Page 255 - Civil Engineering Formulas
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SOIL AND EARTHWORK FORMULAS            189

               The 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
                                     P         r  2 3/2
                                 z    2   1   2                   (8.14)
                                    
z         z

                                          2
             where     vertical stress at a point, lb/ft (kPa)
                   z
                  P   total concentrated surface load, lb (N)
                   z   depth of point at which   acts, measured vertically downward
                                        z
                      from surface, ft (m)
                  r   horizontal distance from projection of surface load P to point at
                      which   acts, ft (m)
                            z
               For values of r/z between 0 and 1, the Westergaard equation gives stresses
             appreciably lower than those given by the Boussinesq equation. For values of
                                                            2
             r/z greater than 2.2, both equations give stresses less than P/100z .
             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:
                                            3
                                      3
                   unit weight of soil, lb/ft (kg/m ) (saturated unit weight, dry unit
                     weight, or buoyant unit weight, depending on conditions)
                P   total thrust of soil, lb/linear ft (kg/linear m) of wall
                H   total height of wall, ft (m)
                   angle of internal friction of soil, degree
                 i   angle of inclination of ground surface behind wall with horizontal;
                     also angle of inclination of line of action of total thrust P and pres-
                     sures on wall with horizontal
               K   coefficient of active pressure
                 A
               K   coefficient of passive pressure
                 P
                               2
                c   cohesion, lb/ft (kPa)
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