materials the facility or difficulty of excavation must be especially considered in laying out the gradients; larger excavations must be adopted in the loose material, and smaller ones in stiff clays, hard rocks, &c., than would have been adopted, had no such variety in the strata existed. Considerable economy in the construction of a railway will result by judiciously taking into account all these circumstances. These important subjects, in the railway mania of 1845, were by far too little, or not at all considered, partly through pressing too much business on a few of the chief engineers, by which they were compelled to confide these important works to unskilful apprentices and other incompetent persons, and partly through the ignorance or wilful negligence of others, who had the audacity to put themselves forth, and were accepted by the public as chief engineers at that time, by which a worse than useless expenditure of several millions of the money of the shareholders in these projects was incurred, their blunders being now obvious even to illiterate agricultural labourers.

CHAPTER III.

ON SETTING OUT THE SURFACE WIDTHS OF RAILWAYS.

After the centre of a railway has been marked out, as directed in Chap. II., the line must again be carefully levelled, and the stumps that mark the line must be numbered and entered consecutively in the level book in a vertical column, with the corresponding depths of cuttings or embankments in a second column (see Level Book, p. 187); these depths are estimated from the formation level, which is commonly about 2 feet below the intended line of the rails; the 2 feet are afterwards to be filled up with gravel to form the permanent way.

The line is now prepared for setting out the surface widths, the simplest case of which is when the surface of the ground is level as well as coincident with the formation level of the intended railway. In this case it is only required to set out half the width of the formation level on each side of the centre stump, perpendicular to the direction of the railway, adding to each half width the intended width of the side fence, and putting down stumps to mark the half widths and breadths of the fences. When the surface of the ground is above or below the formation level, which is commonly the case, the widths must be set out by the following Problems:

PROBLEM I.

To set out the width of a railway cutting, when the surface of the ground is laterally level, and at a given height above the formation level, the ratio of the slopes* being given.

In the annexed cross section of the cutting, RS is the horizontal surface of the ground; AB the formation level;

=

*The ratio of the slopes is the proportion that the batter C a bears to the depth A a. When this ratio is as 1 to 1, Ca Aa; when it is as 1 to 1, Ca 1 times A a. This ratio varies according to the nature of the material through which the cutting is made, being less in rocky or clayey ground, and greater in soft or sandy ground.

Ꭱ Ꮯ

a Mb D

A C, BD the side slopes; M the middle stump, and M m = Aa Bb the perpendicular depth of the cutting. Multiply the depth Mm by the ratio of the slopes, to which add the half width Am or a M, the sum is half the surface width of be set out from M to C;

Am B

after which set out R C for the width of the fence. The same operation must be repeated on the other side of M.

EXAMPLE.

Let the width of the formation level AB = 33 feet, the depth Mm of the cutting 30, the ratio of the slopes as 1 to 1, and the width of the side fences each 6 feet; required the width of the cutting, and the width of land included by the fences.

=

30 x 13345+16= 613 feet MCMD=&width of cutting; and 61+6 = 673 = MR = MS = width of land. The double of this is the whole width.

Construction of the cross section. Draw A B = width of formation level 33 feet; perpendicular to A B, at its middle point m, draw Mm = given depth = 30 feet; through M parallel to AB draw CD, making CM, MD each = AB+ 1x Mm; join AC, BD; then ABDC is the cross section required.

NOTE 1. The numbers in the column, marked "computed half-widths" in the level book are found by this Problem.

NOTE 2. If the figure ABDC be inverted, it will represent the cross section of an embankment; for setting out the width of which the same method obviously applies, as that just given for a cutting.

NOTE 3. Let w= = AB = breadth of the formation level, d = Mm = depth of cutting or embankment, and r— ratio of the slopes. Then 1:r::d: dr=aC; hence dr+=MC=MD; 2 (dr+†w)=2 dr+w= CD = surface width of cutting or embankment; and (AB + CD) × Mm = (dr + w) d area of the cross section.

PROBLEM II.

The same things being given as in the last Problem, to set out the width of the cutting, when the ground is laterally sloping, the lateral fall of the ground in a given horizontal distance being also given.

Let CD be the sloping surface of the ground, ABDC the cross section of the cutting, and Р D' a horizontal line passing through the centre stump, M, M D' being the computed half width of the cutting.-Fix the levelling instrument so that by turning the telescope 2, 3 or more chains of the line may be

seen, on both sides of it; set up a levelling staff at M, and another at q, not exceeding a chain's distance from M, observing

the level readings

on both staves, the difference of which is equal to pq; measure the sloping and horizontal distances Mq, Mp with a tape line in feet. Then take the computed half width MD',

(found by the last Problem) and multiply it by M q, reserving the product; multiply the difference of the stave readings p q by the ratio of the slopes, add and subtract the product to and from the horizontal distance Mp, reserving the sum and difference; lastly, the reserved product, being divided by the reserved sum, will give the corrected half width M C, and by the reserved difference the corrected half width M D.

EXAMPLE.

The depth of the cutting at M is 22 feet, the bottom width A B = 36, the sloping distance Mq = 25, the level distance Mp 24, the difference of the readings of the staves p q = 7 feet, and the ratio of the slopes as 1 to 1. What are the cor

=

rected half widths M C and MD?

22 × 11 + 36 = 33 + 18 = 51 feet computed half width

24

7 × 1 = 101

25

1275 reserved product.

reserved sum = 341) 1275 (36.95 feet = cor.

width M C.

reserved diff. = 13) 1275 (94-44 feet cor. width M D.

Construction of the cross section.-Draw B D' M m, as in Prob. I.; lay off the given horizontal distance Mp = 24 feet; draw pq parallel to M m, and equal to the difference of the stave readings at M and q; through M, q draw CD, meeting AC in C and B D' prolonged in D; then ABDC is the required cross section.

NOTE 1. The following general formula for the values of MC, MD, is easily remembered, and would perhaps be preferred to the rule, as given above, by those who are accustomed to the use of symbols.

The positive sign is used for MC, and the negative for MD; MD' being = w', pq=h, Mq=s, M p = l, and the ratio of the slopes as r: 1. The investigation of this formula is given in Baker's Railway Engineering, page 35, wherein, when h is very small, s may be taken = l without error.

NOTE 2. If the figure A BDC be inverted, it will obviously represent the cross section of an embankment of like dimensions, the longer distance, in this case, being measured down, and the shorter up the slope.

PROBLEM III.

The same things being given, as in Prob. II., to set out the width, when it consists partly of a cutting and partly of an embankment.

In the annexed figure BDPCA is the cross section of the works of a railway, consisting of the cutting BPD and the embankment APC; AB

is the formation level, M the central stump, and DC the sloping surface of the ground. When the cutting BPD extends over more than half the formation level A B, the corrected half width MD is found as in Prob. II. But to find the

R

m

B

other corrected half width MC, passing under the embankment APC', proceed as follows. Multiply MD by the difference of the width of the formation level A B and the estimated half width MD', and divide the product by MD', and the result will be the corrected half width M C.

EXAMPLE.

Let the bottom width AB = 36 feet, the depth Mm = 4 feet, the ratio of the slopes as 2: 1, and the difference of level readings 7 feet at 25 and 24 feet from the central stump, respectively estimated on the slope and horizontally; required the corrected half width M D, M Č.