Handbook of Civil Engineering Calculations

VERTICAL FORCE ON RECTANGULAR AREA

CAUSED BY POINT LOAD

A concentrated vertical load of 20 kips (89.0 kN) is applied at the ground surface. Deter-
mine the resultant vertical force caused by this load on a rectangular area 3 x 5 ft (91.4 x
152.4 cm) that lies 2 ft (61.0 cm) below the surface and has one vertex on the action line
of the applied force.

Calculation Procedure:

1. State the equation for the total force
   Refer to Fig. 4a, where A and B denote the dimensions of the rectangle, H its distance
   from the surface, and F is the resultant vertical force. Establish rectangular coordinate
   axes along the sides of the rectangle, as shown. Let C = A
   2
   
   
   • H
     2
     , D = B
     2
     
     
     • H
       2
       , E = A
       2
       
       
       • B
         2
       
       
       
       
     
     
     
     
   
   
   
   

• H
  2
  , S = sin-
  1
  H(EfCD)
  05
  deg.
  The force dF on an elemental area dA is given by the Boussinesq equation as dF =
  [IPz
  3
  /(2TrR
  5
  )] dA, where z = H and R = (H
  2


• x
  2


• /)°
  5

. Integrate this equation to obtain
an equation for the total force F. Set dA = dx dy; then

F 6 ABH / 1 1 \

7

= 0

'

25

-^+ ^(C

+

D)

(7)

2. Substitute numerical values and solve for F
   Thus, A = 3 ft (91.4 cm); B = 5 ft (152.4 cm); H= 2 ft (61.0 cm); C = 13; D = 29; E = 38;
   e = sin-
   1
   0.6350 = 39.4°; FIP = 0.25 - 0.109 + 0.086 = 0.227; F = 20(0.227) = 4.54 kips
   (20.194 kN).
   The resultant force on an area such as abed (Fig. 46) may be found by expressing the
   area in this manner: abed - ebhf- eagf+fhcj -fgdj. The forces on the areas on the right
   side of this equation are superimposed to find the force on abed. Various diagrams and
   charts have been devised to expedite the calculation of vertical soil pressure.

FIGURE 4

Surfoce