Winkler’s constant

The App allows calculating the value of the vertical subgrade reaction constant kw for shallow foundations and for piles subjected to vertical loads, and the value of the horizontal subgrade reaction constant kh for piles subjected to horizontal loads.

The required input data, in addition to geometric data, are parameters obtainable from conventional geotechnical tests:

• The relative density Dr or NDL30 resistance values for the calculation of kw and kh in the case of cohesionless soils

• The undrained shear strength Cu value or N1S.P.T. resistance values for the calculation of kw and kh in the case of cohesive soils

• The value of the shear modulus G and Poisson’s ratio ν for the calculation of kw related to piles subjected to vertical loads

The results provided by the application are: the value of the constant kw or kh, and the values of the parameters used for their calculation. Furthermore, after the calculation, it is possible to save data and results in a readable text file, and reopen them later within the application itself.

For shallow foundations, both in the case of cohesionless and cohesive soils, the value of kw is determined referring to standard plate load tests with a square or circular base of radius or diameter b equal to 30 cm, from which the value of the parameter k1 is obtained, which is appropriately converted using formulas that take into account the actual size of the foundation.

For cohesionless soils we use:

k_w = k₁ [ (B + b) / 2b ]²

For cohesive soils:

k_w = k₁ (b / 1.5B )

where b is the size of the standard plate used, and B is the plan width of the foundation.

(outputs the value of the parameter k1 and the value of kw)

For the calculation of kh for piles subjected to horizontal loads, two different approaches are used depending on whether it is cohesionless or cohesive soils. In both procedures, the group effect of the piles is taken into consideration according to the indications of Poulos and Davis (1980), which take into account the reduction of kh.

• Group of 2 piles            k_h,g = 0.5 · k_h

• Group of 3 or 4 piles:       k_h,g = 0.33 · k_h

• Group of 5 or more piles:   k_h,g = 0.25 · k_h

Cohesionless soils

For cohesionless soils, reference is made to the expression by Matlock and Reese (1956), which assumes kh varies linearly with depth, and for which a law of this type is considered:

k_h = n_h · z/B

where the value of nh depends on the state of compaction and the presence or absence of groundwater, and can be obtained from the following expression:

n_h = A · γ/1.35

(The application calculates and outputs the interpolation value of coefficient A, based on the entered Dr or NDL30 data, the value of nh and the value of kh, the value of the multiplier considering the group effect between piles is also returned, as well as the value of kh,g referring to the pile group)

Cohesive soils

In the case of overconsolidated cohesive soils, an uniform trend of kh with depth is considered, and for which a law of this type is considered:

k_h = 0.12 · K*/B

(The application calculates and outputs the value of the coefficient K*, based on the entered Cu or N1S.P.T. data, and the value of kh, the value of the multiplier considering the group effect between piles is also returned, as well as the value of kh,g referring to the pile group)

For the determination of kw, reference is made to the method of Randolph and Wroth (1978), which considers the pile immersed in an elastic medium, and separately examines the interaction with this medium of the lateral surface and the base of the pile, the two solutions are then superimposed.

In the method, kw = ks + kb is therefore calculated. Where: ks represents the stiffness of the lateral spring and kb the stiffness of the spring at the base of the pile.

k_s = 2 · π · L · G_m / ζ

k_b = 4 · r_b · G_b / (1 – ν)

The parameters used by the model are:

r₀ = D/2 represents the radius of the pile

L is the length of the pile

ζ is a coefficient that takes into account the amplitude rm of the deformation field that develops around the pile of radius r0

ξ = G_L / G_b is the ratio between the shear moduli at depth z = L, for piles resting on a layer of high stiffness

ρ = G_m / G_L is the non-homogeneity factor of the lateral soil, for piles immersed in soil with variable stiffness

G_m represents the average value of the transverse modulus of elasticity between the surface and depth L

G_L the average value of the transverse modulus of elasticity at depth L.

r_b the radius at the base

G_b the transverse modulus of elasticity of the material below the base of the pile

(The application calculates and outputs the values of the parameters used by the model, the values of ks and kb, and the value of kw)