Soil Subgrade Reaction in Flexible Foundations

Introduction

When a foundation is not rigid, the stress distribution and the modulus of subgrade reaction depends on its flexural rigidity.

Due to the flexibility of the foundation, the settlements of the ground are not uniform and tend to obtain their maximum value at the center of the foundation slab. If the foundation is very flexible, the stresses in its edges may become zero.

In general, the maximum bending moment that a flexible foundation experiences is significantly lower than that of a rigid foundation.

• Analytical solutions applied to an infinite length beam using the theory of infinite beams on an elastic foundation.

Determination of Foundation Rigidity

A structural foundation is very infrequently either completely rigid or flexible but rather there are intermediate conditions that need to be taken into consideration. According to Hetenyi (1946), to quantify the rigidity of a foundation considering the soil layer properties, a dimensionless parameter (λ) should be calculated as:

where:

• k: the modulus of subgrade reaction using the Winkler model (its calculation will be presented below)
• B: the width of the foundation
• E_b: the foundation’s modulus of elasticity
• I: The foundation’s moment of inertia. For a strap footing with a certain height H, I=BH³/12
• L: the length of the footing

Depending on the value of parameter λ, the flexibility and the subgrade reaction model can be selected as following:

1. λ < π/4 → The foundation can be considered rigid; therefore, the subgrade reaction is calculated using the solutions for rigid foundations.
   
2. π/4 < λ <π → The stiffness of the foundation is intermediate. It cannot be characterized either as rigid or flexible. The stress distribution is determined using the Winkler’s Model.
3. λ > π → The foundation is flexible. The subgrade reaction can be determined either using the Winkler model or through analytical solutions for infinite length strip footings.

Footings of Intermediate Rigidity

For foundations of intermediate rigidity, the modulus of subgrade reaction is initially calculated via the following equation (Vesic, 1961):

where:

• E: the elastic modulus of the ground
• ν: the Poisson ratio of the ground

Then, the parameter λ is derived to determine whether the foundation is indeed of intermediate rigidity (Equation 1).

If π/4 , the foundation’s subgrade reaction is evaluated using the Winkler’s method with the derived modulus k.

The analysis is conducted by solving the differential equation of equilibrium of a beam on a Winkler’s foundation:

Where q is the distributed load along the beam (kN/m), B is the width of the beam (m), E_b is the beam’s modulus of elasticity (ΜPa), I is the beam’s moment of inertia (m⁴) and y is the deformation of a spring at a given point (m).

Flexible Footings

When it comes to flexible footings the methodology remains the same. The modulus of subgrade reaction is calculated via Equation 2 and parameter λ is derived subsequently.

In this case, λ > π, therefore, either the Winkler model is utilized as presented above, or the foundation can be assumed of infinite length and thus, analytical solutions can be used through the theory of “infinite beams” on an elastic foundation.

Concentrated Point Load on Infinite-Length Foundation

Based on the analytical solutions, the settlements of the foundation and its section forces are derived as a function of a point’s distance from the concentrated load P as presented in Figure 2.

where:

Concentrated Moment on Infinite-Length Foundation

The same principle applies in the case of a concentrated moment acting on a certain point of an infinite length foundation. The settlements and the section forces are also derived as a function of a point’s distance from the concentrated moment Μ, as shown in Figure 3.

The resulting settlement, bending moment and shear force at a certain point x is derived through the following equations:

Complex Load Conditions on Infinite-Length Foundation

When an “infinite length” foundation is subjected to multiple types of loads (concentrated loads and moment loads), the superposition principle can be utilized and each load should be taken into account separately as shown in the example below.

Calculation Example

A strip beam is founded on a soil layer. In this example, the settlement and the section forces that will develop are calculated.

Based on the characteristics of the foundation and the ground properties, the relative foundation-soil stiffness will be derived to determine which method will be utilized.

The structural properties of the foundation and the elastic properties of the ground are presented in Table 1 and Table 2, respectively.

Table 1: Structural and geometrical characteristics of the Strap Beam foundation

Table 2: The elastic properties of the soil layer

Based on these assumptions, the subgrade coefficient is calculated via Equation 2, as:

The parameters λ and λ’ are then derived using Equations 1 and 7, as:

As a result, the foundation can be considered flexible. Therefore, analytical solutions that derive from theory of infinite beams on an elastic foundation will be utilized.

For this example calculation, it is assumed that both a vertical and a moment load are acting on the foundation (Figure 4). Therefore, the analysis of complex loads acting on a certain point will be employed.

References

Gazetas G., Anastasopoulos I., Garini, E. (2013). Soil-Structure Interaction. National Technical University of Athens, Greece.

Hetényi, M. (1946). Beams on elastic foundations. University of Michigan Press. Ann Arbor. MI.

Kavvadas (2008). Foundations of civil engineering infrastructure. National Technical University of Athens, Greece.

Vesic, A.B. (1961). Beams on elastic subgrade and Winkler’s hypothesis. Proc. 5 th. Int. Conf. on Soil Mech. Found. Engrg., Paris.