(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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