INTRODUCTION
Soil has very complex mechanical behavior, because of its nonlinear, stressdependant,
anisotropic and heterogeneous nature. Hence, instead of modeling the subsoil
in all its complexity, the subgrade is often replaced by a much simpler system
called a subgrade reaction model. Winkler (1867) proposed
a model that assumes the ratio between contact pressure, P, at any given point
and the associated vertical settlement, y, is linear and given by the coefficient
of subgrade reaction, K_{s}:
In this model, the subsoil is replaced by fictitious springs whose stiffness is equal to K_{s}.
Winkler (1867) assumed the soil medium as a system of
identical but mutually independent, closely spaced, discrete and linearly elastic
springs. But, the simplified assumptions which this approach is based on caused
approximations.
One of the basic limitations is that, this model cannot transmit shear stresses which are derived from lack of spring coupling. Also, stressstrain behavior is assumed to be linear.
Many researches like Biot (1937), Terzaghi
(1955), Vesic (1961), Horvath
(1983a, b), Vallabhan and Daloglu
(1999, 2000) and Vallabhan and
Das (1989) have investigated the effective factors and determination approaches
of K_{s}.
Geometry, dimensions of the foundation and soil layering are assigned to be the most important effective parameters on K_{s}. Generally, the value of subgrade modulus can be obtained in the following alternative approaches:
• 
Plate load test 
• 
Consolidation test 
• 
Triaxial test 
• 
CBR test 
Terzaghi (1955) made some recommendations about K_{s}
for 1×1 ft rigid slab placed on a soil medium. However, the procedure to compute
a value of K_{s} to use in a larger slab was not specified. Biot
(1937) solved the problem for an infinite beam with a concentrated load
resting on a 3D elastic soil continuum.
He found a correlation between continuum elastic theory and Winkler model,
that the maximum moments in the beam are equated. Vesic
(1961) matched the maximum displacement of the beam in both models and tried
to develop a value for K_{s} with matching bending moments. He obtained
the equation for K_{s} to use in the Winkler model.
Other works by FilonenkoBorodich (1940),
Hetenyi (1946), Pasternak (1954) and Vlasov
and Leont 'ev (1960) attempt to make the Winkler model more realistic by
assuming some forms of interaction among the spring elements, that represent
the soil continuum.
In this study, effective parameters on modulus of subgrade reaction in clayey soils are studied with use of numerical modeling. The studied parameters include:
• 
Size effect of foundation 
• 
Effect of foundation's shape 
• 
Effect of depth of foundation 
• 
Effect of rigidity of foundation 
Calibration of model: In this way for calibrating model, the studied
case is based on results of 0.3 m plate load test on clayey soil by Consoli
et al. (1998). Soil profile to 4 m deep shown in Fig.
1, in which cone measurements and the values of maximum shear modulus (G_{0})
obtained from crosshole and downhole tests are presented.
It is clear that both the CPT and shear modulus are fairly constant, which indicates that the homogeneity of the soil layer does not depend on strength. The water table is at about 4.0 m depth. The soil layer in finite element software is classified as low plasticity clay (CL) according to the unified soil classification system. Grain size data indicates 44% sand, 32% silt, 24% of clay.
The average bulk unit weight of soil ranged between 17.7 and 18.2 kN m^{3},
the moisture content (ω) was typically 24.526%, the degree of saturation
(S) ranged around 78% and the void ratio (e) varied between 0.80.86.

Fig. 1: 
Soil profile to 4 m deep, containing CPT tip resistance and
shear moduli (G _{0}) values ( Consoli et al.,
1998). (a) q _{e} (Mpa) and (b) G _{0} (MPa) 
Atterberg limits of the material were: liquid limit of 43% and plastic limit of 22% which yields a Plasticity Index (PI) of 21%.
The soil parameters used in Mohrcoulomb model in plaxis analysis are presented in Table 1.
The assumed dimensions of the model are shown in Fig. 2.
The comparison between obtained results from 3D analysis and the results by
Consoli et al. (1998) is shown in Fig.
3.
As shown in Fig. 3ac, there is little
difference between numerical modeling results and mentioned case results and
it shows appropriateness of numerical model.
Soil properties of tested models: In this study four kinds of clayey
soils are considered, which their parameters are based on Bowles
(1997) recommendations. The selected parameters for Mohrcoulomb soil behavior
are shown in Table 2.

Fig. 2: 
Side dimensions of model 

Fig. 3: 
Comparison of calibration results. (a) 0.3 m plate; (b) 0.45
m plate and (c) 0.7 m concrete footing 
RESULTS
Size effect of foundation on K_{s}: Vertical settlement analysis of foundations with different widths (0 to 18 m) was performed with plaxis 3D software. The vertical settlement (y) for each analysis was obtained and the loadsettlement graphs were plotted, then the secant modulus of each graph (K_{s}) was determined.
The Terzaghi's equation for estimating the modulus of subgrade reaction with plate load test results is:
In Eq. 2, K_{s }is subgrade reaction modulus for
prototype foundation, K_{s1 }is subgrade reaction modulus based on plate
load test results and B is the side dimension of foundation.

Fig. 4: 
Decreasing of K_{s} with side dimension of foundation 

Fig. 5: 
Normalized K_{s} with normalized width 
Based on obtained results, the modulus of subgrade reaction (K_{s}) decreased as the side dimension of plate increased (Fig. 4). With normalizing the obtained K_{s} with plate load tests’ K_{s }in its own soil type; all graphs in Fig. 5 are almost the same.
The decreasing manner of K_{s} with increasing foundations’ width is the same between all kinds of clayey soils according to their consistencies, but as it can be seen in Fig. 6, as the consistency of clayey soil decreases, the difference between Terzaghi's equation and obtained K_{s} is increased.
It can be concluded from this figure that the Terzaghi's equation is not suitable for low consistent clayey soils. The first drop of K_{s} in this figure shows that the Terzaghi's equation, as Bowles mentions, deteriorates when foundation dimension is 3 times of plate dimension. As shown in Fig. 4, for a constant side dimension, the modulus of subgrade reaction increases as soil consistency does. According to Fig. 6, for a dimension higher than 3 times of plate dimension, the modified equation could be used:

Fig. 6: 
Consistency effect of clayey soil on K_{s} values 
Table 3: 
Modification coefficient 

In this equation, n is the modification coefficient and the other parameters were previously discussed. Different amounts of n are presented in Table 3.
Effect of foundations’ shape on K_{s}: As it is shown in previous section, it seems that Terzaghi's equation is suitable for stiff clay; therefore this type of soil is selected in foregoing analysis.
Terzaghi’s equation, for estimating the modulus of subgrade reaction of rectangular foundation based on plate load test results, is:
In Eq. 4, K_{s} is the modulus of subgrade reaction for prototype foundation, K_{s1} is modulus of subgrade reaction based on plate load test results and m is the side dimensions ratio of rectangle foundation (B/L).
For investigating the shape effect of foundation on modulus of subgrade reaction, the selected value of m is 1 for square foundation and 1.5 to 5 for rectangle and 10 for strip foundation with same side dimension (B) in all kinds of shape. The selected dimensions are presented in Table 4.
The normalized results of numerical analysis with plate's data are presented
in Fig. 7.

Fig. 7: 
Variation of modulus of subgrade reaction with shape 
Table 4: 
Selected dimensions for comparison between shapes of foundations 

It is shown that the modulus of subgrade reaction
for square shape is the highest value, because with increasing the dimensions
of foundation, the value of settlement with constant load intensity in square
foundation becomes the lowest. Therefore, the modulus of subgrade reaction for strip foundation is the lowest
and for square foundation is the highest value.
As shown in Fig. 7, as the width of each foundation shape is increased, the modulus of subgrade reaction is decreased.
Due to obtained results, it can be concluded that there is uncertainty about the Terzaghi’s equation, because this equation ignores the effect of foundation’s loading area and is based only on dimensions ratio (m).
As shown in Fig. 8, this uncertainty is correct and with increasing dimensions in constant m, the modulus of subgrade reaction has a very high drop down.
Maybe Terzaghi's equation is suitable for the rectangle foundation with 30 cm side dimension; because his equation result is near to 30 cm plate’s result. This difference between rectangle foundations with constant m and constant load intensity, which is because of settlement increase with increasing the dimensions of foundation, is ignored in Terzaghi’s equation.
As a result, the modified equation for rectangular and stripe foundations are presented:

Fig. 8: 
Variation of K_{s} with different side dimension ratio
(m) 
In these equations B_{p} is plate's width, L_{rf} is the length of rectangular foundation, B_{sf} is width of stripe foundation and other parameters were previously described.
In
Fig. 9a
c, the appropriateness of suggested
equations in stripe and rectangular foundations is presented.
Figure
9a presents the accuracy of suggested equation against Terzaghi's equation
in stripe foundation. In
Fig. 9b and
c weakness
of Terzaghi's formula is presented against analysis results and suggested equation.
As mentioned, Terzaghi's equation is not suitable for using in stripe and rectangular
foundations of high width.
Embedment depth effect of foundation on K_{s}: As shown in Fig. 10, with increasing the depth of embedment, the experienced stress by soil is increased and the magnitude of foundation settlement with constant load intensity is decreased, therefore the value of subgrade modulus is increased.
But there is another aim of investigating this effect which is estimating the modulus of subgrade reaction in depth, with results of plate load test in surface. For this reason the numerical analysis is performed on stiff clay and the selected embedment depths are 0 to 5 m.
Equation 7 provides the best fit for correlation between modulus of subgrade reaction in surface (K_{s0}) and in depth (K_{sd}), as shown in Fig. 11.
where, q_{0} is load intensity in surface (kN m^{2}), D is depth in meter and γ is the unit weight of the soil in (kN m^{3}).

Fig. 9: 
Comparison between obtained results (a) suggested and Terzaghi's
equation; (b) Stripe foundation and (c) 2 m width Rectangular foundation 
Rigidity effect of foundation on K_{s} determination: When the flexural rigidity of the footing is taken into account, a solution is based on some forms of a beam on an elastic foundation; this may be the classical Winkler solution in which the foundation is considered as a bed of springs.
As shown in literature (Bowles, 1997) there is a parameter
that determines if a foundation should be analyzed on the basis of the conventional
rigid procedure or as a beam on elastic foundation:

Fig. 10: 
Effect of embedment depth of foundation on K_{s} 
Where:
B: 
Width of foundation (m) 
E: 
Elasticity modulus of foundations material (KN m^{2}) 
K_{s}: 
Modulus of subgrade reaction obtained from 0.3 m plate bearing test (kN
m^{3}) 
The criterion of selecting the foundation as rigid or flexible is:
• 
Rigid members: λL<(π\4) 
• 
Flexible members: Lγ>π 
L: 
Length of foundation (m) 
According to this criterion, the maximum height of foundation for being flexible
was produced and the numerical analysis was done on stiff clay soil with different
foundation’s width.

Fig. 12: 
Flexural effect of foundation on K_{s} 

Fig. 13: 
Comparison between Flexible K_{s} and Rigid K_{s} 
As shown in Fig. 12, as the width of flexible foundation is increased, like the rigid foundation, the modulus of subgrade reaction is decreased.
The ratio between flexible K_{s} and rigid K_{s} is presented in Fig. 13 for different points of foundation. As shown, the modulus of subgrade reaction in corner has the highest value and the side modulus of subgrade reaction is the middle value while the center value is the lowest one. This difference is known as dishing effect of flexible foundation. Figure 14 presents a dishing shape of numerical model after analysis.
As shown the corners settlement of flexible foundation has minimum value and the center point has a maximum value. As can be concluded from Fig. 11 there is little difference between modulus of subgrade reaction in rigid foundation and in center of flexible one.

Fig. 14: 
Dishing shape of footing 
CONCLUSIONS
At present study, shallow foundations on clayey soils are modeled with use of finite element software to investigate the validation of Terzaghi's formula. The effects of different parameters on subgrade reaction modulus are studied and the following results are obtained: Modulus of subgrade reaction is decreased as the side dimension of foundation is increased. It seems that the Terzaghi's equation as Bowles shows deteriorates when foundation dimension is 3 times of plate dimension. As the consistency of clayey soil is decreased the difference between obtained modulus of subgrade reaction and Terzaghi's one is increased. The modulus of subgrade reaction has a direct relation with clayey soils consistencies. It means that as the consistency of soil decreases the modulus of subgrade reaction decreases. With constant width and with constant load intensity, the modulus of subgrade reaction in strip foundation has the lowest value while the square foundation has the highest value. Due to obtained results, Terzaghi's equation for estimating the modulus of subgrade reaction for rectangular foundation is not recommended for use in both rectangular and strip foundation. As depth of embedment of foundation is increased, the modulus of subgrade reaction is increased and that’s because of decreasing the foundations settlement value.
The following equation is proposed for estimation the modulus of subgrade reaction in specific depth of embedment from plate load test results in surface:
Flexural effect of foundation is evaluated. The modulus of subgrade reaction in center of flexible foundation has the same value with same width rigid foundation.
The highest value of modulus of subgrade reaction is in the corner of flexible foundation and the side points have middle value of modulus of subgrade reaction in compare with center point and corner points.