Basics of Beam on Elastic Foundation


To analyze the response of a footing sitting on a soil foundation as shown in Figure 1, a solution based in the form of a beam on elastic foundation (i.e., so-called Winkler foundation) can be used. In this approach, the soil is represented by a bed of elastic springs while the footing is represented by a beam (Figure 2). Based on the FUNDAMENTAL BEAM THEORY in mechanics of materials, the problem is solved using finite elements.



Figure 1 footing on a soil foundation




Figure 2 Beam on elastic foundation


This beam on elastic foundation structure system is characterized by the properties of the beam and the springs. The beam has a Young’s modulus E (e.g., 21,700 MPa for concrete footing) and cross-section of b x h (used to calculate moment of inertial). The springs used in the finite element solution have a stiffness, which is determined by the footing geometry and the soil modulus of sub-grade reaction, ks. Some typical values for ks are as follows (from Principles Of Foundation Engineering, Das 1998):

Soil Type
Ks in SI Unit
(KN/m3)
Ks in English Unit
(lb/in3)
Sand

(dry or moist)
Loose
8,000 ~ 25,000
29 ~ 92
Midium
25,000 ~ 125,000
91 ~ 460
Dense
125,000 ~ 375,000
460 ~ 1,380
Sand

(Saturated)
Loose
10,000 ~ 15,000
38 ~ 55
Medium
35,000 ~ 40,000
128 ~ 147
Dense
130,000 ~ 150,000
478 ~ 552
Clay
Stiff
12,000 ~ 25,000
44 ~ 92
Very Stiff
25,000 ~ 50,000
92 ~ 184
Hard
>50,000
>184

The deflected shape of the beam and the distribution of soil pressure under the footing are dependent on the beam (footing) rigidity relative to the spring (soil) stiffness and types/locations of applied loads.

The stiffness of the beam relative to the foundation soil is governed by the beam’s Young’s modulus and cross section (b x h) and the soil modulus of sub-grade reaction ks (or beam EI and soil ks). By changing these parameters, the relative stiffness can be changed. If the beam (footing) is rigid relative to the spring (soil), the deflected shape and pressure distribution will be close to a straight line. More flexible beams will display a curved deflection and pressure distribution.

In the case of an eccentrically loaded footing, the deflected shape and the pressure distribution will be asymmetric.

Want to get some feeling of how the foundation responds? Please see OUR EXAMPLE or try some HOMEWORK problems.