2. 1 ELASTIC FOUNDATIONS
The theory of elastic foundations has attracted considerable
attention due to its useful application in various technical
disciplines besides foundation engineering.
The problems of elastically supported structures are of interest in
solid propellant rocket motors, aerospace structures, construction
projects in cold regions, and several other fields.
While in some problems, the structure and the elastic support,
generally referred to as the foundation, can be physically identified,
in many others the concept of structure and foundation may be of
an abstract nature.
3. The problem of foundation–structure interaction is generally solved
by incorporating the reaction from the foundation, into the
response mechanism of the structure, by idealizing the foundation
by a suitable mathematical model.
Even if the foundation medium happens to be complex in some
problems, in a majority of cases, the response of the structure at
the contact surface is of prime interest and hence, it would be of
immense help in the analysis, if the foundation can be represented
by a simple mathematical model, without foregoing the desired
accuracy.
4. 2 SOIL–STRUCTURE INTERACTION EQUATIONS
The foundation–soil system subjected to external loads is shown in
Figures 1 and 2 depending on the geometry of the foundation that
is, beam or a plate.
Most of the footings can be considered as either beams (one-
dimensional) or plates (two-dimensional: rectangular, squares,
circular, annular or other shapes).
5. Figure 1: Beam on an elastic foundation
Figure 2: Plate on elastic foundation
6. 2.1 Beams on Elastic Foundations
Figure 1: Beam on an elastic foundation
Neglecting friction between beam and the soil medium, the
governing equation can be written from bending theory as
7.
8. The other parameters that can be defined for beams using classical
bending theory are as follows
9. The conventions from bending theory for bending moment (BM)
and shear forces (SF) are shown below
Figure 3: Convention sketch for bending theory of beams
10. 2.2 Plates on Elastic Foundations
The assumptions usually made in the theory of bending of thin
plates will be deemed to apply to this case. Friction and adhesion
between the plate and the surface of the elastic foundation is
neglected.
Figure 2: Plate on elastic foundation
13. Although Equation (4.17) is known as the equation of bending of
thin plates, it can be applied to the analysis of most rectangular
plates. Further, the soil properties of an elastic foundation as shown
in Figure 2 become
14. After w(x, y) has been determined from Equation (4.17) and the
boundary conditions, the reactions q(x, y) can be found from
Equation (4.17). The moments and shearing forces in the plate
(Figure 3) can be computed using formulae of the theory of
bending of plates as follows
Figure 3: Convention sketch for plate bending.
15.
16. Following Kirchhoff, the shearing forces Nx, Ny, and the torque H
at the plate edges are usually replaced by the reduced shearing
forces Qx and Qy which, for a rectangular plate, are
17. Equations (4.17) and (4.18) are valid for plates with other
geometries such as circular, annular, and so on, since Lapacean
operator is invariant except that its expansion in other
coordinate systems has to be taken for solving the equation.
Figure 4
19. If the load is axisymmetric, the θ coordinate can be omitted in
Equation (4.23a), resulting in
However, the bending moments and shear forces take the
following forms in such a situation
Bending moments
21. 2.3 Soil Reaction
To solve the final form of soil–structure interaction Equations
(4.14) and (4.17), the soil reaction, q(x), has to be incorporated in
those equations which are dependent on the beam/plate and soil
characteristics and the bond at the interface.
Assuming frictionless contact, and complete bond at the interface
between the beam/plate and the soil, q(x) can be expressed in
terms of soil displacements (mainly vertical displacement for
vertical loads) using different foundation models.
22. 3 ELASTIC MODLES FOR SOIL BEHAVIOIR
3.1 Winkler Model
Figure 5: Load on Winkler’s foundation
The earliest formulation of the foundation model was due to
Winkler, who assumed the foundation model to consist of closely
spaced independent linear springs, as shown in Figure 5.
23. If such a foundation is subjected to a partially distributed surface
loading, q, the springs will not be affected beyond the loaded
region.
For such a situation, an actual foundation is observed to have the
surface deformation as shown in Figure 6.
25. Hence by comparing the behavior of theoretical model and actual
foundation, it can be seen that this model essentially suffers from
a complete lack of continuity in the supporting medium.
The load deflection equation for this case can be written as
q = k w
Where,
k is the spring constant and is often referred to as the
foundation modulus (subgrade modulus), and
w is the vertical deflection of the contact surface.
(4:24)
26. It can be observed that Equation (4.24) is exactly satisfied by an
elastic plate floating on the surface of a liquid and carrying some
load which causes it to deflect.
The pressure distribution under such a plate will be equivalent to
the force of buoyancy, k being the specific weight of the liquid.
With this analogy in view, the first solution for the bending of
plates on a Winkler-type foundation was presented by Hertz
(1884).
Also, in such a foundation model the displacements of the loaded
region will be constant whether the foundation is subjected to a
rigid stamp or a uniform load as can be seen from Figure 5.
However, the displacement for these cases are quite different in
actual foundations as can be noted from Figures 6(a) and (b).
27. 3.2 Two-parameter Elastic Models
The term "two-parameter" signifies that the model is defined by two
independent elastic constants.
The development of these two-parameter soil models has been
approached along two distinct lines.
The first type continues from the discontinuous Winkler model and
eliminates its discontinuous behaviour by providing mechanical
interaction between the individual spring elements.
Such physical models of soil behaviour have been proposed by
Filonenko-Borodich (1940, 1945), Hetenyi (1946), Pasternak
(1954) and Kerr (1964) where interaction between the spring
elements is provided by one of elastic membranes, elastic beams or
elastic layers capable of purely shearing deformation.
28. 3.2.1 Filonenko Borodich model
The model proposed by Filonenko-Borodich (1940) acquires continuity between
the individual spring elements in the Winkler model by connecting them to a thin
elastic membrane under a constant tension T (Figure 7) by considering the
equilibrium' of the membrane-spring system.
Figure 7: Surface displacement of the Filonenko-Borodich model
Stretched membrane
29. It can be shown that for three-dimensional problems (e.g., rectangular or
circular foundations) surface deflection of the soil medium due to a pressure q
is given by
is Laplace's differential operator in rectangular Cartesian coordinates. In the
case of two-dimensional problems (e.g. a strip foundation), above equation
reduces to
Where
Where
31. Hetenyi (1946, 1950) achieved the continuity in the Winklers’s foundation
model by embedding an elastic beam in the two-dimensional case and an
elastic plate in the three dimensional case (Figure 8), with the stipulation that
the hypothetical beam or plate deforms in bending only. In this case the
relation between the load q and the deflection of the surface w can be
expressed as
3.2.2 Hetenyi model
Figure 8: Surface displacement of the Hetenyi model
Plate in bending
32.
33. 3.2.3 Pasternak model
By providing for shear interaction between the Winkler’s spring elements,
Pasternak presented a foundation model as shown in Figure 9. The shear
interaction between the springs has been achieved by connecting the ends of
the springs to a beam or a plate (as the case may be), consisting of
incompressible vertical elements, which hence deform in transverse shear
only. The corresponding equation relating the load, q, and deflection, w, can
be derived as
35. Pasternak proposed another foundation model consisting of two layers of
springs connected by shear layer in between as shown in Figure 10. The
relation between the load q, and the deflection w of the surface of the
foundation can be expressed as
37. 4 Evaluation of Spring Constant in Winkler’s Soil Model
4.1 Coefficient of Elastic Uniform Compression – Plate Load Test
The idea of modeling soil as an elastic medium was first introduced by Winkler
and this principle is now referred to as the Winkler soil model. The subgrade
reaction at any point along the beam is assumed to be directly proportional to
the vertical displacement of the beam at that point. In other words, the soil is
assumed to be elastic and obeys Hooke’s Law. Hence, the modulus of subgrade
reaction (ks) for the soil is given by
38. The test set up is also shown in Figure 11.
Figure 11: Plate load test set up.
39. The plate should obviously be as large as possible, consistent with being able to
exert the vertical forces required. The standard plate is either a circular shape of
760 mm diameter or a square shape 760 × 760 mm, 16 mm thick, and requires
stiffening by means of other circular/square plates placed concentrically above
it. Invariably, a large plate does not settle uniformly. The settlement must,
therefore, be monitored by means of three or four dial gauges equally spaced
around the perimeter in order to determine the mean settlement. Supports for
these dial gauges should be sited well outside the zone of influence of the
jacking load which is measured by a proving ring. When choosing a diameter of
plate to use for the test, due consideration should also be given to the limited
zone of influence of the loaded plate.
Typically, the soil will only be effectively stressed to a depth of 1.25–1.50 times
the diameter of the plate. This limitation can be overcome to some extent by
carrying out the plate test at depth in pit, rather than on the surface. Small
diameter plates are often used to overcome the practical difficulties of providing
the requisite reaction/vertical forces.
40. Figure 12 shows a typical plot of q against w that would be obtained from a
plate bearing test.
Figure 12: q–w curve obtained from plate load
test
41. It has been observed that in general Poisson’s ratio varies from about 0.25 to
0.35 for cohesionless soils and form about 0.35 to 0.45 for cohesive soils
which are capable of supporting foundation blocks. Hence, in the absence of
any test data Poisson’s ratio can be assumed as 0.3 for cohesionless soils and
0.4 for cohesive soils without causing any appreciable error in the analysis
and design of foundations
4.2 Poisson’s Ratio of the Soil Medium
