Additional width at bottoms of cuttings, from 0 to 9 feet.

Arches over the railway are seldom made of the minimum spans shown by the foregoing tables, except in the case of tunnels. Bridges over narrow gauge lines are usually of the following spans:

over a single line, from 16 to 18 feet;

over a double line, from 28 to 30 feet.

5. Breadths of Slopes of Earthwork.-Let h denote the central depth of the piece of earthwork, whether cutting or embankment; b, the half-breadth of its base, or formation;

8, the rate of slope of the earthwork; that is, s horizontal to 1 vertical;

r, the rate of sidelong slope of the natural ground, if any; that is, r horizontal to 1 vertical;

B, the required breadth of the slope of the earthwork.

CASE I.-In ground level across, B = s h.

CASE II.-In ground that slopes away from the base,

CASE III.—In ground that slopes towards the base, but without intersecting it;

CASE IV.-In ground that intersects the base between the centre line and the edge of the earthwork,

SECTION VI.-RULES RELATING TO MENSURATION OF EARTH

WORK.

1. Sectional Areas of Earthwork.-Figs. 52, 53, and 54 repre

sent examples of cross-sections of pieces of earthwork, in each of

which D E is the base, A B the

natural surface, and D A and E B are the slopes.

Figs. 52 and 53 represent cut

tings; to represent embankments, conceive them to be turned upside

down.

M

N

B

H

E

Fig. 54 represents a piece of earthwork, of which one side, QE B, is in side cutting, and the other, Q D A, in embankment. The following are the symbols used in the rules:

Natural slope of the ground, r (horizontal) to 1 (vertical).

Slope of the earthwork,

s (horizontal) to 1 (vertical).

In many measurements of earthwork having sections such as figs. 52 and 53, it is convenient to suppose the slopes produced till they meet at K, and to calculate or measure the following quantity:

To find by direct measurement in a longitudinal section of earthwork, draw a line parallel to the formation line of the work, and at the vertical distance - below it in cuttings, or above it in

b

8

embankments. Depths measured from that line to the surface of the ground will be augmented depths.

RULE I.-When the ground is level across;

A = triangle A B K - triangle D E K = 8 k2

Or otherwise,

RULE IA.

=

b2

A = rectangle D G H E + 2 triangle A D G 2b h + sh2. RULE II.—When the ground has an uniform sidelong slope, not intersecting the base, as in fig. 53,

A triangle A B K - triangle DEK=

=

$2

k2

62

RULE III. To find the augmented depth in ground level across, of a cross-section of earthwork equal to a given cross-section in sidelong sloping ground; take a mean proportional between the augmented depths measured from K vertically to the two edges A and B respectively; that is to say, in fig. 53, parallel to D E, draw A M and B P, cutting the vertical centre line in M and P; then make k = √(K M.K P);

k':

:

and the area may be found by Rule I., as follows:

2. Volumes or Quantities of Earthwork.— -RULE I.-When a series of equidistant cross-sections are given, see p. 72, Article 5; also the rules there referred to, A, B, C, pages 64 to 66.

RULE II.—When the piece of earthwork to be measured is a "prismoid," as shown in page 74, fig. 12, use the rule given in that page below the figure.

The most simple algebraical expression of that rule, as applied to the present case, is as follows:-The prismoidal piece of earth to be measured is to be considered as formed by a wedge of a crosssection such as A B K in fig. 52 or fig. 53, from which is taken away a wedge of uniform cross-section such as D E K.

Let x denote the length of the piece of earth; k1 and k2, the values of the augmented depth C K at its two ends; then,

The last formula is specially suited for calculation by the aid of

a table of squares.

When the ground is level across, the co-efficient of the first term becomes simply:

= 8.

The quantity in brackets by which the length x is multiplied is the mean sectional area.

If the measurements are in feet, the preceding rules give quantities in cubic feet. To reduce these to cubic yards divide by 33 = 27.

RULE III.-When earthwork on sidelong ground occurs on a sharp curve. By the rules of pages 142, 143, calculate the halfbreadths (A L, B N, fig. 53) required for the two slopes; take their difference, and divide it by three times the radius of the curve; the quotient is to be added to or subtracted from 1, according as the greater half-breadth lies from or towards the centre of the curve. The result will be a factor by which the area A B K in fig. 53—that is, the first of the two terms of the formula in Rule II., page 144-is to be multiplied. From the product subtract the area DEK; the remainder will be an area modified for curvature; then proceed as in Rule I. of this Article.

L

PART IV.

RULES AND TABLES RELATING TO DISTRIBUTED FORCES AND MECHANICAL CENTRES.

1. Specific Gravity (as stated at page 102) is the ratio of the weight of a given bulk of a given substance to the weight of the same bulk of pure water at a standard temperature. In Britain the standard temperature is 62° Fahr. = 16°·67 Cent. In France it is the temperature of the maximum density of water = 3°.94 Cent. 39°-1 Fahr.

In rising from 39°1 Fahr. to 62° Fahr., pure water expands in the ratio of 1.001118 to 1; but that difference is of no consequence in calculations of specific gravity for engineering purposes.

RULE I.-To find the specific gravity of a solid body that is heavier than water approximately, by experiment. Weigh it in air, and again weigh it immersed in pure water.

Divide the weight in air by the loss of weight when immersed (or buoyancy); the quotient will be the specific gravity.

RULE II.-When the body is lighter than water, weigh it in air; then load it with a piece of a substance heavier than water, and large enough to make the light body sink, and weigh them in water together. Also weigh the heavy body separately, in air and in water. Subtract the buoyancy of the heavy body from the buoyancy of the two bodies together; the remainder will be the buoyancy of the light body separately; by which its weight in air is to be divided as before.

RULE III. To find approximately the specific gravity of a liquid; weigh some convenient solid body in air, in pure water, and in the given liquid; divide the buoyancy or loss of weight in the given liquid by the buoyancy in water; the quotient will be the required specific gravity.

RULE IV. To find approximately the specific gravity of a solid body that is soluble in water; ascertain its buoyancy in some liquid which does not dissolve it, and whose specific gravity is known; divide the weight in air by the buoyancy in that liquid, and multiply the quotient by the specific gravity of the liquid.

The approximate character of all those rules arises from their not taking account of the buoyancy due to the pressure of the air, whether on the body weighed or on the weights; but for ordinary practical purposes the error so occasioned is immaterial.