However, their curve was criticized as basing their estimation of the initial modulus of subgrade reaction in the study of Vesic  that is inappropriate for piles socketed into rocks .
(3) It was found that the p-y curves near the bedrock were strongly influenced by the material properties of the bedrock; there were significant increases in the ultimate soil resistance per unit length and in the stiffness (modulus of subgrade reaction).
Before calculation of the effective length of load distribution, the modulus of subgrade reaction should be determined first.
And the modulus of subgrade reaction k is computed by the most commonly used expression for the relationship between k and [E.sub.s] [17,18]:
Terzaghi  showed that the modulus of subgrade reaction (k) was not an intrinsic physical parameter of the soil, but inversely proportional to foundation width.
Here, they assumed a constant value for modulus of subgrade reaction and obtained the moments and deflections along the length of the piles.
where [eta] and k are factors for shape of pile (i.e., for circular cross-section = 1) and initial modulus of subgrade reaction
, respectively; D is pile diameter and z is the depth; A is a coefficient based on loading conditions (static or cyclic); and [P.sub.u] is determined based on wedge type and flow failure which takes from the smaller from the following equations:
Consequently, runway design requirements and pavement evaluation are largely affected by potential deflection, which primarily relates to California bearing ratio (CBR) or modulus of subgrade reaction
of soil subgrade of the runway (Ashford, Wright 1992; Horonjeff, McKelvey 1994).
Table 2 presents calculated values of the proportionality coefficient [beta] for the following input data: diameter of tire contact area a = 0.15 m, modulus of elasticity of cement concrete E = 35 x [10.sup.6] kN/[m.sup.2], Poisson's ratio of cement concrete v = 0.15, modulus of subgrade reaction
k = 100 x [10.sup.3] kN/[m.sup.3] and k = 200 x [10.sup.3] kN/[m.sup.3], thickness of concrete slab h = from 0.16 to 0.30 m.
The design procedure established by the American Association of State Highway and Transportation Officials (AASHTO) is only moderately sensitive to subgrade support values (in terms of "k" or modulus of subgrade reaction
Modulus of Subgrade Reaction
. The most popular model in determining the modulus of subgrade reaction
(k) is Winkler model.
At the AASHO Road Test, the elastic modulus of subgrade reactions
k gross (obtained with 30-in[760-mm-] diameter plate) in units of pounds per cubic inch (pci) is taken to be an average 60 pci (16 kPa/mm).