(4) See the bottom chart of Figure 10 for influence values for
stresses at various depths produced by the loads within each annular space.
The product I x A multiplied by the load intensity equals vertical stress.
(5) To determine a profile of vertical stresses for various depths
beneath a point, the target need not be redrawn. Obtain influence values
for different ordinates Z/R from the influence chart.
e. Horizontal Stresses. Elastic analysis is utilized to determine
horizontal stresses on unyielding walls from surcharge loads (see Chapter
7.02, Chapter 3), and pressures on rigid buried structures. (See basic
formulas for simple loads in Figure 2.) For more information, see Reference
5, Elastic Solutions for Soil and Rock Mechanics, by Poulos and Davis.
f. Shear Stresses. Elastic solutions generally are not applicable when
shear stresses are critical, as in stability problems. To determine if a
stability analysis is required, determine the maximum shear stress from
elastic formulas and compare this stress with the shear strength of the
soil. For embankment loads in Figure 2, maximum shear stress in the
foundation is exactly or approximately equal to p/[pi] depending upon the
shape of the load and point in question. If the maximum shear stress equals
shear strength, plastic conditions prevail at some point in the foundation
soil and if the load is increased, a larger and larger portion of the
foundation soil passes into plastic equilibrium. In this case, failure is
possible and overall stability must be evaluated.
2. LAYERED OR ANISOTROPIC FOUNDATIONS. Actual foundation conditions
differ from the homogeneous isotropic, semi-infinite mass assumed in the
Boussinesq expressions. The modulus of elasticity usually varies from layer
to layer, and soil deposits frequently are more rigid in the horizontal
direction than in the vertical.
a. Westergaard Analysis. The Westergaard analysis is based on the
assumption that the soil on which load is applied is reinforced by closely
spaced horizontal layers which prevent horizontal displacement. The effect
of the Westergaard assumption is to reduce the stresses substantially below
those obtained by the Boussinesq equations. The Westergaard analysis is
applicable to soil profiles consisting of alternate layers of soft and stiff
materials, such as soft clays with frequent horizontal layers of sand having
greater stiffness in the horizontal direction. Figures 11 (Reference 1), 12
(Reference 6, An Engineering Manual for Settlement Studies, by Duncan and
Buchignani), and 13 (Reference 1) can be used for calculating vertical
stresses in Westergaard material for three loading conditions. Computations
for Figures 11, 12, and 13 are made in a manner identical to that for
Figures 3, 4, and 7, which are based on the Boussinesq equations. For
illustration see Figure 8.
b. Layered Foundations. When the foundation soil consists of a number
of layers of substantial thickness, having distinctly different elastic
properties, the vertical and other stresses are markedly different from
those obtained by using the Boussinesq equation. (See Figure 14, Reference
7, Stresses and Displacement in Layered Systems, by Mehta and Veletsos, for
influence values of vertical stresses in a two-layer foundation with various
ratios of modulus of elasticity. See Figure 15 for an example.)
7.1-175
Change 1, September 1986
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