Soil Behaviour and Hyperbolic Model IGC 2009, Guntur, INDIA

SOIL BEHAVIOUR AND HYPERBOLIC MODEL

R.B. Arun Murugan P.G. Student, Division of Soil Mechanics and Foundation Engineering, Department of Civil Engineering, Anna University Chennai, Chennai–600 025, India. E-mail: [email protected] V.K. Stalin Assistant Professor, Division of Soil Mechanics and Foundation Engineering, Department of Civil Engineering, Anna University Chennai, Chennai–600 025, India. Email: [email protected]

ABSTRACT: Many empirical correlations have been obtained to predict the Engineering properties with the help of Index properties. These empirical equations are not only used to predict the engineering properties, but also can be used as a measure of validating experimental values. Of the available mathematical correlations, hyperbolic model has been widely used by many researchers to establish the stress-strain, time-settlement, time-swelling and time-shrinkage characteristics of fine grained soils. In this investigation an attempt is made to re-examine the suitability of hyperbolic model for widely varying plasticity characteristics, initial moisture content and initial density. Using hyperbolic model few equations are proposed in this paper for predicting the stress-strain behaviour, suction-water content and time-shrinkage behaviour of soil for any liquid limit, initial moisture content and density. The validity of the proposed equations have been attempted and found to be in good agreement experimental values.

1. INTRODUCTION strain behaviour of soil, shrinkage and suction characteristics of soil based on hyperbolic model for varying initial Understanding and prediction of index and engineering conditions such as initial moisture content, density, clay characteristics of soils have their own significance in the content etc. geotechnical engineering practices and need hardly be stressed. Empirical correlations are available to predict the properties like compression index swell potential, 2. MATERIALS permeability coefficient, shear strength etc., using index Natural soils were collected from nearby Chennai area at 1.5 m properties. Kondner (1963) proposed a functional form based depth from ground level by making open trench. Commercial on hyperbolic stress–strain function, which developed later soil bentonite was collected from local market. The physical by Duncan & Chang (1970) for static and qusai-static properties of natural soils and bentonite are shown in Table 1 behaviour of soil. Al-Shayea et al. (2002) used hyperbolic (a) and (b). The soils were selected such that they exhibit model for simulating stress-strain response of soils by wide range of plasticity characteristics. varying confining pressure. Sridharan & Prakash (1985) studied from the Terzahi’s theory of consolidation and Ta showed that T/U and T relation was rectangular hyperbola ble over a fairly wide range of degree of consolidation. In the 1: plot of T/U versus T, a straight line was fitted for the range (a) of 60% to 90% degree of consolidation with a high precision, Ph for calculating the coefficient of consolidation. Stalin et al. ysi (2004) observed from the shrinkage test conducted on cal bentonite + sand mixtures that the time-shrinkage curves Pr generally follow hyperbolic relationship irrespective of the op soil type and amount of coarser fraction. Dakshinamurthy ert (1978) proposed a new empirical equation to predict the ies swelling of expansive clays using hyperbolic relationship. of The swelling experiments were conducted by Sivapullaiah et So al. (1996) on bentonite + sand mixtures at a constant density ils and IMC. They found that irrespective of percentage of sand Ben- in bentonite, the time-swell relationship follow hyperbolic Soil- Soil Soil- Soil Soil description tonite relationship especially during the primary and secondary (1) (2) (3) (4) (1) stages. In this paper, attempts are made to predict the stress–

120 Soil Behaviour and Hyperbolic Model

Specific Gravity 2.68 2.60 2.65 2.66 2.77 “X-Y” variables of strain–stress (kN/m2) is shown in LL % 70 45 24 60 70 Figure 1. PL % 34 20 18 21 41 PI % 36 25 6 39 30 Free Swell Index % 76 40 10 150 294 Maximum 3 17.28 17.8 19.4 18.09 15.2

Dry Density, kN/m 2 m

OMC,% 14.5 13.5 12.2 16.95 24.25 / N

Compressibility k

CH CI CL CH CH ,

Classification s s e r Ta t ble S 1: (b) Ph ysi Strain cal Pr Fig. 1: Stress–Strain Relationship for Soil–4 op ert 2. The X’ and ‘Y’ variables were transformed and ies repotted as ‘X’ Vs ‘(X/Y)’. Because the initial of portion of the curve would not impart in the So linearization of the curve, initial portion of the curve ils is ignored. The slopes and intercept of that plot was determined. A typical plot of (strain/stress) % Vs Ben- Soil Soil Soil Strain hyperbolic relationship after ignoring initial Properties tonite (5) (6) portion of stress–strain curves is shown in Figure 2. Linearised equation is Specific Gravity 2.71 2.6 2.7 2.8 Liquid Limit % 62.5 69 72 430 Y = X / (c + mX) Plastic Limit % 27.5 31 40 48 Plasticity Index % 35 38 32 382 Shrinkage Limit % 9.3 12 10 7.5

Free Swell Index % 78.0 70 70 350 2 m Maximum Dry / 3 16.5 15.9 16 12.8 N k

Density, kN/m , s

OMC,% 23.36 25 28 35 s e r Compressibility t S

CH CH CH CVH /

Classification n i a r t S 3. METHODS Unconfined compressive strength, suction and shrinkage tests were conducted on natural and commercial soils for Strain varying initial moisture content and density. The stress–strain curve, suction–water content curve and time–shrinkage Fig. 2: Strain Vs Strain /Stress Relationship for Soil–4 curves were analysed for linearization, so as to propose a more generalized equation for predicting the same. Where,

3.1 Procedure Involved in the Determination of Soil c = Intercept of (X/Y) Vs X plot Properties Using Hyperbolic Model m = Slope of hyperbolic plot of (X/Y) Vs X plot 1. The values of ‘X’ and ‘Y’ variable (For example 3. The linear portion of the transformed plot (i.e. ‘X’ strain (X)-stress (Y) or time (X)–shrinkage (Y)) is Vs ‘(X/Y)’) was critically analyzed using best fitting obtained from different engineering properties for curve technique by ignoring some initial points and varying initial conditions such as density, IMC, clay the same alone was considered for analysis and content etc., from experimental results. A typical proposing a more generalized empirical equations.

121 Soil Behaviour and Hyperbolic Model 4. RESULTS AND DISCUSSIONS The stress–strain values predicted based on above equation (1) are in close agreement with experimental values 4.1 Stress–Strain Behaviour of Soil irrespective of soil type, density and IMC as shown in Considering stress–strain curve as rectangular hyperbola, the Figure 3. curves are re-plotted into (strain/stress) Vs strain curve with high correlation coefficient of 0.95 to 0.91. The soil with low liquid limit (LL) is not following a hyperbolic relationship. Hence, such curves observed for low clay content or low liquid limit or dry side of optimum were ignored. Further, it 2 m /

is noticed from table 2 that the slope (m) and intercept (c) of N k the linear plot ((ε/σ) = m (ε) + c), keeps decreasing with , s s initial moisture content, initial density and liquid limit. e r t

Hence, the slope and intercept are independently related to S ratio of liquid and initial moisture content. A generalized equation is proposed to predict the stress–strain behaviour of Strain soil as, Fig. 3: Comparisons of Experimental and Predicted Values σ = ε/(2.518 (LL/IMC)–1.24 × ε + 0.544 (LL/IMC)–3.23) (1) of Stress–Strain Curve for Soil – (2)

Table 2: Best Fit Linear Equations of Transformed Plot of 4.2 Suction–Water Content Behaviour of Soil Stress–Strain Curves of Soil for Varying IMC and Density An empirical equation is proposed to predict the suction– Density IMC Soil Equation water content curve of any soil irrespective of LL and IMC (kN/m3) (%) 16.0 11.0 (ε/σ) = 0.273 (ε) + 0.002 using hyperbolic relationship. Treating suction–water content 16.5 12.0 (ε/σ) = 0.282 (ε) + 0.003 curve as rectangular hyperbola, the curve is re-plotted into Soil - 1 17.28 14.5 (ε/σ) = 0.221 (ε) + 0.004 suction Vs (suction/water content) with high correlation 16.5 18.0 (ε/σ) = 0.937 (ε) + 0.001 coefficient of 0.999. The slope (m) and intercept (c) of the 16 19.75 (ε/σ) = 0.892 (ε) + 0.006 linear plot ((ψ/w) = m (ψ) – c) is shown in Table 3. 16.5 10.75 (ε/σ) = 0.349 (ε) + 0.003 17.84 13.5 (ε/σ) = 0.219 (ε) + 0.007 Soil - 2 Table 3: Best fit Linear Equations of Transformed Plot of 17.0 17.90 (ε/σ) = 0.7 (ε) + 0.015 Suction–Water Content Curves of Soils for Varying Initial 16.5 19.90 (ε/σ) = 0.950 (ε) + 0.019 Moisture Content (IMC) 17.95 16.6 (ε/σ) = 0.525 (ε) + 0.003 18.095 16.95 (ε/σ) = 0.499 (ε) + 0.003 Soil IMC% Equation Soil - 4 17.95 17.5 (ε/σ) = 0.635 (ε) + 0.003 Soil - 1 70 (ψ/w) = 0.025 (ψ) – 0.904 17.7 18.5 (ε/σ) = 0.622 (ε) + 0.003 Soil - 4 40 (ψ/w) = 0.049 (ψ) – 2.176 15.2 24.25 (ε/σ) = 0.2163 (ε) + 0.011 Bent (1) 67 (ψ/w) = 0.033 (ψ) – 4.195 14.5 29 (ε/σ) = 0.572 (ε) + 0.036 50 (ψ/w) = 0.035 (ψ) – 3.566 Soil - 5 90 % Bent 15.4 19.5 (ε/σ) = 0.075 (ε) + 0.007 40 (ψ/w) = 0.041 (ψ) – 3.476 (1) + 10 % 16.0 23.75 (ε/σ) = 0.261 (ε) + 0.03 32 (ψ/w) = 0.042 (ψ) – 1.919 Sand 15.4 26.5 (ε/σ) = 0.577 (ε) + 0.045 70 % Bent 17.6 18 (ε/σ) = 0.312 (ε) + 0.023 63 (ψ/w) = 0.026 (ψ) – 2.093 53 (ψ/w) = 0.029 (ψ) – 2.114 (1) + 30 % 17.25 20 (ε/σ) = 0.288 (ε) + 0.025 Soil - 6 Sand 16.5 22 (ε/σ) = 0.787 (ε) + 0.063 43 (ψ/w) = 0.037 (ψ) – 2.308 18.5 12.5 (ε/σ) = 0.3075 (ε) + 0.001 33 (ψ/w) = 0.050 (ψ) – 4.560 50 % Bent 18.95 15 (ε/σ) = 0.451 (ε) + 0.024 (1) + 50 % 64 (ψ/w) = 0.022 (ψ) – 1.251 18.5 16.5 (ε/σ) = 0.687 (ε) + 0.056 52 (ψ/w) = 0.034 (ψ) – 2.579 Sand Soil - 7 17.5 18.5 (ε/σ) = 1.360 (ε) + 0.076 43 (ψ/w) = 0.038 (ψ) – 2.152 33 (ψ/w) = 0.047 (ψ) – 3.696

It is observed that the slope is decreasing with increasing IMC and intercept is increasing with increasing IMC. Further, both slope and intercept are decreases with increasing clay content (LL). Hence the slope and intercept are independently related to product of LL and IMC. Using this correlation an empirical equation is proposed to predict the suction–water content curve of soil.

122 Soil Behaviour and Hyperbolic Model

–0.97 –1.4 w = ψ/(90.94(LL × IMC) × ψ – 20297 (LL × IMC) ) (2) 60 (t/Vsh) = 0.11 (t) + 544.3 50 (t/V ) = 0.133 (t) + 759.5 Based on above equation (2), the suction–water content Soil - 7 72 sh 40 (t/Vsh) = 0.149 (t) + 987.5 values were predicted and found to be in close agreement 30 (t/Vsh) = 0.16 (t) + 1071 with experimental values irrespective of soil type and IMC as 70 (t/Vsh) = 0.169 (t) + 648.3 shown in Figure 4. Bentonite (1) 70 49 (t/Vsh) = 0.243 (t) + 107.4

28 (t/Vsh) = 0.625 (t) + 355.7

430 (t/Vsh) = 0.093 (t) + 747.5

Bentonite (2) 430 301 (t/Vsh) = 0.082 (t) + 723.8

172 (t/Vsh) = 0.142 (t) + 722.6

% 70% Bent (1)

, 49 34 (t/Vsh) = 0.235 (t) + 442 t + 30 % Sand n e t 50% Bent (1) n 35 25 (t/Vsh) = 0.397 (t) + 604.7 o + 50 % Sand C

r 30% Bent (1)

e 21 15 (t/V ) = 0.933 (t) + 140.6

t sh

a + 70 % Sand

W 70% Bent (2) 245 172 (t/V ) = 0.091 (t) + 772.3 + 30 % Sand sh 2 Suction, kN/m 50% Bent (2) 170 119 (t/V ) = 0.116 (t) + 796.5 Fig. 4: Comparison of Experimental Vs Predicted Values of + 50 % Sand sh 30% Bent (2) Suction–Water Content Curve for Soil – (1) 85 60 (t/V ) = 0.293 (t) + 503.1 + 70 % Sand sh 4.3 Time-Shrinkage Behaviour of Soil The time–shrinkage values were predicted based on above Earlier Stalin et al. (2004) have proposed an empirical equation (3) and found to be in close agreement with equation to predict the time–shrinkage behaviour of soil experimental values irrespective of soil type (LL) and irrespective of soil type. In that equation influence of IMC is IMC as shown in Figure 5. not included. The existing equation could not be used successfully to predict the shrinkage of soil with IMC lower than LL water content. In order to overcome this difficulty, in this investigation an attempt is made bring the effect of m m

IMC in the existing equation. The hyperbolic shape of time– , e vertical shrinkage curve is re-plotted into time–(time/vertical g a k

shrinkage) with high correlation coefficient of 0.94 to 0.99. n i r

The slope (m) and intercept (c) of linear plot ((t/Vsh) = m (t) + h S c) with high correlation coefficient is shown in Table 4. It is l a c i observed that the slope is decreasing with increase in IMC t r and intercept is increasing with increase in IMC. From the e V closer observation of graph, it is noticed that the slopes of Time, min linear equations are decreasing with increase in liquid limit. Hence, attempt is made to correlate the slope and intercepts Fig. 5: Comparisons of Experimental and Predicted Values with liquid limit and IMC of respective soils. Using this of Time–Vertical Shrinkage Curve for Bentonite - (2) correlation an empirical equation is proposed to predict time –shrinkage behaviour of soil. 5. CONCLUSIONS

–0.35 -0.11 Vsh = t/(4.784 (LL × IMC) × t + 2693 (LL × IMC) ) (3) The following conclusions may be drawn from this study, 1. Stress–Strain characteristics of low plastic clays and Table 4: Best Fit Linear Equations of Transformed Plot of clays whose initial moisture content is on the dry of Time–Shrinkage Curves of Soils for Varying Liquid Limit optimum side are not following hyperbolic relationship. (LL) and Initial Moisture Content (IMC) 2. The slope and intercepts of linearised transformed plot of LL IMC Soil Equation strain–(strain/stress) relationship are decreasing with (%) (%) increasing liquid limit, IMC and density, irrespective of 50 (t/Vsh) = 0.116 (t) + 1626 soil type and hence a generalized equation is proposed to Soil - 5 62.5 40 (t/V ) = 0.24 (t) + 183.7 sh predict the stress – strain characteristics of clays. 30 (t/Vsh) = 0.296 (t) + 819.9 –1.24 –3.23 60 (t/Vsh) = 0.095 (t) + 1175 σ = ε / (2.518 (LL / IMC) × ε + 0.544 (LL / IMC) ) 50 (t/V ) = 0.087 (t) + 1754 Soil - 6 69 sh 3. Suction–Water Content curves were re-plotted into 40 (t/Vsh) = 0.208 (t) + 556.9 suction Vs (suction/water content) and the slope and 30 (t/Vsh) = 0.26 (t) + 1292 intercepts of the transformed plot is linearly decreasing

123 Soil Behaviour and Hyperbolic Model with LL and IMC. Accordingly, a generalized equation is Duncan J.M. and Chang C.Y. (1970). “Nonlinear Analysis of proposed to predict the suction–water content Sstress and Strain in Soils”, Journal of the Soil Mechanics characteristics for any LL and IMC as and Foundations Division, ASCE, 96, No. SM 5, pp. w = ψ/(90.94(LL × IMC)–0.97 × ψ – 20297 (LL × IMC)-1.4) 1629–1653. 4. An empirical equation is proposed to predict time - Kondner R.L. (1963). “Hyperbolic Stress–Strain Response: vertical shrinkage behaviour for any liquid limit and IMC Cohesive Soils”, Journal of the Soil Mechanics and as Foundations Division, ASCE, 89, No. SM1, pp. 115–143. –0.35 –0.11 Vsh = t/(4.784 (LL × IMC) × t + 2693 (LL × IMC) ) Sridharan, A and Prakash, K. (1985). “Improved Rectangular Hyperbolic Method for the Determination of Coefficient of Consolidation”, Geotechnical Testing Journal REFERENCES GTJODJ, Vol. 8, No. 1, March 1985, pp. 37–40. Al-Shayea, S. Abduljuwad, R. Bashir, Al-Ghamedy and I. Sivapulliah, P.V., Sridharan, A. and Stalin, V.K. (1996). Asi (2002). “Determination of Parameters for a “Swelling Behaviour of Soil Bentonite Mixtures”, Hyperbolic Model of Soils”, Proceedings of the Canadian Geotechnical Journal, Vol. 33, pp. 808–814. institution of civil engineers, Geotechnical Engineering Stalin, V.K., Anuradha, P. and Ambily A.P. (2004). “A 156, April 2003 issue GE 2. pp. 105–117. Study on the Control of Shrinkage Potential of Expansive Dakshanamurthy, V. (1978). “A New Method to Predict Clays and Their Predictions”, 15th South East Asian Swelling using a Hyperbolic Equation”, Geotechnical Geotechnical Society Conference, November 2004, Engineering, 9, pp. 29–38. Bangkok, Thailand, Vol. 1, pp. 867–870.

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