An earthwork contract was awarded to excavate and backfill the foundation of a proposed 50-ft by 50-ft office building.  The existing grade elevation is 437.5-ft while the sub-base for the below grade basement is 432.5-ft.  The concrete contractor requires a 3-ft perimeter walkway in order to place the concrete formwork. 

Soil conditions are classified as OSHA Type B soils which require 1H:1V side slopes. The bank volume (yd³) to be stockpiled and used for backfill is most nearly:

Solution:The foundation excavation can be described as an inverted truncated pyramid.  Compute the earthwork volume using the buildings dimensions and add a 3-ft perimeter walkway around the building for the workers erecting the concrete formwork. Equation for the volume of a truncated pyramid:

Volume = V₁ = h/3 (A₁ + A₂ + √(A₁ x A₂))

Compute depth of foundation = h = 437.5-ft - 432.5-ft = 5.0-ft

A1 = Area of the base of truncated pyramid = (50 + 3 + 3) (50 + 3 + 3) = 3,136-ft²

A2 = Area of the top of truncated pyramid = (50 + 3 + 3 + 5 + 5) (50 + 3 + 3 + 5 + 5) = 4,356-ft²

V1 = 5/3 (3,136 + 4,356 + (√(3,136 x 4,356)

V1 = 18,647-ft³= 690-yd³

Compute the volume of the basement and subtract this from the total excavation to determine the volume of backfill material.

Building Volume = 50'  x 50'  x 5' =  12,500-ft³  =   463yd³

Backfill stockpile required = 690-yd³ - 46³-yd³ = 227-yd³