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:
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, P is the applied load and y the deformation in a given direction.
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), B is the width of the beam, Eb is the beam’s modulus of elasticity (ΜPa) and I is the beam’s moment of inertia (m4).
Despite its popularity, the Winkler Method has specific shortcomings, including:
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, ν.
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
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