Stress Distribution in Soils
Step 4: Two possibilities are addressed in this step; (a) for uniform loading and (b) for random loading
(a) For uniform loading count the number of influence areas enclosed by the plan of the loaded area as N. In case of uniform loading, the increase in vertical pressure is as follows:
z qI NB
s = (3.9)
(b) For random loading over the area, determine the average pressure on each of the squares covered.
1 1
1 n z q I nB
s =
Â
(3.9a)where
IB = influence value
q1= average pressure or loading on the area n1= number of influence areas covered by q1.
3.5 WESTERGARD’S SOLUTION [IS 8009 (PART I) 1976]
Boussinesq assumed soil as an ideal material. Only normally consolidated clays can be considered as such a material. In overconsolidated clays and in laminated clays with thin soil layers of infinite rigidity in between the strata, the lateral strain is restricted by the rigidity of these layers. Hence the ratio of the modulus (Eh/Ev) is very large in fact, equal to infinity. Westergard [5] gave a solution for such problems in 1938. Later in 1948 Taylor evolved a chart for use of the above theory for practical purposes as shown in Fig. 3.7. This chart is used in the same manner as Newmark’s chart [1].
Fig. 3.7 Influence chart for vertical stress according to Westergard’s theory (IS 8009 Part 1, Fig. 19).
O
Scale distance hz A B
n =
1 2 2 2
Influence value = 0.001
The unit distance AB marked in the chart is hz, where h = [(1 – 2m)/(2 – 2m)]1/2 and m is the Poisson’s ratio for the soil.
3.6 METHODS FOR SPECIAL CASES [1]
Stress distribution for much more complex conditions is available. Three such methods are stated below.
3.6.1 Fröhlich’s Concentration Factor Method [IS 8009 (Part I) 1976]
In the case of sands, experimental evidence indicates that (Eh/Ev) is considerably less than 1.
Published experimental evidence suggests that sz for a rectangular loaded area can be computed from the semi-empirical equation that Frohlich worked out in 1934 for a material with elastic anisotropy and poor tensile strength. For this purpose, a concentration factor is used. For the case where the modulus of elasticity increases linearly with depth, a concentration factor of 4 is recommended. Charts similar to Newmark’s chart, as shown in Fig. 3.8, are available for the application of this theory.
Fig. 3.8 Influence chart for vertical stress with Fröhlich concentration factor m¢ = 4 (IS 8009, Part 1—1976, Fig. 20).
B A
Influence value
= 0.005
o
7 cm
3.6.2 Burmister’s Method for Layered Soils
Many natural soil deposits occur in layers. In 1943 Burmister [6] derived expressions for stresses and displacements for two- and three-layered elastic systems. These are available in reference books for application to such cases [2].
3.6.3 Stress Distribution below Embankments
It has been found from field measurements that the stress distribution below the foundation of earth embankment, as calculated by assuming the embankment, to be a trapezoidal loading made of a rectangle and a triangle, gives higher pressures than the actual values. However, the fact that such application of Boussinesq’s solution (as a plane strain problem) does not lead to a correct assessment was shown by Perloff [2]. It is well known that shear distortions occur at the interface between the embankment and underlying soil. (Embankments have been strengthened by reinforcing mats at fill foundation surfaces.) Whereas the vertical stress in Boussinesq’s solution is independent of E and m. In the field, the values have been found to depend on m also. Charts for correct estimation of these pressures are available in many references [2].
3.7 STRESS DISTRIBUTION FOR LOAD APPLIED BELOW THE SURFACE When vertical loads are applied below the surface, on excavating a depth of soil, the stresses at a corresponding depth D are reduced due to two reasons. Firstly, the overburden is removed and secondly, the application of load below the base alters its distribution. (In both cases, the vertical stresses at a point z below the load will be less than that for the loading at the surface.) This problem of excavation analysis has been dealt with by many investigators and solutions in the form of tables and charts are also available for easy calculation of stresses [2].
Another problem of importance is the distribution of stresses due to loaded piles. The Boussinesq’s equation cannot be used for this case since it assumes that the load acts on the boundary of the half space only. Piles transfer the loads in the interior of the elastic half space.
The solution was published by Mindlin in 1936 [7]. On the basis of Mindlin, Poulos and Davis [8] developed charts for influence factors for calculating settlements of piles and piers.
3.8 PRESSURE DISTRIBUTION AT BASE OF STRIP FOOTINGS Figure 3.9 shows the distribution of vertical normal stresses and Fig. 3.10, the distribution of shear stresses under a uniformly distributed strip load. For a strip load, the principal stresses at the centre line below the strip can be expressed as follows.
s1 = (q/p)(q + sin q) s2 = (q/p)(q – sin q)
t = (1/2)(s1 – s2) = (q/p) sin q; max. value = q/p (3.10) where
q = load per unit area
q = central angle as shown in Fig. 3.10.
3.8.1 Summary of Findings
As shown in Fig. 3.10, shear stresses increase to a maximum when q = 90° at a depth of about B/2. At this point, its value will be (q/p) or 0.32q. Subsequently, it decreases with depth. On the
other hand, the vertical stress decreases steadily with depth as shown in Fig. 3.9. The vertical stress reduces to one-third its surface value at 2B, which can be considered as the zone of major influence of footings.
3.9 HEAVE IN CLAY DUE TO EXCAVATION
The decrease of load due to excavation and the consequent heave in clay soils is an important problem for soil engineers. As already mentioned in Section 3.7, this can be carried out by the
“excavation analysis” developed by Baladi in 1968, which takes into account the effect of the material surrounding the excavation on the stresses and deformation [2]. An approximate method is to consider excavation as an unloading with E value greater than for loading. However, heave due to excavation in clays when given access to water should be estimated on the basis of oedometer rebound curves.
Fig. 3.9 Isobars for vertical (normal) stresses under a strip load.
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 q
3.10 STRESSES ON RETAINING WALLS DUE TO LOADS BEHIND THE WALLS
There are many situations where the effect of loads behind walls are to be evaluated. We may use Boussinesq’s equation for their solution. An empirical method given by Terzaghi and Peck [9]
for a line load (like a railway line or water tank behind a wall) is shown in Fig. 3.11. This problem will be dealt with in more detail in Chapter 17 on retaining walls.
Fig. 3.10 Shear stress distribution under a strip load as a function of q/p.
B
Fig. 3.11 Loads behind retaining walls: (a) Line load; (b) concentrated load.
C¥ p
3.11 APPROXIMATE METHODS FOR EVALUATION OF VERTICAL STRESSES IN SOILS
There are many approximate methods to evaluate the magnitude of pressures in soils. They are very much used in practice for rough solutions. Some of them are the following [10].