A Simple Modeling Approach for Estimation of Soil ...

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A Simple Modeling Approach for Estimation of Soil Deformation Behaviour of Natural Expansive Soils Using the Modulus of Elasticity as a Tool S.K. Vanapalli1 and H.H. Adem1 Downloaded from ascelibrary.org by OTTAWA, UNIVERSITY OF on 06/18/14. Copyright ASCE. For personal use only; all rights reserved.

1

Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada; PH (613) 562-5800 x. 6638; FAX (613) 562-5173; email: [email protected]

ABSTRACT This paper details the fundamental concepts of a proposed approach along with the step-by-step procedural details for modeling time-dependent deformation of natural expansive clay soils. Estimation of the soil suction variations within the active zone depth and the corresponding soil modulus of elasticity with respect to the suction variation form the key components of the proposed approach. The soil suction variations within the active zone of the expansive soil profile are estimated using the soil-atmosphere commercial model VADOSE/W. The soil deformation (shrinkage/swelling) is estimated based on the soil structure constitutive relationship for unsaturated soils considering the variation of modulus of elasticity with respect to suction. The proposed approach, which is referred to as the modulus of elasticity based method (MEBM), reasonably well reproduced the deformations of previously published field study results of a natural expansive soil caused by changes in suction over time taking account of environmental factors. INTRODUCTION Expansive clays belong to a special category of unsaturated soils that experience large swelling strains on wetting. Significant swelling strains are also typically observed in saturated expansive clays due to unloading by erosion or excavation. Structures and foundations constructed in or on expansive soils are prone to significant damage due to large swelling strains. Significant research has been undertaken during the last century to better understand the swell and shrinkage behavior of expansive soils (e.g., Katz 1918, Terzaghi 1925, 1926, 1931, 1943, Hamilton 1963, Chen 1975, Fredlund and Morgenstern 1976, Gens and Alonso 1992, Lytton 1994, Briaud et al. 2003, Ng et al. 2003, Fityus et al. 2004, Vu and Fredlund 2006, and Nelson et al. 2012). These soils are a challenge even to date to the geotechnical engineers as the swell and shrinkage behavior with respect to environmental factors cannot be reasonably well predicted or estimated. Karl Terzaghi’s (1925, 1926, 1931) pioneering modeling and experimental studies showed that clay swelling and shrinkage are essentially elastic deformations caused by the clay’s affinity for water. Terzaghi (1926) investigated the mechanics of the swelling of a gelatin gel, as model for clay, using the thermodynamic principles. Empirical relationships were proposed considering many parameters that influence the swelling behavior of gel such as the concentration of gel, the size of gel micropores and the temperature. In addition, from this study, the swelling pressure

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was found to be merely due to the elastic expansion of the solid phase, previously held under compression by the surface tension of the water. The swelling pressure of the gel represents the “free energy” of the system and can be entirely converted into mechanical work. The heat developed in connection with a change in free energy with unrestricted expansion is exclusively due to liquid friction as the gel expands. Terzaghi (1931) explained the fundamental swelling behavior of a two-phase system of liquid and solids (i.e., water and soil) using two different scenarios as examples. In the first scenario, the initial water content w0 of the submerged two-phase system is reduced to a value w1 by applying external pressure p per unit area (Figure 1-a, volume reduction by aa′bb′ ). In the second scenario, it is assumed the water content is reduced to w1 by drying instead of applying the external pressure p (Figure 1-b); assuming that no air has invaded into the system (two-phase system). The volume reduction in first scenario is due to compression (i.e., consolidation) by load and in the second it is associated with shrinkage by evaporation. When the applied external pressure p is removed, water will flow into the system accompanied by swelling in the first scenario (i.e., Figure 1-a). To initiate swelling in the second scenario, the surface a′b′ of the system represented in Figure 1-b is flooded with water. In both the scenarios, swelling starts with the identical water content w1 and arrangement of the particles. In the first scenario, a load of p per unit area of surface is necessary to prevent the material from expanding. In the second scenario (Figure 1-b) there must be a force of equal intensity (as in scenario 1) acting on the surface a′b′ in the system. This force is exerted by the surface tension of the capillary water (i.e., suction). The difference between the pressure in the external water (free water) and in the interstitial water causes water to flow through the surface into the two-phase system until the influence of hydraulic gradient ceases. In other words, the system expands until the suction is zero. The water flow is independent of the physiochemical reaction inside the two-phase system. This leads to a conclusion that the physiochemical effect could not have had any influence on the swelling behaviour of the system. Terzaghi (1931) suggested that the most important factor that contributes to swelling is the negative pressures (suction) associated with the capillary water in the interconnected pores of the clay macrostructure. Terzaghi (1925, 1931) also maintained that the swelling of the soil produced by eliminating the surface tension of the capillary water (suction) is identical with the expansion produced by the removal of the external load. He explained that every soil which is capable of swelling contains a solid phase under a pressure equal to the tension in the liquid phase. He concluded that the swelling capacity of any soil is dependent on the elastic properties of the solid phase. The physiochemical reactions between the solid and the liquid phase (formation of adsorption compounds within the system) can at best only play a minor role. These fundamental studies of Terzaghi (1925, 1926, 1931) though not as widely cited as his other research studies in the conventional geotechnical literature; it has significantly contributed to our present state-of-the-art of understanding of the swelling behavior of expansive soils. A simple approach for modeling the timedependent soil deformation (swelling/ shrinkage) of natural expansive clay soils is proposed in this paper using the modulus of elasticity as a tool following Terzaghi’s (1931) suggestions. The soil suction variations within the active zone of the expansive soil profile are estimated using the soil-atmosphere model VADOSE/W.

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The soil deformation is estimated based on the soil structure constitutive relationship for unsaturated soils considering the variation of modulus of elasticity with respect to suction. The paper also briefly presents validation of the proposed approach using a field study measurements previously published in literature.

(a) Reducing the water content (b) Reducing the water by applying external load content by drying Figure 1. Capillary pressure and swelling process (after Terzaghi, 1931) FUNDAMENTAL CONCEPTS OF MODELING APPROACH Background Terzaghi (1936) effective stress equation provided a rational framework for explaining the mechanical (i.e., volume change and shear strength) behavior of saturated soils. σ ′ = σ − uw (1) where σ ′ = the effective stress; σ = the total stress; and u w = the pore water pressure. Bishop (1959) extended Terzaghi’s equation (Eq. 1) for explaining the effective stress in unsaturated soils as follows: σ ′ = (σ − ua ) + χ (ua − uw ) (2) where ua = the pore air pressure; (σ − ua ) = the net normal stress; and χ = the empirical parameter that represents a portion of the suction (ua − uw ) that contributes to the effective stress. The parameter χ depends on the degree of saturation, stress path and soil type. Many investigators (Coleman 1962, Burland 1965, Matyas and Radhakrishna 1968) questioned the application of the Bishop’s (1959) effective stress principle for unsaturated soils. Limitations of the effective stress principle have been cited in the literature (Blight 1965, Fredlund and Morgenstern 1976). To overcome the limitations associated with the use of single effective stress equation, it has been suggested that the behaviour of unsaturated soils should be described extending the independent stress state variables approach. Fredlund and Morgenstern (1976) proposed constitutive equations for volume change in unsaturated soils using two independent stress state variables, namely; net normal stress (σ m − ua ) and matric suction (ua − uw ) . A separate set of material properties was then introduced for each of the stress state variables: (3a) d ε v = m1s d (σ m − ua ) + m2s d (ua − uw ) w w (3b) dθ = m1 d (σ m − ua ) + m2 d (ua − uw ) where d ε v and dθ = the incremental volumetric strain and the incremental volumetric water content, respectively; m1s and m2s = the coefficient of total volume change with respect to changes in mechanical stress and with respect to changes in

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suction, respectively; m1w and m2w = the coefficient of the pore-water volume change with respect to changes in mechanical stress and changes in suction, respectively; and σ m = the mean mechanical stress. From a stress field analysis, Fredlund and Morgenstern (1977) demonstrated that only two stress state variables (i.e., (σ m − ua ) and (ua − uw ) ) are necessary to define the stress state in an unsaturated soil. Null experiments (i.e., ∆σ = ∆ua = ∆uw ) supported the proposed theoretical framework for stress state variables. A total of four volume change coefficients ( m1s , m2s , m1w and m2w ) are needed for the description of volume change behavior of unsaturated soils. The coefficients can be obtained from void ratio and water content constitutive surfaces of the soil (Figure 2) (Fredlund and Rahardjo 1993). 1 de 1 = at 1 + e0 d (σ m − ua ) 1 + e0 1 de 1 = m2s = am 1 + e0 d (ua − u w ) 1 + e0 m1s =

(4a) (4b)

m1w =

Gs G dw = s bt 1 + e0 d (σ m − ua ) 1 + e0

(4c)

m2w =

Gs G dw = s bm 1 + e0 d (ua − uw ) 1 + e0

(4d)

where at and am = the coefficient of compressibility with respect to a change in mechanical stress and a change in suction, respectively; and bt and bm = the coefficient of water content changes with respect to a change in mechanical stress and a change in suction, respectively. The compressibility coefficients ( at , am , bt and bm ) are the slopes of the void ratio and water content constitutive surfaces at a point (Figure 2). These constitutive surfaces can be obtained from the consolidation tests or triaxial tests with suction control. However, such tests are usually time consuming, costly, and may not be reasonable in engineering practice. Vu and Fredlund (2006) proposed a method to calculate the four coefficients of soil volume change. The void ratio constitutive surface of unsaturated soil is estimated in terms of the compressive indices obtained from the conventional oedometer tests. However, this method estimates unreasonably large soil deformations at low net normal stresses and/or low suctions. This method was evaluated and verified only for Regina expansive clay. In addition, the solution of volume change equations with several nonlinear functions for unsaturated soil properties is difficult to obtain. Constitutive relationship for estimating expansive soil deformation over time The coefficients of volume change ( m1s , m2s , m1w and m2w ) have been defined to describe the volume-mass relationships under any set of stress conditions. However, when predicting soil deformation, a change in soil volumetric strain is of main interest; therefore, only the coefficients m1s and m2s are required. In addition, if changes in total stress are negligible, only m2s coefficient is required (Fredlund et al., 1980). This assumption reduces the constitutive relationship for the soil structure as:

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d ε v = m2s d (ua − uw )

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(5) This infers that the soil suction changes have a direct bearing on the volume change behavior of unsaturated expansive soils as per the discussions presented in Terzaghi (1925, 1931) on the swelling process.

(a) (b) Figure 2. Constitutive surfaces: (a) void ratio constitutive surface, (b) water content constitutive surface (modified after Fredlund and Rahardjo, 1993) The soil deformations due to changes in environmental factors often occur near the ground surface within the active zone of the natural expansive soil deposit. Due to this reason, the soil deformations in expansive soils may be assumed to be predominant in the vertical direction. In other words, the loading condition can be assumed to be the K0-loading. The volumetric strain d ε v is equal to the vertical strain d ε y while the soil is not permitted to deform laterally (i.e., d ε x = d ε z = 0). Equation (5) can therefore be written in terms of the vertical strain as follows: (6) d ε y = m2s d (ua − uw ) (7) in which: m2s = (1 + µ ) / ( H (1 − µ )) where H = the elastic modulus with respect to suction; and µ = the Poisson’s ratio. The summation of the vertical strain changes for each increment provides the final vertical strain of the soil (8) ε y = ∑ dε y To calculate the vertical deformation of expansive soils with respect to time, the soil profile within the active zone has to be subdivided into several layers. The vertical deformation for each arbitrary layer ∆hi (i.e., i-th layer) associated with the time increment is computed by multiplying the vertical strain ε y at the mid-layer for the time increment with the thickness of the layer hi . ∆hi = hi  m2s ∆ (ua − uw ) 

(9)

i

where ∆ (ua − u w ) change in soil suction for each time increment. To calculate the coefficient of total volume change with respect to changes in suction m2s , the modulus of elasticity with respect to change in suction H is estimated in terms of the modulus of elasticity for the soil structure E as: H = E / (1− 2µ ) (10)

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The relationship between the soil elasticity moduli H and E may be more complex for unsaturated soils. However, this relationship (Eq. 10) which is valid for saturated soils has been applied to unsaturated soils extending assumptions suggested by Wong et al. 1998. A model for predicting the variation of modulus of elasticity Eunsat considering the influence of soil suction was established by Vanapalli and Oh (2010) for both fine and coarse grained soils. The model is mainly based on using the soilwater characteristic curve (SWCC) as a tool.   (u − u w ) Eunsat = Esat 1 + α a (S )β  ( Pa 101.3)  

(11)

where Eunsat and Esat = the modulus of elasticity under unsaturated and saturated condition, respectively; Pa = the atmospheric pressure (i.e., 101.3 kPa); S = the degree of saturation; and α and β = the fitting parameters. Knowing the relationship between the modulus of elasticity and the suction for a given soil type, the elasticity modulus of the soil can be estimated for any given equilibrium suction value over time. SOIL-ATMOSPHERIC INTERACTIONS Environmental factors significantly influence the deformation behavior of natural expansive soils. These factors have an impact on the soil suction, which is the predominant stress state variable that governs the soil deformation behavior. In other words, the soil suction profile within the active zone is a representation of a state of balance of environmental factors (i.e., soil-atmospheric interactions) and soil-water storage processes. Several commercial programs are available for estimating the soil suction profiles considering both water flow in unsaturated soils and soil-atmospheric interactions. Such techniques are valuable, simple and economical in comparison to the direct measurement of in-situ suction which is unreliable. Most of the commercially available programs require the following as input information: (i) material properties, namely: the SWCC, the coefficient of permeability function, soil thermal conductivity, mass, and specific heat capacity; (ii) atmospheric parameters such as precipitation, temperature, wind speed, relative humidity, and net radiation; and (iii) geometrical boundary conditions including the ground water table. The output information of these programs includes the modeled data of temperature, water content, and, most importantly, soil suction fluctuations within a soil profile. No definitive or universal recommendation can be provided with respect to a program that would ensure reasonable assurance that water flow predictions are accurate (Bohnhoff et al. 2009). A finite element program (VADOSE/W) is used herein for estimating the suction changes over time within the active zone of the soil profile. VADOSE/W couples the heat transport and mass (i.e., water and vapor) transfer in unsaturated soil, together with the water and energy balance. It has been established that VADOSE/W is capable of simulating saturated and unsaturated flow behavior where the complex soil-atmosphere interaction is of particular interest. More details on VADOSE/W are available in Geo-Slope International Ltd. (2007).

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Different case studies related to deformation behavior of expansive soils published in the literature have been used for testing the validity of the proposed modeling approach which is referred to as the modulus of elasticity based method (MEBM) (Vanapalli and Adem 2012, 2013, Adem and Vanapalli 2013). Due to limitations of space, only one field study is presented in this paper to illustrate the step-by-step procedure and its validation for prediction of the natural expansive soil deformations over time. Figure 3 shows the 11 m high cut slope in Zaoyang, Hubie, China, about 230 km northwest of Wuhan, monitored by Ng et al. (2003) over one month (13 August to 12 September 2001) to investigate the complex soil-water interaction associated with the rainfall infiltration. Several parameters such as climatic conditions, soil properties, and soil cracks influencing the deformation behavior of expansive soils were considered to model this comprehensive and wellinstrumented slope. The upper soil layer with a thickness varying from 1.0 m to 1.5 m was rich in cracks and fissures due to the swelling and shrinkage phenomenon associated with expansive soils. The daily climate data obtained from the weather station in Wuhan city with two artificial rainfall events created by Ng et al. (2003) were applied as a climate boundary at the slope surface (Figure 3). Figure 4 provides a flow chart to detail the step-by-step procedure for modeling the deformation of expansive soil slope over time using the proposed approach (i.e., MEBM). 100

Average daily rainfall (mm)

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VALIDATION OF THE MODELING APPROACH

80 60 40 20 0 13/8

18/8

23/8

28/8

2/9

7/9

12/9

Date (2001)

Figure 3. Cross-section of the research slope along with the details of the two rainfall events during the monitoring period (modified after Ng et al. 2003) Based on the estimated changes in suction within the soil profile over time, the vertical deformation of the soil with respect to time can be calculated at any depth. Figure 5 shows the comparison of the estimated vertical soil deformations with the field measurements at different depths along the mid-slope section R2. Critical review of this figure shows that the patterns of predicted and measured vertical soil deformations immediately after the commencement of the first

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rainfall event were quite different. The predicted vertical deformation was relatively high with an immediate response to the rainfall infiltration in comparison to the measured values. This can be attributed to the difficulty to simulate the performance of the soil in the field considering the high intensity of the initial soil cracks. In the field, cracks provide an easy pathway for rainwater to infiltrate into the soil. The rainwater bypasses the regions between cracks as it ingresses directly through cracks before the soil deformation can take place. In other words, the existence of cracks changes the soil properties which, in turn, change the soil-water interaction over time. Conversely, the results of modeling simulation were mainly based on uniform properties over time. In addition, the field observations although indicate that the intensity and the depth of cracks are large at the upper part of the slope R1 and decrease towards the lower part of the slope R3, the soil layers and their properties in the simulation were assumed to be uniform for the entire slope. Nevertheless, a reasonable agreement between the measured and the predicted soil deformations was obtained after the end of the first rainfall event. The maximum soil deformations obtained from the measurements could be captured using the modeling approach. This is because soil suction decreased with an increase in the degree of saturation, and the influence of cracks on the soil performance hence diminished.

Figure 4. Flowchart for the modeling approach 35

Vertical deformation of soil ( mm )

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Measured deformation at 0.1 m Measured deformation at 0.5 m Measured deformation at 1 m Predicted deformation at 0.1 m Predicted deformation at 0.5 m Predicted deformation at 1 m

30 25 20 15 10 5 0 0

5

10

15

20

25

30

Time ( days )

Figure 5. The predicted and the measured deformation

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SUMMARY AND CONCLUSION The fundamental concepts and the step-by-step procedure for modeling the volume change behaviour of a natural, unsaturated, expansive soil with respect to time using the modulus of elasticity based method (MEBM) are presented in this paper. The modeling approach is based on the theoretical concepts of unsaturated soils. It involves integrating the numerical modeling results of the VADOSE/W model (i.e., soil suction profiles over time) and the simple constitutive equation for soil structure. According to Terzaghi (1925, 1926, 1931), the soil swelling merely represents elastic expansion produced by lowering of the capillary pressure (suction). Hence, the proposed model was simplified to calculate the time-dependent soil deformation (swelling/ shrinkage) in terms of the soil suction variations and the corresponding soil modulus of elasticity with respect to the suction variations. The model was then validated using a cut-slope in expansive soil in China which was originally investigated by Ng et al. (2003). The estimated soil deformation values are in a good agreement with the measured values. REFERENCES Adem, H.H. and Vanapalli, S.K. (2013). “Constitutive modeling approach for estimating the 1-D heave with respect to time for expansive soils.” Int. J. Geotech. Eng., Accepted for publication (to be published in April 2013 issue) Bishop, A.W. (1959). “The principle of effective stress.” Teknisk Ukeblad, Norwegian Geotechnical Institute, 106(39), 859-863. Blight, G.E. (1965). “A study of effective stresses for volume change.” in Moisture Equilibria and Moisture Changes in Soils Beneath Covered Areas, Butterworths, Sydney, 259-269. Bohnhoff, G.L., Ogorzalek, A.S., Benson, C.H., Shackelford, C.D. and Apiwantragoon, P. (2009). “Field data and water-balance predictions for a monolithic cover in a semiarid climate.” J. Geotech. Geoenviron. Eng., ASCE, 135 (3), 333-348 Briaud, J.L., Zhang, X. and Moon, S. (2003). “The shrink test-water content method for shrink and swell prediction.” J. Geotech. Geoenviron. Eng., ASCE, 129(7), 590-600. Burland, J.B. (1965). “Some aspects of the mechanical behaviour of partly saturated soils.” in Moisture Equilibria and Moisture Changes in the Soils Beneath Covered Areas, Butterworths, Sydney, 270-278. Chen, F.H. (1975). “Foundations on Expansive Soils.” 1st ed., Elsevier, New York. Coleman, J.D. (1962). “Stress/strain relations for partly saturated soils.” Géotechnique, 12(4), 348-350. Fityus, S.G., Smith, D.W. and Allman, M.A. (2004). “An expansive soil test site near Newcastle.” J. Geotech. Geoenviron. Eng., ASCE, 130(7), 686-695. Fredlund, D.G. and Morgenstern, N.R. (1976). “Constitutive relations for volume change in unsaturated soils.” Can. Geotech. J., 13(3), 261-276. Fredlund, D.G. and Morgenstern, N.R. (1977). “Stress state variables for unsaturated soils, J. Geotech. Eng. Div., ASCE, 103(GT5), 447-466.

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Fredlund, D.G. and Rahardjo, H. (1993). “Soil Mechanics for Unsaturated Soils.” John Wiley & Sons. Fredlund, D.G., Hasan, J.U. and Filson, H. (1980). “The prediction of total heave.” Proc. 4th Int. Conf. on Expansive Soils, Denver, CO, 1, 1-17. Gens, A. and Alonso E.E. (1992). “A framework for the behaviour of unsaturated expansive clays.” Can. Geotech. J., 29(6), 1013-1031. Geo-Slope International Ltd. (2007). “Vadose Zone Modeling with VADOSE/W 2007: An Engineering Methodology.” 3rd ed., Geo-Slope, Calgary. Hamilton, J.J. (1963). “Volume changes in undisturbed clay profiles in western Canada.” Can. Geotech. J. , 1(1), 27-42. Katz, J.R. (1918). “ Die gesetze der quellung.” Kolloidchem. Beihefte, 19, 1-182 Lytton, R.L. (1994). “Prediction of movement in expansive clays.” Proc. settlement ’94 Conf., Geotechnical Special Publication, ASCE, Reston, Virginia, 2(40), 1827-1845. Matyas, E.L. and Radhakrishna, H.S. (1968). “Volume change characteristics of partially saturated soils.” Géotechnique, 18(4), 432-448. Nelson, J., Thompson, E., Schaut, R., Chao, K., Overton, D. and Dunham-Friel, J. (2012). “Design procedure and considerations for piers in expansive soils.” J. Geotech. Geoenviron. Eng., ASCE, 138(8), 945-956. Ng, C.W.W., Zhan, L.T., Bao, C.G., Fredlund, D.G. and Gong, B.W. (2003). “Performance of an unsaturated expansive soil slope subjected to artificial rainfall infiltration.” Géotechnique, 53(2), 143-157. Terzaghi, K. (1925). “Principles of soil mechanics: I – Phenomena of cohesion of clays.” Engineering News-Record, 95(19), 742-746. Terzaghi, K. (1926). “The mechanics of adsorption and swelling of gels.” 4th Colloid Symposium Monograph, Chem. Catal. Co., New York, 58-78. Terzaghi, K. (1931). “The influence of elasticity and permeability on the swelling of two-phase systems.” Colloid Chemistry, J. Alexander, ed., 3, New York, Chem. Catal. Co., 65-88. Terzaghi, K. (1936). “The shear resistance of saturated soils.” Proc. 1st Inter. Conf. on Soil Mechanics and Foundation Engineering, Cambridge, MA, 1, 54-56. Terzaghi, K. (1943). “Theoretical Soil Mechanics.” Wiley, New York. Vanapalli, S.K. and Adem, H.H. (2012). “Estimation of the 1-D heave in expansive soils using the stress state variables approach for unsaturated soils.” Keynote paper. Proc. 4th Int. Conf. Problematic Soils, Wuhan, China. Vanapalli, S.K. and Adem, H.H. (2013). “Estimation of the 1-D heave of a natural expansive soil deposit with a light structure using the modulus of elasticity based method.” Keynote paper. 1st Pan-American Conference on Unsaturated Soils, Colombia. Vanapalli, S. K. and Oh, W. T. (2010). “A model for predicting the modulus of elasticity of unsaturated soils using the soil-water characteristic curve.” Int. J. Geotech. Eng., 4(4), 425-433. Vu, H. Q. and Fredlund, D. G. (2006). “Challenges to modeling heave in expansive soil.” Can. Geotech. J., 43(12), 1249-1272. Wong, T.T., Fredlund, D.G. and Krahn J. (1998). “A numerical study of coupled consolidation in unsaturated soils.” Can. Geotech. J., 35(6), 926-937.

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