The following points highlight the main theories and methods used for determining vertical stress in soil mass. Also learn about the comparison of different methods for determination of vertical stress in soil mass.
1. Boussinesq Theory for Vertical Stress due to Concentrated Load:
Boussinesq in 1885 gave a solution for stress distribution in a homogeneous and isotropic subgrade subjected to a vertical concentrated load (point load) on the ground surface.
Boussinesq showed that for the point load acting at the ground surface, the polar stress σr at a point P as shown in Fig. 8.1 is given by –
σr = (3Q/2π) (cosθ/R2) …(8.1)
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where Q is the magnitude of the concentrated load, R is the radial distance of point P from the point of application of the load, and θis the angle made by the radius R with the axis of the load. But –
cosθ = z/R …(8.2)
where z is the depth of point P below the ground level. Therefore –
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R = z/cosθ …(8.3)
Substituting this value of R in Eq. (8.1), we have –
σr = (3Q/2π) (cosθ/z/cosθ)2 = (3Q/2πz2)cos3θ …(8.4)
Vertical stress at point P is given by –
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σz = σr cos2θ = (3Q/2πz2)cos5θ …(8.5)
Again
where r is the horizontal radial distance of point P from the axis of the load. Substituting this value of cos θ in Eq. (8.5), we get –
Simplifying and rearranging, we get –
where IB is the Boussinesq influence factor for vertical stress and is given by –
It may be noted that the Boussinesq influence factor for vertical stress is a function of the ratio r/z and is independent of soil properties. It is maximum when r = 0, that is, below the concentrated load, and is equal to 0.4775. Values of IB are computed for different values of r/z and are presented in Table 8.1. The vertical stress computed by the Boussinesq theory for a point load is infinite when z = 0, that is, at the ground surface, where the load is applied.
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The following are the assumptions in the Boussinesq theory:
a. The soil is homogeneous and isotropic.
b. The soil mass is semi-infinite; that is, it extends infinitely in all directions below a level surface.
c. The soil may not be elastic, but obeys Hooke’s law; that is, the stress-strain relationship is linear.
d. The soil is weightless and unstressed before application of the load.
e. The load is applied at the ground surface.
i. Pressure Bulb:
From the Boussinesq equation for vertical stress, it may be observed that the vertical stress decreases with the increase in depth from the ground surface. It also decreases with the increase in the distance from the axis of the load in the lateral direction at any depth. The zone of soil within which the vertical stress is significant and causes significant deformation of the soil, affecting the safety and stability of the structure, is known as pressure bulb. The soil outside the pressure bulb is assumed to have negligible stresses.
An isobar is a curve joining points of equal vertical stresses in the soil mass. Isobar is a spatial curved surface in the shape of an electric bulb or an onion. For a concentrated load acting at the ground surface, the isobar is symmetrical on either side of the axis of the load. A pressure bulb is an isobar of stress intensity 0.1Q, where Q is the magnitude of the concentrated load. The coordinates of the points on the pressure bulb can be obtained by substituting σz = 0.1Q in the Boussinesq equation for vertical stress due to a point load. Thus,
By substituting the different values for z, the corresponding values of r/z and hence r can be obtained.
The resulting coordinates of r and z are plotted to get the pressure bulb of intensity 0.1Q, as shown in Fig. 8.3. It may be observed from the pressure bulb that r is zero directly below the point load and increases with the increase in depth below the ground surface.
The radial distance r attains a maximum value of 0.938 when z = 1.25. It then decreases with further increase in depth and becomes zero again at a depth of 2.185. The left-hand side of the pressure bulb can be drawn using the same value of r at different depths but with a negative sign. It is also possible to draw isobars of stress intensity 0.2Q, 0.5Q, and 0.75Q using a similar procedure; these isobars are shown in Fig. 8.4.
ii. Vertical Stress Distribution on a Horizontal Plane:
The vertical stress distribution on a horizontal plane at any given depth, say z = 1 m, due to concentrated load, can be computed by substituting different values of r in Eq. (8.7) for the same depth z = 1 m. From Eq. (8.7),
Thus, the vertical stress at a depth of 1 m can be computed in terms of Q by substituting different values of r in the above equation. The vertical stress distribution on a horizontal plane at depth 1 m is shown in Fig. 8.5. It may be seen that the vertical stress is maximum below the axis of the load equal to 0.4775Q and decreases rapidly to 0.27Q at r = 0.5 m and to 0.08Q at r = 1 m on the either side of the axis of the load.
iii. Vertical Stress Distribution on a Vertical Plane:
It is possible to determine the vertical stress distribution due to a point load on a vertical plane below the axis of the load by substituting r = 0 in Eq. (8.7). The vertical stress distribution on a vertical plane below the axis of the load is shown in Fig. 8.6. It starts from infinity, immediately below the load, and decreases very rapidly to an intensity of about 2Q within a depth of 0.5 m and to about 0.5Q at a depth of 1 m below load level. The distribution is asymptotic with the x-axis within this depth, whereas it is asymptotic with the y-axis over the remaining depth.
iv. Vertical Stress Due to Line Load:
A line load acts over negligible width for infinite length. The concept of line load is useful as it is extended to determine the vertical stress below strip footing.
Consider a line load of infinite length having intensity q’ per unit length. Consider a point P having coordinates (x, y, z) with respect to line load, at which the vertical stress is to be determined, as shown in Fig. 8.7. The point P is thus at a depth z below the load level and at a distance x normal to the length of the line load. The y-axis is along the length of the load.
Consider an elemental length δy of the load. The total elemental load over this small length can be considered as concentrated load and Eq. (8.6) of Boussinesq’s theory can be used to determine the elemental vertical stress at point P as –
The total vertical stress at point P due to the entire line load is obtained by integrating Eq. (8.11) as –
Solution of Eq. (8.12) gives the vertical stress, at point P, due to line load as –
where q’ is the load per unit length and IB is the Boussinesq influence factor given by –
2. Westergaard’s Theory for Vertical Stress:
The Boussinesq theory assumes that the soil mass is isotropic. Actual sedimentary soils are generally anisotropic. Thin layers of sand are usually embedded in a homogeneous clay deposit. Westergaard’s theory assumes that thin sheets of rigid material are sandwiched in a homogeneous soil mass. These thin sheets are closely spaced and are of infinite rigidity; hence, they are incompressible. These thin sheets of sand permit only downward displacement of the soil mass without lateral deformation.
As per Westergaard’s theory, vertical stress due to a point load is given by –
where σz is the vertical stress below the load Q at depth z at a radial distance r, Iw is Westergaard’s influence factor, and µ is the Poisson’s ratio of the soil. When µ = 0
The vertical stress below the axis of the load (r = 0), when µ = 0, will become –
Thus, Westergaard’s theory gives the vertical stress a 50% higher value than that given by the Boussinesq theory at any depth below the axis of the load.
3. The 2:1 Distribution Method:
For calculation of settlements, vertical stress is often obtained by assuming that the load is dispersed at a slope of 2 Vertical: 1 Horizontal from the base of the footing. Consider a rectangular footing ABCD of width b and length 1 subjected to a uniform pressure q. It is required to determine the vertical stress at depth z below the footing.
As per 2:1 distribution method, the resisting width AB at the foundation level will increase to EF at depth z, as shown in Fig. 8.34(a) and the resisting width BC at the foundation level will increase to FG at depth z, as shown in Fig. 8.34(b).
The resisting area ABCD at the foundation level will increase to EFGH at depth z, as shown in Fig. 8.34(c). Thus the load, which is acting over the area 1 × b foundation level, is dispersed at 2V:1H over depth z, so that the resisting area at depth z will be (l + z) x (b + z) at depth z.
The vertical stress at depth z below the rectangular footing is therefore determined from –
The 2:1 distribution method enables easy and quick determination of vertical stress at any depth below a footing. The method is approximate but the error involved is not considerable and preliminary estimation of settlement of footings can be done using this method.
Comparison of Different Methods for Determination of Vertical Stress:
Figure 8.36 shows the variation in vertical stress with depth as computed by the Boussinesq theory and the 2:1 approximate method. It may be observed from Fig. 8.36 that the vertical stress by the Boussinesq theory gives maximum and minimum values at the center and at the edge of the footing, respectively. The vertical stress computed by the 2:1 distribution method lies between these two extreme values at any depth. The average vertical stress is also shown in Fig. 8.36 with depth.
The vertical stress computed by the 2:1 distribution method is closer to the stress at the edge of the footing computed by the Boussinesq theory for depth > 0.5B. It approaches the average vertical stress at a depth of 0.2B. We know that the vertical stress at the edge of a square footing is about 50% of that at the center. Hence, the 2:1 distribution method underestimates vertical stress, especially for depths less than about 1.5B, where stresses are significant. Hence, the use of the 2:1 distribution method causes errors on the unsafe side.