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The interaction between a foundation system and the underlying ground is the defining factor of the stress-dependent behavior of the subgrade soil, also known as **soil subgrade reaction**.

The stress-deformation characteristics of the soil on which a structure is founded, are not a straightforward matter. Deriving an estimation of the actual stress distribution underneath a footing is challenging. Subsequently, over the years, simplified methodologies aiming to derive the subgrade reaction have been developed. These methods are based on the assumption that the reaction applied to a footing is proportional to the settlement of the underlying soil (elastic behavior).

The most commonly considered methods employed for the modeling of the response of the soil beneath a foundation include:

- An
**elastic half-space**and utilization of simplified correlations and coefficients. This method uses analytical solutions to calculate the stresses and deformations, by taking into account the geometry of a foundation, the loads, the elastic modulus of the soil (E) and the Poisson ratio (ν). - Simulation of soil behavior via a series of isolated springs. This approach is also known as the
**Winkler Method**(1867). Due to its simplicity and the fact that it takes into consideration the soil-structure interaction, it is particularly popular and widely used especially among structural engineers. - Simulation and evaluation of soil behavior using
**numerical modeling**. This approach utilizes the Finite Element or the Finite Difference Method (FEM or FDM) and the soil is replaced by a number of elements with specified geometry that are connected via nodes. Given the problem geometry, the constitutive models employed and the boundary conditions, a series of partial differential equations for each element are solved to derive the stresses and deformations of the soil.*This approach is far more advanced and tackles most of the assumptions and simplifications made by the other methods, however, it is not always possible and/or cost efficient to create and assess complex numerical models in industry projects.*

The Winkler method is considered the oldest and most widely applied method to model the subgrade reaction. This approach replaces the soil medium with a system of finite, discrete, linear elastic springs as depicted in **Figure 1**.

When a spring is loaded, it experiences a linear deformation which is dependent on the modulus of subgrade reaction:

Where ** K** is the Winkler’s spring constant or modulus of subgrade reaction,

It is particularly important to note that **K constant is not a soil property** since it is dependent on the foundation characteristics (dimensions and stiffness), as well as the loads applied to the foundation.

The Winkler Method is suitable for the determination the section forces of the foundation but it is not considered reliable to derive the settlements of the substratum. The differential equation of a beam’s equilibrium on a Winkler’s foundation can be expressed as:

Where ** q** is the distributed load along the beam (kN/m),

Despite its popularity, the Winkler Method has specific shortcomings, including:

- The Winkler model assumes a linear stress-strain behavior of the ground which is a rough simplification as shown in
**Figure 2**. - As the springs are independent and do not interact, the deformation of the ground is constricted in the loaded region, an assumption that is unrealistic since the soil is a continuous material as it deforms as depicted in
**Figure 3**. - It is challenging to determine the actual value of the modulus of subgrade reaction,
**K**. Moreover, there is not a representative value of**K**since the deformation of the ground is not uniform underneath the footing as depicted in**Figure 3**.

Assuming an elastic half-space, the soil underneath a foundation is represented by a continuous, isotropic, elastic and homogenous half-space.

The soil behavior can be characterized by two soil properties, the modulus of elasticity, ** E**, and the Poisson ratio,

The stress distribution beneath a footing is highly dependent on the characteristics of the soil. In a cohesionless soil, the maximum stress is expected in the middle section of the foundation, while, in a cohesive soil, the maximum stresses are concentrated near the edges of the footing.

Nevertheless, based on elasticity theory, the modulus of subgrade reaction is still considered uniform and, therefore, it is assumed that the stresses are linearly distributed underneath the foundation. A schematic of the actual and the simplified subsurface reaction on a rigid footing is depicted in **Figure 4**.

It is worthwhile to note that the stress distribution patterns presented in **Figure 4** assume that a vertical load is applied to the foundation. When horizontal loads or bending moments are applied, the stress distribution is significantly affected. An example of a linear stress distribution developed under a foundation subjected to a combination of a vertical load and a bending moment, is shown in **Figure 5**.

Terzaghi, K. (1955). *Evaluation of coefficients of subgrade reaction*. Géotechnique 4.

Bowles, J. E. (1988). *Foundation analysis and design*. New York: McGraw-Hill.5th Edition

In the following weeks, geoengineer.org will be adding educational material on the following topics. Stay tuned!

**Soil Subgrade Reaction in Rigid Foundations***Soil Subgrade Reaction in Flexible Foundations**Effect of Embedment on Soil Subgrade Reaction**Soil Subgrade Reaction in Piles**Soil Subgrade Reaction in Tunnels*-
*Simplified Relationships for Soil Subgrade Reaction in Various Applications*

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