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CHAPTER VI.

Quantities of Materials.

I. Volumes from Cross Sections.

3. Volumes of Continuous Prismoids.

a. Volumes from Plotted Sections.

Deduction of Formula.

To develop a form of the Prismoid formula which will cover the case of obtaining by one operation the volume of a number of continuous prismoids, let us suppose that we desire to find the total volume of earth excavation in a railroad section several hundred feet in length, cross-sections of which have been taken as usual at each one hundred feet.

Let us suppose there are n of these prismoids each of which is 100 ft. long, and that beginning with the first end section we denote the successive cross-sections by the letters A0, A1, A2, A3, ..., An1 and An, the subscript of each A denoting the number of prismoids up to that section and n being an even number.

It is evident that with L = 100 and treating each A having an even subscript as the end section of a prismoid, 2 × L = 2 × 100 or 200 ft. long, the intermediate As having an odd subscript would correspond to the Am of Eq. 1, Pg. 62.

Taking then the prismoid whose end sections by this arrangement are A0 and A2 and making A1, the middle section, we have for the volume in cubic yards of the first prismoid

V1 = (2 × 100) ÷ (6 × 27) (A0 + 4 A1 + A2) ... (8)

Taking the next prismoid whose end sections are evidently A2 and A4, and whose middle section is A3, and denoting its volume in cubic yards of V2 we have

V2 = (2 × 100) ÷ (6 × 27) (A2 + 4 A3 + A4) ... (9)

Similarly for the third prismoid for whose volume we should have

V3 = (2 × 100) ÷ (6 × 27) (A4 + 4 A5 + A6) ... (10)
and for the last prismoid
Vn = (2 × 100) ÷ (6 × 27) (An2 + 4 An1 + An) ... (11)

Since the factor (2 × 100) ÷ (6 × 27) is common to all we should have the total volume of n prismoids.

V1 + V2 + V3 + ... + Vn = (2 × 100) ÷ (6 × 27) (A0 + 4 A1 + 2 A2 + 4 A3 + 2 A4 + ... + 4 An1 + An) ... (12)
or by reduction and arranging
Total Vol. = V Cu. Yds. = (2 × 100) ÷ (3 × 27) (A0 + 2 A1 + A2 + 2 A3 + A4 + ... + An (A0 + An) ÷ 2) ... (13)
or
Total Vol. = V Cu. Yds. = 2.4691 (A0 + 2 A1 + A2 + 2 A3 + A4 + ... An (A0 + An) ÷ 2) ... (14)
which is the formula desired. It is at once seen that this general formula can be written as
Total Vol. = V Cu. Yds. = 2.4691 (Sum Even A's + Twice Sum Odd A's ½ Sum Extreme A's) ... (15)

To employ this formula let us suppose that the railroad section whose volume is desired is say 600 feet long. Having selected a scale, say 1 inch to 8 feet, we proceed to plot the consecutive cross-sections, superimposing them one on another as shown in Fig. 3 of Plate IV, and denoting each cross-section by the letter A with the proper subscript showing its number from the first end section.

Since we have plotted our cross-sections to a linear scale of 1 in. = 8 ft. each square inch of actual area of plotted cross-section will represent 8² or 64 square feet, and introducing this factor in the general Eq. 14 we have the particular formula for this scale

V = 2.4691 × 64 × (A0 + 2 A1 + A2 + 2 A3 + A4 + 2 A5 + A6 (A0 + A6) ÷ 2) ... (16)
or
V = 158.02 (A0 + 2 A1 + A2 + 2 A3 + A4 + 2 A5 + A6 (A0 + A6) ÷ 2) ... (17)

In Eq. 17 it is seen that if the entire quantity inside the parenthesis should reduce to an actual area of 1 Sq. Inch the resulting equation would be

V = 158.02 × (1) = 158.02 Cu. Yds. ... (18)
which shows that if we plotted all our cross-sections to the scale of 1 inch = 8 feet, that every square inch of actual area of cross-section so plotted which is represented by the quantity within the parenthesis represents a Vol. of 158.02 Cu. Yds.

It is evident that if we adjust the Planimeter to that Setting which will give a Reading of 1580.2 Vernier Units when tracing an actual area of 1 square inch, and then proceed to trace the various plotted cross-sections continuously and in the manner indicated by the quantities inside the parenthesis of Eq. 17, the resulting Reading of the Planimeter at the completion of the entire tracing when multiplied by the value of the Relative Vernier Unit, which in this case is 0.1, will be the total volume in cubic yards of the 6 prismoids or the entire total volume of the section between the end sections A0 and A6.

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