The soil reaction coefficient, or **Winkler** coefficient, is normally referred to as K or K_{S} [kgf/cmc]. In **Winkler**‘s model, the subsurface is characterized by a linear relationship between the failure of a point (s) and the contact pressure (p) at the same point: p = K s

K is the “background constant” or “coefficient of subgrade reaction”. From the physical point of view, the Winkler medium can be assimilated into a bed of elastic springs independent of each other. The reaction coefficient of the soil is by definition the ratio between load and subsidence. In real soil, subsidence depends, in addition to the applied load and the properties of the soil, on the shape and size of the foundation and the stratigraphy of the soil. **The reaction coefficient is therefore not a property of the soil and cannot be defined with only reference to the soil, but must also refer to the size and shape of the foundation.**

The most appropriate method for deriving K is to calculate the subsidence s of the foundation with the most appropriate method, taking into account the applied load, the geometry of the foundation, the stratigraphy of the soil and the characteristics of the individual layers, and then derive K as the ratio between the average pressure applied p and the subsidence s. Alternatively, and especially if the terrain is relatively uniform, first approximation assessments can be made according to the following procedures.

For a **homogeneous elastic** medium, the failure of a foundation of width B which applies pressure p to the supporting ground is given by:

s= [p · B · (1-ν^{2}) · I]/E K=p/s= E · (1-ν^{2}) · I/B

where:

E = elastic modulus

ν = Poisson’s coefficient

I = coefficient of influence depending on the shape of the foundation and the thickness of the compressible layer.

**There is also the relationship :**

E=[E_{ed} · (1+ν) · (1-2ν)]/(1-ν)

For ν = 0,20 is E = 0,90 E_{ed}, while for ν = 0,30 it is E = 0,74 E_{ed}.

Since the edometry module (E_{ed}) is a more easily assessable datum, it is advisable to refer directly to it.

The oedometric module is given by:

E_{ed} = (∆s · H)/∆p

where H is the thickness of the compressible layer.

therefore:

K = ∆p/∆s =E_{ed} / H

If H is less than B for a square foundation or 1.5B for an elongated foundation, K can be derived directly from this expression.

If, on the other hand, the thickness of the compressible layer is large, considering that most of the subsidence is given by the ground up to a depth equal to about B for a square foundation and 1.5B for an elongated foundation, as a first approximation we can assume:

∆s = (∆p · B)/ E_{ed} for a square foundation and ∆s = (∆p · 1.5B)/E_{ed} for an elongated foundation

from which:

k = ∆p/∆s = E_{ed}/B for a square foundation, k = ∆p/∆s = E_{ed}/(1.5 · B) for an elongated foundation

It is therefore possible to go back to an indicative value of K by having the edometric modulus E_{ed}, obtainable both from edometric compression tests, but also from empirical correlations, such as:

E_{ed} = 5 · qc = 50 qu for clays

E_{ed} = 2,5 qc + 75 kg/cmq for clean sands

E_{ed} = 2,0 qc + 50 kg/cmq for fine sands and silty sands

qc = resistance to the tip of the static penetrometer

qu = simple compressive strength

**Vesic (1961)** suggests the following (simplified) relationship:

k = [(1/B) · E]/(1-ν^{2}) where ν is the Poisson coefficient.

A different way of arriving at an approximate value of K is to refer to the value k_{1s} determined with a load test on a square plate of side b = 30 cm. Being fixed shape and size of the plate k_{1s} depend only on the characteristics of the soil. In general, it is not convenient to actually perform a load test on a plate, both for the cost and because with clay soil the test should be long-lasting. It is advisable to refer to the typical values of k_{1s} for the various types of soil provided in the literature. The Terzaghi gives values of k_{1s} for sands correlated with the state of densification and for clays correlated with the simple compressive strength qu.

**Correlation k**_{1s}** -relative density for incoherent soil (sand) according to Terzaghi:
**Values of k

_{1s}in tons/cu.ft for square plates, 1 ft · 1 ft, or beams 1 ft wide, resting on sand.

1 tons/cu.ft = 0,032 kg/cmc; 1 kg/cmc = 31 tons/cu.ft

Loose sand: qc = 20-40 kg/cmq ; Nspt = 5 – 10

Medium sand: qc = 40-120 kg/cmq; Nspt = 10 – 30

Dense sand: qc > 120 kg/cmq; Nspt > 30

Terzaghi also provided a correlation between k_{1s} and Nspt: k_{1s} = Nspt/7,8

**Correlation k**_{1s}** -qu for overconsolidated cohesive soil (overconsolidated clay) according to Terzaghi:
**Values of k

_{1s}, in tns/cu.ft for square plates, 1 ft · 1 ft and k1 for long strips, 1 ft wide, resting on pre-compressed clay

*: Higher values should be used only if were estimated on the basis of adequate test results. For rectangular plates with a width 1 ft ed length of L ft :

k_{1} = k_{1s }· (L + 0,5 )/1,5 L = k_{1} · (1+0.5/L)/1.5

To move from the value referred to the sample plate to the value referred to a real foundation, the shape and size of the foundation must be taken into account: An over consolidated cohesive soil, for the depths affecting a direct foundation, can be assimilated in first approximation to a homogeneous elastic medium, and the subsidence s is approximately proportional to B. Since K is inversely proportional to s it is inversely proportional also to B, so you have:

K = k_{1s }· b/B for a square foundation, K = k_{1s }· b/(1,5 · B) for an elongated foundation

In the sands, assuming that rigidity increases with depth, Terzaghi proposes the relation:

K = k_{1s } · [(B + b )/ 2B]^{2}

and, as B increases, K tends to a limit value:

K = 0.25 k_{1s}

This criterion goes well with normal-established sands where the tip resistance (q_{c}) grows approximately linearly with depth. In reality, for small depths, sands often show a constant resistance to the tip (and therefore rigidity) with the depth and the above criterion does not seem applicable in this case. When evaluating the value of K to be attributed to the foundation it is always advisable to consider how the consistency of the clay soil or the density of the sandy soil varies with depth and therefore in particular if the q_{c} increases or decreases with depth within a significant depth. In the case of an extended foundation (slab), the reference width B shall be chosen according to the criterion shown in the following figure.

(a) and (b) bulbs of pressure beneath concentrated load Q, equally spaced both ways, acting on rectangular concrete mat; (c) concentrated loads, and (d) line loads acting on mat foundation.

Rp= Tip resistance with CPT test

γ = Volume weight

φ = Internal friction angle

Cu = Undrained cohesion

mv = Coeff. of volumetric compressibility

σ = Compressive strength.

**The calculation codes that allow calculating this coefficient are: KH included in the geoutility, MP (Piles and Micropiles of foundation), Loadcap, Additional module of Foundations in CA,.**

**On geostru geoapp there are applications for calculating horizontal and vertical reaction coefficients:**

__Horizontal reaction coefficient of foundation piles__

__Winkler’s constant__