Foundation Size Effect on Modulus of Subgrade Reaction in Clayey Soil Reza Ziaie Moayed Assist. Prof., Head of Dept. of Civ. Engrg., Imam Khomeini International Univ., P.O. Box 34149-16818, Qazvin, Iran
[email protected]
Masoud Janbaz Geotechnical Engineer P.O. Box 1469818144, Tehran, Iran
[email protected]
ABSTRACT There are several mathematical models such as two parameter models by Vlasov (1966), Vallabhan and Das (1987, 1988, 1989), three parametric models by Straughan (1990) and others which utilize the modulus of subgrade reaction. One of the most popular models in determining the modulus of subgrade reaction is Winkler model. In this model the subgrade soil assumes to behave like infinite number of linear elastic springs that the stiffness of the spring is named as the modulus of subgrade reaction. This modulus is dependant to some parameters like soil type, size, shape, depth and type of foundation and etc. The direct method to estimate the modulus of subgrade reaction is plate load test that it is done with 30-100 cm diameter circular plate or equivalent rectangular plate. After that we have to extrapolate the test value for exact foundation. In the practical design procedure, use of Terzaghi 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 we tried to model the effect of size of foundation on clayey subgrade with use of finite element software (plaxis 3D) to investigate the Terzaghi formulation on determination of subgrade reaction modulus.
INTRODUCTION Soil medium has very complex and erratic mechanical behavior, because of the Nonlinear , stressdependant, anisotropic and heterogeneous nature of it. Hence, instead of modeling the subsoil in its threedimensional 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.
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Winkler (1867) 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 : Ks = p / y 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 represent the shear stresses in the ground and its foundation interface due to 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 including Biot (1937), Terzaghi (1955), Vesic (1961) and US Navy (1982) have investigated the effective factors and determination approaches of Ks. Geometry and dimensions of the foundation and soil layering are assigned to be the most important effective parameters on Ks. In general it is assumed that the value of subgrade modulus can be obtained using the following alternative tests: 1- Plate load test, 2- Consolidation test, 3- Triaxial test, 4- CBR test
PREVIOUS STUDIES TO EVALUATE VALUE OF KS 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. Several studies by Filonenko-Borodich (1940), Heteneyi (1950), and Pasternak (1954), and others, have attempted to make the Winkler model more realistic by assuming some form of interaction among the spring elements that represent the soil continuum.
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: Ks =Ksp (B1/B) B1 = side dimension of square base used in the plate load test to produce Ks B = side dimension of full-size foundation Ksp = 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.
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According to Terzaghi (1955) this equation deteriorates when B/B1 ≥ 3, another uncertainty according to Bowles (1997) is that this equation is problematic 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. In present paper, according to these uncertainties, with use of finite element software (Plaxis 3D), the effect of side dimension of foundation on modulus of subgrade reaction, Ks, are investigated and the obtained results are compared with Terzaghi equation.
CALIBRATION AND ANALYSIS METHOD The calibration of three dimensional modeling is based on the results of plate load tests on clayey soil by Nilo Consoli, Fernando Schnaid and Jarbas Milititsky (1998). The soil parameters that used in Mohr-Coulomb soil behavior model are obtained from this article as shown in table 1. Table1: Soil parameters Parameter Modulus of elasticity (E) Friction angle (Φ) Cohesion (C) Poisson ratio Soil unit weight
Value 10 (MPa) 26° 17 (kPa) 0.25 (assumed) 17.7 (kN/m³)
The element used in analysis of three dimensional models is the 15-node wedge element that is composed of 6-node triangles in x and y directions 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 finite element model of analysis is presented in Figure 1. The ratio between thickness to side dimension of footing is considered 1/12 and is constant in all of models. The comparison between obtained result from 3D analysis and the result by Console et al. is shown in Figure 2. As it is shown, the good accordance is exists between the finite element results and in-situ plate load test data.
Figure 1: Finite element model of analysis
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RESULTS 45 vertical settlement analyses are modeled on commercial Plaxis 3D software. The vertical settlement (y) for each analysis were obtained according to the constant contact pressure (p) and plotted, Then the secant modulus of each graph (Ks) is determined. Full results are shown in Table 2. Based on obtained results, the modulus of sub-grade reaction (Ks) is decreased as the side dimension of plate increased, but the final value Of both method (Terzaghi equation and finite element analysis) showed that the Terzaghis one is about %33 lower than 3D results. The results of statistical analysis on relationship between
0.3 m plate
100
Settlement%
80 60 40 Plaxis results Plt results
20 0 0
20
40
60
80
100
Load%
0.45 m plate
100
Settlement %
80 60 40 Plaxis results PLT results
20 0 0
50
100
Load %
0.7 m concrete footing
100
Settlement %
80 60 40 plaxis obtained PLT results
20 0 0
50 Load %
Figure 2: Comparison of calibration results
100
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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). 14000.0 12000.0 Ks (KN/m³)
10000.0 Ks (kN/m³) = 5004 (B)-0.73
8000.0 6000.0 4000.0
Obtained…
2000.0 0.0 0
5
10
15
20
Footing Size (m)
Figure 3: Obtained results and statistical equation Ks (kN/m³) = 5004 * (B)–0.73 14000.0 12000.0 Ks = 3759 (B)-1
Ks (KN/m³)
10000.0
Ks = 5004 (B) -0.73
8000.0 6000.0
Obtained… Terzaghi…
4000.0
2000.0 0.0 0
5
10
15
20
Footing Size (m)
Figure 4: Comparison between obtained results and Terzaghi’s Equation The differences between the equation obtained here and Terzaghi’s equation is shown in Figures 4 and 5. It can be seen that the ratio between Ks (obtained) and Ks (Terzaghi) increased with increasing the side dimension of footing.
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6 3.000 Ks (Obt) / Ks (Terzaghi)
2.500
2.000 1.500 1.000 0.500 0.000 0
5
10
15
20
Footing size (m)
Figure 5: The ratio between obtained Ks and Terzaghi’s results The results obtained for the value of Ks from 3D finite element analyses in this study are higher than those predicted by Terzaghi’s formula. If we format the obtained equation according to the Terzaghi equation we have: Ks =1.33* Ksp (Bp / B) 0.73 Ks = Ksp (Bp/B) (Terzaghi equation) Where all terms were previously defined.
CONCLUSION In this article a 3D finite element analysis of shallow foundation carried out on clayey soil and the following results are obtained: The statistical correlation between modulus of subgrade reaction (Ks) and side dimension of footing (B) is obtained as follows. Ks =1.33* KP * (Bp/B) 0.73 The comparison between Terzaghi’s famous equation for clayey subgrade and the statistical equation for 3D finite element analysis obtained. The modulus of subgrade reaction values obtained from Terzaghi’s equation for prototype footing has lower value than from 3D finite element results 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 on the safe side.
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7 Table 2: Complete results B Footing Size(m) 0.3 0.45 0.6 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 16 17 18
S (m)
Obtained
[in mid point] 0.039 0.067 0.08 0.099 0.109 0.125 0.144 0.16 0.175 0.188 0.202 0.216 0.23 0.243 0.259 0.271 0.283 0.296 0.311 0.327 0.344 0.363 0.376 0.389 0.408 0.429 0.441 0.46 0.497 0.538 0.549 0.558 0.569 0.577 0.59 0.6 0.619 0.639 0.662 0.681 0.7 0.72 0.749 0.794 0.84
Ks (KN/m³) 12820.5 7462.7 6250.0 5050.5 4587.2 4000.0 3472.2 3125.0 2857.1 2659.6 2475.2 2314.8 2173.9 2057.6 1930.5 1845.0 1766.8 1689.2 1607.7 1529.1 1453.5 1377.4 1329.8 1285.3 1225.5 1165.5 1133.8 1087.0 1006.0 929.4 910.7 896.1 878.7 866.6 847.5 833.3 807.8 782.5 755.3 734.2 714.3 694.4 667.6 629.7 595.2
Terzaghi Ks Obt/Ks equation Terzaghi 12530.0 1.023 8353.3 0.893 6265.0 0.998 3759.0 1.344 3007.2 1.525 2506.0 1.596 2148.0 1.616 1879.5 1.663 1670.7 1.710 1503.6 1.769 1366.9 1.811 1253.0 1.847 1156.6 1.880 1074.0 1.916 1002.4 1.926 939.8 1.963 884.5 1.998 835.3 2.022 791.4 2.032 751.8 2.033 716.0 2.034 683.5 2.034 653.7 2.034 626.5 2.070 578.3 2.150 537.0 2.200 501.2 2.262 469.9 2.268 442.2 2.275 417.7 2.300 395.7 2.340 375.9 2.384 358.0 2.455 341.7 2.536 326.9 2.593 313.3 2.660 300.7 2.686 289.2 2.706 278.4 2.713 268.5 2.735 259.2 2.755 250.6 2.771 234.9 2.841 221.1 2.848 208.8 2.850
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REFERENCES 1. Daloglu, Ayse T. and C. V. Girija Vallabhan (2000) “Values of K for slab on winkler foundation” Journal of Geotechnical and Geoenvironmental Engineering, May, pp. 463-471. 2. Bowels, J. E. (1997) Foundation Analysis and Design (fifth edition), Mc Graw-Hill. 3. Nilo C. Consoli, F. Schnaid, and J. Milititsky (1998) Interpretation of plate load tests on residual soil site. 1998. Journal of geotechnical and geoenvironmental engineering. ASCE. September. pp. 857-867. 4. D'Apollonia, D. J., E. D'Apollonia, R. F.Brissette (1968) Settlement of spread footing on sand. JSMFE DIV. ASCE, SM3. 5. Terzaghi, K. (1955) Evaluation of coefficient of subgrade reaction, Geotechnique, Vol. 5, No. 4, pp. 297326. 6. Vesic, A. S. (1961) Beams on elastic subgrade and the Winkler's hypothesis," 5th ICSMFE, Vol. 1, pp. 845-850.
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