Calculus

Factoring and Multiples

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Calculating Areas Before volumes can be calculated, the areas of excavation (either horizontal or vertical) must be established. Horizontal areas are associated with cross sections cut through the earthwork volume and vertical surfaces associated with Digital Terrain Model (DTM) volumes. Horizontal areas usually refer to either the area extent of the excavation and horizontal areas enclosed by contour elevation lines. These areas are defined by a series of discreet points along their boundaries. Areas are calculated by connecting these points in a series of continuous triangles that extend across the area. Given the northing and easting of each of the three vertices of each triangle, and the lengths of each of the three sides of the triangles, each triangle's area can be calculated as follows: A = sqrt[s * (s - a) * (s - b) * (s - c)]

Where,

A = the area of the triangular area (square feet)

a, b, c, = the lengths of the three sides of the triangle (feet)

s = (a + b + c) / 2

An alternate method involves measuring the lengths of parallel lines traversing the area from one side to the opposite side at a constant interval. The area of each strip is calculated by multiplying the line's length by the distance interval between the lines. The sum of all the strips' areas gives the overall surface area. Computer and CADD programs use either method, but can perform many more operations, allowing for a higher degree of accuracy. Calculating Volumes There are several ways of calculating bank (in-place) earth and soil volumes. The simplest is the Depth Area Method (DAM), which involves multiplying the thickness of the strata to be excavated by the surficial area of the strata. This can be done with any reasonable accuracy only for strata that is consistently thick and whose area extent is known. It is perfectly suitable for estimating the amount of topsoil to be stripped at a consistent depth (usually 6 inches). It is also applicable for estimating the volume of regular (square or rectangular with vertical sideslopes) excavations of a consistent depth below a relatively flat surface. Remember to use the sloped surficial area of the excavation zone, not the projected plain area. Failure to do so will introduce additional errors into the volume calculation. For example, a plane acre of a slope with a 25% grade (approximately 14 degrees to the horizontal) will have a surficial area of about 1.03 acres. Volumes are calculated as follows: V = T * A * (1/27)

Where,

V = volume (cubic yards)

A = surficial slope area (square feet)

T = thickness of strata or even cut (feet)

The Grid Method (also known as the Borrow Pit Method) extends DAM to an excavation of varying depths. Borrow-pit leveling calculates the excavation volume by applying a grid to the excavation area. The grids can be staked to squares of 10, 20, 50, 100, or more feet depending on the project size and the accuracy desired. For each grid square, final elevations are established for each corner of every grid square. These are subtracted from the existing elevations at the same location to determine the depth of cut or height of fill at each corner. For each grid square an average of the depths/heights of the four corners is multiplied by the area of the square to determine the volume of earthwork associated with the grid area. The total earthwork volume for the project is calculated by adding the volumes of each grid square in the excavation area. Volumes are calculated as follows: V = ((D1 + D2 + D3 + D4) / 4) * A * (1/27)

Where,

V = volume (cubic yards)

A = area of the grid square (square feet)

D = depth of cut/fill at each grid corner (feet)

The End Area Method (EAM) utilizes the areas of parallel cross sections at regular intervals through the proposed earthwork volume. These cross sections are can be spaced at intervals of 25, 50, 100, or 200 feet depending on the size of the site and the required accuracy. They are aligned perpendicular to a baseline that extends the entire length of the excavation area. There are several types of cross sections, which can be drawn by hand or generated by CADD. For flat terrain or level excavation, a level section is suitable. Irregular sections are used for most excavations in rough terrain. Transition or side-hill sections occur when passing from excavation cut to embankment fill. Cross-sectional areas are calculated with either the triangular area method described above (if the cross sections are geometrically simple) or by the Length Interval Method for more complicated cross sections. Volumes are calculated as follows: V = L * ((A1 + A2) / 2) * (1/27)

Where,

V = volume (cubic yards)

A = areas of the adjacent cross sections (square feet)

L = distance between cross section along the baseline (feet)

The Prismoidal Formula (PF) allows for greater accuracy than EAM. It is especially useful when the ground is not uniform or significantly irregular between cross sections. PF adds an additional cross-sectional area midway between the two cross sections defining the volume being calculated. Note that this cross section is calculated separately and is not an average between the two end areas. Volumes are calculated as follows: V = L * ((A1 + (4 * Am) + A2) / 6) * (1/27)

Where,

V = volume (cubic yards)

A1, A2 = areas of the adjacent cross sections (square feet)

Am = area of the midway cross section (square feet)

L = distance between cross section along the baseline (feet)

The Contour Area Method (CAM) uses the area of the excavation elevation contour lines to determine volumes. From a topographic map of the site, the areas enclosed by regular contour intervals are measured. This area measurement can be done by hand with a planimeter, electronically by a digitizer, or directly with a CADD program. If the horizontal areas enclosed by each contour line are large relative to the elevation difference between the two contour elevations, averaging the two areas and multiplying the average by the height difference can determine volumes. However, for relatively small earthworks (like spoil piles and borrow areas), volumes can be calculated based on the formula for the volume of a truncated pyramid: V = (h / 3) * (B1 + B2 + sqrt[B1* B2] ) * (1/27)

Where,

V = volume (cubic yards)

B1, B2 = areas of the contour elevation lines (square feet)

H = elevation difference between the contour elevations (feet)

*************************************************************************** Average End Area Method

Assumes volume between two consecutive cross sections is the average of their areas multiplied by the distance between them V = L(A1 + A2)รท(2*27)

V = volume (yd3)

A1 and A2 = end areas of cross-sections 1 & 2 (ft2)

L = distance between cross-sections (feet)

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