Computation of Volume from Cross-sections

Profile levelling is carried out along the centre line of the alignment of the project and cross-sectional levels are taken at regular intervals to compute earth work. Then if the cross-sections are determined the volume of earth work can be found.

First the calculation of cross-sectional area is discussed.

(a)   If section is level

Let 'h' be the depth at the centre line of the alignment and 1 : n be the side slopes.

Then

(b)   If it is a multilevel section

Suppose the coordinates of points be (x1, y1), (x2, y2), ..., (xn, yn), then arrange the coordinates in the following order

Then area of the figure

= 2[Σ Product of pair of coordinates connected by continuous lines - ΣProduct of coordinates connected by dotted lines]   ...(18.9)

The above formula can be proved easily by taking a simple instance of a quadrilateral. Suppose the coordinates of A, B, C and D be (x1, y1), (x2, y2), (x3, y3) and (x4, y4). Then area of ABCD

= Area of a AB b + Area of b BC c + Area of c CD d - Area of a AD d.

= 2  [x₁y₂ - x₁y₁ + x₂y₂ - x₂y₁ + x₂y₃ - x₂y₂ + x₃y₃ - x₃y₂ + x₃y₄ - x₃y₃ + x₄y₄ - x₄y₃

- x₁y₄ + x₁y₁ - x₄y₄ + x₄y₁]

= (x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₁ + x₁y₄)

[Note terms with same subscript appear in pairs and cancel each other]. Therefore equation is proved.