The modulus of elasticity decreases andapproaches a constant value.

The proposed function (eq. [8]) is useful to predict the var-iation of the modulus of elasticity of unsaturated sandy soilsin all the three different zones of the SWCC.

The moduli of elasticity values for Sollerod sand andLund sand are low in comparison with conventional sand,

Fig. 14. SWCC, variation of modulus of elasticity, and immediatesettlement with matric suction from model footing tests in coarse-grained sand.

Fig. 15. SWCC, variation of modulus of elasticity, and elastic set-tlement with matric suction from model footing tests in Sollerodsand.



such as the coarse-grained sand (Fig. 14). This behavior maybe attributed to the relatively small size of the model foot-ings used for testing.

Variation of elastic settlement with matric suctionElastic settlements of the model footing gradually de-

crease with an increase in modulus of elasticity values asmatric suction increases in the boundary effect zone. In thetransition zone, decreasing trends of elastic settlements canbe observed in the lower suction region. However, elasticsettlements gradually start increasing as the suction ap-proaches the residual zone. Such a behavior can be attrib-uted to the gradual decrease of modulus of elasticity in thiszone. The elastic settlements in the residual zone are almostconstant irrespective of the increase in matric suction values.It is of interest to note that the elastic settlement at zero ma-tric suction (i.e., saturated condition) and 10 kPa (i.e., resid-ual condition) values are almost the same for the testedcoarse-grained sand (see Fig. 14).

Figure 23 shows the variation of elastic settlement withrespect to matric suction for different applied stress values

Fig. 17. Relationship between the applied stress versus settlementin coarse-grained sand (B � L = 37.5 mm � 37.5 mm) (Li 2008).

Fig. 18. Comparison between measured and predicted modulus ofelasticity for 37.5 mm � 37.5 mm model footing in coarse-grainedsand.

Fig. 16. SWCC, variation of modulus of elasticity, and elastic set-tlement with matric suction from model footing tests in Lund sand.

Table 4. Fitting parameter a for the three sands stu-died.

Sand type Footing size a

Coarse-grained 100 mm � 100 mm 1.5150 mm � 150 mm 2.5

Sollerod 22 mm � 22 mm 2.5Lund 22 mm � 22 mm 0.5

Oh et al. 911


in the range of 40 to 400 kPa for the 100 mm � 100 mmmodel footing tested in coarse-grained sand. For the appliedstress of 400 kPa, the minimum elastic settlement is3.50 mm at a matric suction value of 5 kPa. This elastic set-tlement value is the same as the value measured for an ap-plied stress of 100 kPa at a matric suction value of 0.5 kPa.The SWCC (see Fig. 7) used in the present analysis was

Fig. 21. Variation of modulus of elasticity with the parameter, a,for 150 mm � 150 mm footing in coarse-grained sand.

Fig. 22. Variation of modulus of elasticity with the parameter, a,for 150 mm � 150 mm footing in coarse-grained sand.

Fig. 19. Variation of the fitting parameter, a, with respect to modelfooting width for coarse-grained sand.

Fig. 20. Failure mechanism of soils under model footing with re-spect to the size of soil particles. (a) large footing; (b) small foot-ing.

Fig. 23. Variation of elastic settlement with respect to various ap-plied stresses for the 100 mm � 100 mm model footing in coarse-grained sand.



measured without the application of any stress. The SWCCbehavior of coarse-grained soils such as sands will not beinfluenced by the applied stress.

ConclusionsA semi-empirical model is proposed in this paper for pre-

dicting the variation of the modulus of elasticity with re-spect to matric suction using model footing test results. Theconclusions obtained from this study are as follows:

(1) The predicted moduli of elasticity and elastic settlementvalues using the proposed model are approximately thesame as the measured values for the three sands studied.

(2) The fitting parameter values of a = 2.5 and b = 1 areexpected to provide a reasonable estimation of modulusof elasticity behavior of unsaturated sandy soils in mod-eling the stress versus elastic settlement behavior basedon the results of the present research program. In addi-tion, it is also expected that a value of a between 1.5and 2 provides conservative elastic settlements in engi-neering practice. However, more test results both in la-boratory and in situ conditions using different footingsizes taking account the influence of embedment of foot-ings into the soil would be valuable.

(3) The proposed function is useful to predict the variationof modulus of elasticity and elastic settlement of unsatu-rated sandy soils in all three zones of the SWCC (i.e.,boundary effect, transition, and residual zone).

(4) The elastic settlement behavior of foundations in satu-rated conditions is similar to that of residual or dry con-ditions for the coarse-grained sands tested.

The technique presented in this paper is encouraging formodeling studies and practicing engineers to predict im-mediate or elastic settlement in unsaturated sandy soils witha reasonable degree of accuracy. The study shows that elas-tic settlements in shallow foundations can be significantlyreduced even if low matric suction values are maintained insandy soils.

ReferencesAgarwal, K.B., and Rana, M.K. 1987. Effect of ground water on

settlement of footings in sand. In Proceedings of the 9th Eur-opean Conference on Soil Mechanics and Foundation Engineer-ing, Dublin, Ireland, 31 August – 3 September 1987. A.A.Balkema, Rotterdam, the Netherlands. pp. 751–754.

Bao, C., Gong, B., and Zhan, L. 1998. Properties of unsaturatedsoils and slope stability of expansive soils. In Proceedings ofthe 2nd International Conference on Unsaturated Soils, UNSAT98, Beijing, China, 27–30 August 1998. International AcademicPublishing House, Beijing, China. Vol. 1. pp. 71–98.

Davis, A.G., and Poulos, H.G. 1968. The use of elastic theory forsettlement prediction under three dimensional conditions. Geo-technique, 18: 67–91.

Fredlund, D.G., Xing, A., Fredlund, M.D., and Barbour, S.L. 1996.Relationship of the unsaturated soil shear strength to the soil-water characteristic curve. Canadian Geotechnical Journal,33(3): 440–448. doi:10.1139/t96-065.

Garven, E., and Vanapalli, S.K. 2006. Evaluation of empirical pro-cedures for predicting the shear strength of unsaturated soils. InProceedings of the 4th International Conference on UnsaturatedSoils, Carefree, Ariz., 2–6 April 2006. ASCE Geotechnical Spe-

cial Publication 147. American Society of Civil Engineers, Re-ston, Va. pp. 2570–2581.

Khalili, N., and Khabbaz, M.H. 1998. A unique relationship for c

for the determination of the shear strength of unsaturated soils.Geotechnique, 48(5): 681–687.

Kumbhojkar, A.S. 1993. Numerical evaluation of Terzaghi’s Ng.Journal of Geotechnical Engineering, 119(3): 598–607. doi:10.1061/(ASCE)0733-9410(1993)119:3(598).

Lade, P.V. 1988. Model and parameters for the elastic behavior ofsoils. In Proceedings of the International Conference on Numer-ical Methods in Geomechanics, Innsbruck, Austria. Edited by G.Swoboda. Balkema, Rotterdam, the Netherlands.

Lade, P.V., and Nelson, R.B. 1987. Modeling the elastic behaviorof granular materials. International Journal for Numerical andAnalytical Methods in Geomechanics, 11(5): 521–542. doi:10.1002/nag.1610110507.

Lancelotta, R. 1995. Geotechnical engineering. Balkema, Rotter-dam, the Netherlands.

Li, X. 2008. Laboratory studies on the bearing capacity of unsatu-rated sands. Master’s thesis, University of Ottawa, Ottawa, Ont.

Maugeri, M., Castelli, F., Massimino, M.R., and Verona, G. 1998.Observed and computed settlements of two shallow foundationson sand. Journal of Geotechnical and Geoenvironmental Engi-neering, 124(7): 595–605. doi:10.1061/(ASCE)1090-0241(1998)124:7(595).

Mohamed, F.M.O., and Vanapalli, S.K. 2006. Laboratory investiga-tions for the measurement of the bearing capacity of an unsatu-rated coarse-grained soil. In Proceedings of the 59th CanadianGeotechnical Conference, Vancouver, B.C., 1–4 October 2006.Canadian Geotechnical Society, Richmond, B.C. pp. 219–226.

Oberg, A., and Sallfours, G. 1997. Determination of shear strengthparameters of unsaturated silts and sands based on the water re-tention curve. Geotechnical Testing Journal, 20(1): 40–48.doi:10.1520/GTJ11419J.

Oh, W.T., and Vanapalli, S.K. 2008. Modelling the stress versussettlement behavior of model footings in saturated and unsatu-rated sandy soils, In Proceedings of the 12th International Con-ference of International Association for Computer Methods andAdvances in Geomechanics, Goa, India, 1–6 October 2008.pp. 2126–2137.

Poulos, H.D., and Davis, E.H. 1974. Elastic solutions for soil androck mechanics. John Wiley and Sons, New York.

Simons, N.E., and Som, N.N. 1970. Settlement of structures onclay with particular emphasis on London clay. Construction In-dustry Research and Information Association, London. Report22.

Steensen–Bach, J.O., Foged, N., and Steenfelt, J.S. 1987. Capillaryinduced stresses – fact or fiction? In Proceedings of the 9th Eur-opean Conference on Soil Mechanics and Foundation Engineer-ing, Dublin, Ireland, 31 August – 3 September 1987. A.A.Balkema, Rotterdam, the Netherlands. pp. 83–89.

Tekinsoy, M.A., Kayadelen, C., Keskin, M.S., and Soylemez, M.2004. An equation for predicting shear strength envelope withrespect to matric suction. Computers and Geotechnics, 31(7):589–593. doi:10.1016/j.compgeo.2004.08.001.

Timoshenko, S., and Goodier, J.N. 1951. Theory of elasticity.McGraw-Hill, New York.

Vanapalli, S.K., and Fredlund, D.G. 2000. Comparison of empiricalprocedures to predict the shear strength of unsaturated soils usesthe soil-water characteristic curve. In Advances in UnsaturatedSoils, Proceedings of Geo-Denver 2000, Denver, Colo., 5–8 Au-gust 2000. ASCE Special Publication 99. Edited by C.D. Shack-elford, S.L. Houston, and N.-Y. Chang. American Society ofCivil Engineers, Reston, Va. pp. 195–209.

Oh et al. 913


Vanapalli, S.K., and Mohamed, F.M.O. 2007. Bearing capacity ofmodel footings in unsaturated soils. In Experimental UnsaturatedSoil Mechanics. Springer Proceedings in Physics. Springer-Verlag Berlin Heidelberg. Vol. 112. pp. 483–493.

Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E., and Clifton, A.W.1996. Model for the prediction of shear strength with respect tosoil suction. Canadian Geotechnical Journal, 33(3): 379–392.doi:10.1139/t96-060.

Vanapalli, S.K., Sillers, W.S., and Fredlund, M.D. 1998. The mean-ing and relevance of residual water content to unsaturated soils.In Proceedings of the 51st Canadian Geotechnical Conference,4–8 October 1998, Edmonton, Alta. BiTech Publishers Ltd.,Richmond, B.C. pp. 101–108.

Vanapalli, S.K., Fredlund, D.G., and Pufahl, D.E. 1999. The influ-ence of soil structure and stress history on the soil-water charac-teristics of a compacted till. Geotechnique, 49(2): 143–159.

Vesic, A.B. 1973. Analysis of ultimate loads of shallow founda-tions. Journal of the Soil Mechanics and Foundations Division,ASCE, 99(1): 45–73.

Xu, Y.F. 2004. Fractal approach to unsaturated shear strength. Jour-nal of Geotechnical and Geoenvironmental Engineering, 130(3):264–273. doi:10.1061/(ASCE)1090-0241(2004)130:3(264).

Xu, Y., and Sun, D. 2002. A fractal model for soil pores and itsapplication to determination of water permeability. Physica A:Statistical Mechanics and its Applications, 316(1–4): 56–64.doi:10.1016/S0378-4371(02)01331-6.



Semi-empirical model for the prediction of modulus of elasticity for unsaturated soils

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