in this case be taken less than the distance ca' computed by formula 3. As in the preceding case, if the measured distance from cr to the trial point is less than the computed distance, the point should be moved out; if greater, it should be moved in.

Form of Notes in Cross-Section Work.-When each slope stake has been set as just explained, its distance from the center line and the elevation of the stake above or below subgrade are entered in the field book in the form of a fraction. The numerator of this fraction is the distance of the stake above or below subgrade, and the denominator is the distance of the stake from the center line. Thus, if the slope stakes in the preceding example are set at Sta. 131, the complete entry in the notebook will be as follows:

Fig. 4, is 27.2 ft. from the center line of the roadbed and 11.4

ft. above subgrade. Similarly, the fraction

C2.3

indicates

13.5

that the right slope stake m' is 13.5 ft. to the right of the center line and 2.3 ft. above subgrade. These expressions are called slope-stake fractions.

When the ground between the slope stakes and the center stake is irregular, the elevations and distances from the center of the intermediate points where the ground changes abruptly are determined and also entered in the notebook in the form of fractions.

COMPUTATION OF VOLUME

In calculating the cubical contents of earthwork, the volumes between two consecutive cross-sections are considered as prismoids whose bases are such sections as mcm'l'l, Fig. 4, and whose lengths are the distances between the crosssections. These are usually 100 ft., unless the surface of the ground is rough and irregular, when sections at intervals of less than 100 ft. are taken. If A1 and A2 are the areas of the bases of a prismoid, Am the area of a section midway between the bases, and I the perpendicular distance between them, the approximate volume Va of the prismoid, as figured by the endarea method, is

and the true area, as figured by the prismoidal formula, is

Prismoidal Correction.-Formula 1 will usually give fairly good results; for accurate work, however, formula 2 is used. This formula requires that the dimensions of the middle section whose area is Am shall be determined. This may be done by averaging the di

mensions of the bases from which Am might be computed. It is much simpler, however, to figure the approximate volume Va by formula 1, and then, if desired, apply a correction

FIG. 5

equal to the algebraic difference between the volume V and Vai the result obtained will be the same as if formula 2 were used. This difference is called the prismoidal correction.

Correction for a Triangular Prismoid.-Fig. 5 shows a triangular prismoid, the dimensions of which are marked. Its approximate volume as computed by formula 1 is

and the prismoidal correction is

เ

C= 12(b1-b2) (h2-hı)

The true volume of the triangular prismoid is, therefore,
V=Va+C

A study of the correction will show that, if either the bases, or the altitudes of the two end sections are equal, one of the factors (bi — b2) or (h2− h) will become zero, and therefore the correction becomes zero. It shows also that, when one or both of these factors are small, the correction is a correspondingly small quantity; and that, when (as is usually the case) the breadth and height at one section are both smaller or both larger than the breadth and height at the other section, the correction is negative. . Thus, if b2 is less than bi and he is less than h1, then b1-b2 is positive, h2-hi is negative, and, therefore, C is negative. But when C is negative, Va is greater than the true volume V; that is, the method of averaging end areas usually gives a result that is too large. When the difference of the breadths and heights is very large, the correction is very large, and Va is very greatly in error. Thus, for a pyramid, in which both b2 and h2 are zero, the correction is bihil

-(b1-0) (0—hı) =

12

The true volume is bihil, and therefore, the error in the value of Va is one-half or 50%, of the true volume. This extreme case shows the importance of computing the prismoidal correction when the areas of the bases are very unequal. EXAMPLE. The dimensions of the bases of a triangular prismoid are: b1 = 18 ft., hi=8 ft., b2 12 ft., and h2=9 ft. Find the volume of this prismoid, in cubic yards, if the length of the prismoid is 100 ft.

SOLUTION.-The areas of the bases are: A1X18X8=72 sq. ft., and A2= } × 12 × 9 = 54 sq. ft. Substituting these values in the preceding formula for Va, and dividing by 27 to reduce to cubic yards,

Va=100X (72+54)÷27=233.33 cu. yd., nearly Substituting the given values in the formula for C, and dividing by 27 to reduce to cubic yards,

C= X(18-12) × (9-8)÷27=1.85 cu. yd.

Therefore,

V=233.33+1.85=235.18, say 235, cu. yd. Correction for Curvature.-Besides the prismoidal correction, a correction for curvature is sometimes required in calculations of earthwork on a curve.

In Fig. 6, let rri be the curved center line of the roadbed, O the center of this circular curve, and R its radius. Let A1 be the area of the cross-section mnpq, G its center of gravity, and en the horizontal distance from G to the center of the roadbed, which distance is called the eccentricity of the section. Similarly, let A2 be the area of the section minipigi, G1 its

center of gravity, and e2 the eccentricity of that section. The general formula for curvature correction is, then,

If G and G1 lie on the outside of the curved center line of the roadbed, C, is to be added to the volume calculated as for a straight track. If G and G1 are on the inside of this curved center line, the correction C, is to be subtracted.

The expression for C, shows that the larger the eccentricities of the end sections, the larger C, will be, and that, if the radius of the curve is very large, C, will be very small. For curves of very large radii, the correction is usually so small that it may be neglected. When the area of that part rpqt of the end section lying on the inside of the center of the track is approximately equal to the portion of the area rtmn lying

outside of the center, the eccentricity is small, and the correction may usually be neglected, even with curves of short radii. But when the eccentricity is large (as is usually the case in side-hill work), the curvature correction may be a very considerable percentage of the volume, and should not be neglected, especially if the radius of the curve is small. To apply the general formula for curvature correction, the eccentricities ei and e2 are required. These can be determined by using the methods employed in finding the center of gravity of plane figures. The section is divided into triangles and their areas are referred to the vertical axis through the center of the track; then the coordinate of the center of gravity of the total area with regard to this axis is found, which coordinate

FIG. 7

is the eccentricity of the section.

Three-Level Sections. Where the surface of the ground is fairly regular, it is sufficiently accurate to determine the elevation of the center point and the distances and elevations of the two slope stakes. The method assumes that the straight lines cq and cp, Fig. 7, that join the center with the slope stakes are on the surface

of the ground. When this method is used, the sections are called three-level sections.

To calculate the volume of a prismoid whose bases are threelevel sections distant from each other, let, in Fig. 7, the area qcpn A and the area of tmn=T. Then, using the notation of the figure and the sign (') to denote corresponding values at the other base, the approximate volume is