SOIL AND EARTHWORK FORMULAS 189The Westergaard equation applies to an elastic material laterally reinforced
with horizontal sheets of negligible thickness and infinite rigidity, which prevent
the mass from undergoing lateral strain. The vertical stress at a point in the
mass, assuming a Poisson’s ratio of zero, is
(8.14)
wherezvertical stress at a point, lb/ft^2 (kPa)
Ptotal concentrated surface load, lb (N)
zdepth of point at which zacts, measured vertically downward
from surface, ft (m)
rhorizontal distance from projection of surface load Pto point at
whichzacts, ft (m)
For values of r/zbetween 0 and 1, the Westergaard equation gives stresses
appreciably lower than those given by the Boussinesq equation. For values of
r/zgreater than 2.2, both equations give stresses less than P/ 100z^2.
LATERAL PRESSURES IN SOILS,
FORCES ON RETAINING WALLS
The Rankine theory of lateral earth pressures, used for estimating approxi-
mate values for lateral pressures on retaining walls, assumes that the pres-
sure on the back of a vertical wall is the same as the pressure that would
exist on a vertical plane in an infinite soil mass. Friction between the wall
and the soil is neglected. The pressure on a wall consists of (1) the lateral
pressure of the soil held by the wall, (2) the pressure of the water (if any)
behind the wall, and (3) the lateral pressure from any surcharge on the soil
behind the wall.
Symbols used in this section are as follows:
unit weight of soil, lb / ft^3 (kg / m^3 ) (saturated unit weight, dry unit
weight, or buoyant unit weight, depending on conditions)
Ptotal thrust of soil, lb / linear ft (kg / linear m) of wall
Htotal height of wall, ft (m)
angle of internal friction of soil, degree
iangle of inclination of ground surface behind wall with horizontal;
also angle of inclination of line of action of total thrust Pand pres-
sures on wall with horizontal
KAcoefficient of active pressure
KPcoefficient of passive pressure
ccohesion, lb/ft^2 (kPa)zP
z^2 1 (^2)
r
z
2
3/2