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KEYWORDS: Winkler model, Modulus of subgrade reaction, Terzaghi's equation, Foundation, Plate load test ABSTRACT: In the practical design procedure, use of Terzaghi's equation to determine the modulus of subgrade reaction for exact foundation is common, but there are some uncertainties in utilizing such equation. In this paper the effect of size of foundation on clayey subgrade with use of finite element software (Plaxis 3D) is proposed to investigate the validation of Terzaghi's formulation on determination of subgrade reaction modulus. Also the comparison between Vesic's proposed equation, Terzaghi's one and obtained results are presented. 1 INTRODUCTION

Soil medium has very complex and erratic mechanical behavior, because of the nonlinear, stress-dependant, anisotropic and heterogeneous nature of it. Hence, instead of modeling the subsoil in its three-dimensional nature, subgrade is replaced by a much simpler system called a subgrade model that dates back to the nineteenth century. The search in this context leads to two basic approaches which are Winkler approach and the elastic continuum model are of widespread use, both in theory and engineering practice.

Winkler (1867) was assumed the soil medium as a system of identical but mutually independent, closely spaced, discrete and linearly elastic springs and ratio between contact pressure, P, at any given point and settlement, y, produced by load application at that point, is given by the coefficient of subgrade reaction, Ks :

(1)

In fact, in this model subsoil is replaced by fictitious springs whose stiffness equal to Ks.

However, the simplifying assumptions which this approach is based on cause some approximations. One of the basic limitations of it lies in the fact that this model cannot transmit the shear stresses which are derived from the lack of spring coupling. Also, linear stress-strain behavior is assumed. The coefficient of subgrade reaction, Ks, identifies the characteristics of foundation supporting and has a dimension of force per length cubed.

Many researches like Biot (1937), Terzaghi (1955), Vesic (1961) and most recently Vallabhan (2000)... have investigated about the effective factors and determination approaches of Ks. Geometry

Foundation size effect on modulus of subgrade reaction rested on clayey soil

Masoud Janbaz M.S. Candidate of Geotechnical engineering, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin,Iran Reza Ziaie Moayed Assistant Professor, Head of Department of Civil Engineering, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin,Iran

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and dimensions of the foundation and soil layering are assigned to be the most important effective parameters on Ks. In generally, the value of subgrade modulus can be obtained in the following alternative approaches: 1- Plate load test, 2- Consolidation test, 3- Triaxial test, 4- CBR test

Many researchers have worked to develop a technique to evaluate the modulus of subgrade reaction, Ks. Terzaghi (1955) made some recommendations where he suggested values of K for 1*1 ft rigid slab placed on a soil medium; however, the implementation or procedure to compute a value of K for use in a larger slab was not specific. Biot (1937) solved the problem for an infinite beam with a concentrated load resting on a 3D elastic soil continuum. He found a correlation of the continuum elastic theory and Winkler model where the maximum moments in the beam are equated. Vesic (1961) tried o develop a value for K, except, instead of matching bending moments. He matched the maximum displacement of the beam in both models. He obtained the equation for K for use in the Winkler model. Another works by Filonenko-Borodich (1940), Heteneyi (1946), and Pasternak (1954)... attempt to make the Winkler model more realistic bye assuming some form of interaction among the spring elements that represent the soil continuum.

2 ESTIMATION OF KS FOR FULL SIZED FOOTINGS The modulus of subgrade reaction is preferable method among others because of its greater ease

of use and to the substantial savings in computation time. A major problem is to estimate the numerical value of Ks. Terzaghi in 1955 proposed that Ks for full sized footing in clayey subgrade can be obtained from:

(2)

where:

B1 =side dimension of square base used in the plate load test to produce Ks B= side dimension of full-size foundation Kp= the value of Ks for 0.3*0.3 bearing plate or other size load plate Ks= desired value of modulus of subgrade reaction for the full-size foundation According to Terzaghi (1955), this equation deteriorates when B/B1 ≥ 3 ,another uncertainty is

according to Bowles(1997), this equation is not correct in any case, as Ks using a 3 m footing would not be 0.1 the value obtained from a B1= 0.3 m plate. Another equation, which can be used for estimating the modulus of subgrade reaction, is Vesic ones which is based on elastic parameters of soil medium like elasticity modulus of soil, E, and Poisson ratio, μ.

(3)

where:

Ks= modulus of subgrade reaction E= elasticity modulus of soil B= width of foundation μ= Poisson ratio In present paper, according to these uncertainties, with use of finite element software (plaxis 3D),

the effect of the width of foundation on modulus of subgrade reaction, Ks, are investigated and the obtained results are compared with Terzaghi's and Vesic's equations.

3 CALIBRATION AND ANALYSIS METHOD

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The calibration of three dimensional modeling is based on the results of plate load tests on clayey soil by Consoli et al (1998). The soil parameters that used in Mohr-Coulomb soil behavior model are obtained from this article as shown in Table 1.

Table 1- Soil parameters

Parameter Value Modulus of elasticity (E) 10 (MPa)

Friction angle (Φ) 26° Cohesion (C) 17 (kPa) Poisson ratio (ν) 0.25(assumed) Soil unit weight (γ) 17.7(KN/m³)

The element used in analysis of three dimensional model is based on program and is the 15-node

wedge element that is composed of 6-node triangles in x-y-direction and 8-node quadrilaterals in z direction and in addition to this the 8-node element is used to simulate the plates(foundation) behavior. This type of volume element for soil behavior gives a second order interpolation for displacements and the integration involves 6 stress points.

The sides of 3D modeling with side dimension of plate or foundation in X and Z directions are 5B where B is the plate or foundation dimension, and 4B according to the bulb of stress in square footings in Y direction as shown in Figure 1. For higher dimensions of footing, the ratio between thicknesses of footing to side dimension of footing is 1/12 and is constant in all of models. The comparison between obtained result from 3D analysis and the result by Consoli et al. is shown in Figure 2.

Figure 1. Side dimensions of model

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Figure 2.Comparison of calibration results

4 RESULTS 45 vertical plate load tests analysis are performed on plaxis 3D software. The vertical settlement (y) for each analysis were obtained according to the constant contact pressure (P) that is about

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500(KN/m²) and plotted and Then the secant modulus of each graph (Ks) is determined. Full result is shown in Table 2.

Based on obtained results, the modulus of subgrade reaction (Ks) is decreased as the side dimension of plate increased, but the final value of both method(Terzaghi's equation and finite element analysis) showed that the Terzaghi's one is about 0.33 lower than 3D results. The results of statistical analysis on relationship between side dimensions of foundation (B) and (Ks) results showed that the following power function provides the best fit for correlation between plate load test data (Ks) and side dimension of foundation B (Figure 3).

Figure 3.Obtained results and statistical equation

Ks = 5004 (B)-0.73 (4)

Figure 4.Comparison between obtained results and Terzaghi's and Vesic's Equations As it can be seen from Figure 4, the differences between obtained Ks and two famous approaches

in estimating the modulus of subgrade reaction are proposed. As it is shown, the Vesic's approach, which is based on elastic parameters of soil medium, is very conservative while the obtained Ks are between these two methods. It can be seen from Figure 5 that the ratio between Ks (obtained) and Ks (Terzaghi) increased with increasing the side dimension of footing.

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Figure 5.The ratio between obtained Ks and Terzaghi's one

Figure 6.The ratio between obtained Ks and Terzaghi's one in first drop The first drop down of Figure 5 is exaggerated in Figure 6. As can be seen from this figure, The

first drop of Ks in this Figure shows this fact that the Terzaghi's equation as Bowles shows deteriorates when foundation dimension is 3 time of plate dimension. The value of Ks for 3D finite element analysis is higher than Terzaghi's one as it is shown in Figure 5. If we format the obtained equation according to the Terzaghi's equation we have:

Ks = 1.33 Kp (5)

Where all terms were previously defined.

5 CONCULUSION In this article a 3D finite element analysis of plate load test is obtained for clayey soil. According to the obtained results: The statistical correlation between modulus of subgrade reaction (Ks) and side dimension of footing (B) is obtained.

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The comparison between Terzaghi's famous equation for clayey subgrade and the statistical equation for 3D finite element analysis obtained. As side dimension of footing (B) increased the modulus of subgrade reaction (Ks) decreased. The modulus of subgrade reaction that obtained from Terzaghi's equation for prototype footing has lower value than from 3D finite element obtained. The ratio between obtained Ks to Terzaghi's Ks increased with side dimension of footing increased, and it shows that the Terzaghi's equation in high side dimensions is not in safe side.

Table 2- Full results

B Footing Size(m) P (KN/m²) S (m) [in

mid point]

Obtained

Ks(KN/m³)Terzaghi Approach

Ks Obt/Ks Terzaghi

0.30 500.00 0.04 12820.51 12530.00 1.020.45 500.00 0.07 7462.69 8353.33 0.890.70 500.00 0.08 6250.00 5370.00 1.161.00 500.00 0.10 5050.51 3759.00 1.341.25 500.00 0.11 4587.16 3007.20 1.531.50 500.00 0.13 4000.00 2506.00 1.601.75 500.00 0.14 3472.22 2148.00 1.622.00 500.00 0.16 3125.00 1879.50 1.662.25 500.00 0.18 2857.14 1670.67 1.712.50 500.00 0.19 2659.57 1503.60 1.772.75 500.00 0.20 2475.25 1366.91 1.813.00 500.00 0.22 2314.81 1253.00 1.853.25 500.00 0.23 2173.91 1156.62 1.883.50 500.00 0.24 2057.61 1074.00 1.923.75 500.00 0.26 1930.50 1002.40 1.934.00 500.00 0.27 1845.02 939.75 1.964.25 500.00 0.28 1766.78 884.47 2.004.50 500.00 0.30 1689.19 835.33 2.024.75 500.00 0.31 1607.72 791.37 2.035.00 500.00 0.33 1529.05 751.80 2.035.25 500.00 0.34 1453.49 716.00 2.035.50 500.00 0.36 1377.41 683.45 2.025.75 500.00 0.38 1329.79 653.74 2.036.00 500.00 0.39 1285.35 626.50 2.056.50 500.00 0.41 1225.49 578.31 2.127.00 500.00 0.43 1165.50 537.00 2.177.50 500.00 0.44 1133.79 501.20 2.268.00 500.00 0.46 1086.96 469.88 2.318.50 500.00 0.50 1006.04 442.24 2.279.00 500.00 0.54 929.37 417.67 2.239.50 500.00 0.55 910.75 395.68 2.3010.00 500.00 0.56 896.06 375.90 2.3810.50 500.00 0.57 878.73 358.00 2.4511.00 500.00 0.58 866.55 341.73 2.5411.50 500.00 0.59 847.46 326.87 2.5912.00 500.00 0.60 833.33 313.25 2.6612.50 500.00 0.62 807.75 300.72 2.6913.00 500.00 0.64 782.47 289.15 2.7113.50 500.00 0.66 755.29 278.44 2.7114.00 500.00 0.68 734.21 268.50 2.7314.50 500.00 0.70 714.29 259.24 2.7615.00 500.00 0.72 694.44 250.60 2.7716.00 500.00 0.75 667.56 234.94 2.8417.00 500.00 0.81 617.28 221.12 2.7918.00 500.00 0.86 584.80 208.83 2.80

REFERENCES

Biot, M. A. (1937). Bending of an infinite beam on an elastic foundation. Journal of Applied Mechanic Transactions. ASME, 59, A1-A7.

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Geotechnical and Geoenvironmental Engineering. ASCE. September 1998. pp 857-867. Filonenko-Borodich, M. M. (1940). Some approximate theories of the elastic foundation. Uchenyie Zapiski

Moskovskogo Gosudarstvennoho Universiteta Mekhanica, 46, 3-18 (in Russian). Hetenyi, M. (1946). Beams on elastic foundations. The university of Michigan Press, Ann Arbor, Michigan. Pasternak, P. L. (1954). On a new method of analysis of an elastic foundation by means of two foundation

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Terzaghi, K.V., (1955). Evaluation of coefficient of subgrade reaction. Geotechnique, Vol. 5, No. 4, pp 297-326.

Vallabhan, C. V. and Daloglu, T.A., (2000). Values of K for slab on Winkler foundation. Journal of Geotechnical and Geoenvironmental Engineering, pp 463-471.

Vesic, A. S. (1961) .Beams on elastic subgrade and the Winkler's hypothesis. 5th ICSMFE. vol. 1, pp 845-850. Winkler, E. (1867). Die Lehre von Elastizitat and Festigkeit (on elasticity and fixity). Prague, 182.