GRADIENTS.

Gradients are the ascending and descending lines of a railway, dividing the section, showing the natural surface of the ground nearly into equal parts; the portion on the upper side of the line is called cutting, and the lower portion embankment. The earth is removed from one to the other, forming a level surface from one fixed point to another; at this point either another gradient is formed, or it is carried on the side level to the datum line until it changes again at another fixed point.

There is great judgment required in fixing gradients, such as economy in the earthwork and bridges, the limit of ascent and descent, and the restriction in the proper height for bridges over roads, rivers, &c.

In the laying out of gradients public and turnpike roads command particular attention; level crossings are to be avoided as much as possible, or they must be crossed 20 feet above or below them; if then it is impracticable, the road has to be either raised or lowered to meet the level of the rail, as the case may be, and sunk to gain the depth required; if it passes under the railway, the inclination of the approaches in the turnpike raised will be 1 in 30, and other roads 1 in 20.

Problem 18.

To find the rate of inclination of a gradient between two fixed points.

Rule. Divide the distance contained between the two points by the difference of the two heights, the quotient will be the rate of inclination.

Note. The lengths on the horizontal or datum line are usually divided into chains, therefore they must be reduced to feet, the same as the vertical height.

Fig. 1, Plate 35. Given the length A B = 34 chains 45 links, or 2273.70 feet, and the height from datum to A = 109.90 feet, and to B = 122.15 feet.

:

Thus 122.15 109.90 12.25 2273.70 =

185, or 1 in 185

Problem 19.

To find the height at every chain from the datum to the gradient.

Rule. Divide the number of feet in a chain by the rate of inclination; the quotient must then be added to the lower given height if the gradient ascends, and subtracted from the greatest height if the gradient descends, as shown in the middle column, which, subtracted from the heights in the lower column, will give the heights and depths of the cuttings and embankments shown in the upper column.

Thus: 185 66000 = .356 the decimal to be added or subtracted

Problem 20.

Given the height at one end of the gradient, the rate of inclination, and the length, to find the height at the other end of the gradient.

Rule. Divide the given length by the given gradient, the product will be the difference between the two heights, which add or subtract to the given height.

To find the rate of inclination in 1 mile.

Rule. Divide the number of feet in a mile by the given gradient, the product will be the rate required.

Thus: 5280185 = 28.540 or 28 in a mile nearly

Let the next gradient, B to C in Fig. 1, be cast through in the

same manner.

LAYING OUT A LINE RAILWAY ON THE GROUND.

If the plan provided in the first instance is not truly accurate, it is indispensable that a fresh survey should be made before any other proceedings are entered into, as by it in particular the lines and curves are projected, the quantity of land and the earthwork calculated, and the longitudinal section governed.

The line being carefully drawn on the plan, the curves clearly described by their radius, and tangent points drawn, a tracing is given to the surveyor to stake out the same.

In the general observations of Part II. is described the most perfect method of ranging out a straight line, and the care required in chaining, which equally applies to this case. A stump has to be driven in the ground very truly in the line level with the surface at every chain, extending beyond the tangent points where required; at the tangent points it is usual to put a stake on each side the regular one, to distinguish it from the ordinary stakes.

The curves adopted to a line of railway are arcs of circles of different radii; some have the same arc inverted, others there are having two or more arcs combined; all of which should have their tangent points accurately defined, that being the correct point of junction. Some curves are united by a short length of straight line between them.

In laying out curves there are numerous obstacles to contend against, and they require different methods to lay them out. Under all circumstances they should be very carefully plotted to a larger scale; it will afford great assistance to the surveyor in his operations, and enable him to determine on the best method to adopt.

WOODEN CURVES.

The wooden curves are usually made to inches and half and quarter inches up to a very large radius; amongst the many, there may be none to draw the curve required, which seldom happens; when that is the case, it must be remedied by geometric construction. The following rule may be applied generally.

Problem 22.

Fig. 7, Plate 29. It is required to find the radius to a curve, the tangent points B C being fixed, A B and C D the straight portion of the line.

Erect perpendiculars to the lines A B and CD from the

points B and C, their intersection at O will be the centre, and O B and O C the radius.

Should the curve be required to go through a fixed point at E, the tangent points must be extended to B' and C', and by the same rule, as Fig. 34, Plate 2, the centre of the arc will be found.

TO LAY OUT A RAILWAY CURVE.
Problem 23.

Fig. 1, Plate 31. The following method has been most extensively practised, requiring only the chain to form the curve; it requires to be executed with peculiar care, as an error committed in setting out one of the points will extend itself to every succeeding operation, and cause considerable deviation from the true curve.

The only calculation required is the first and second offset, therefore a table is unnecessary. This rule applies to all curves that are set out by this method.

Given A B and C D, portions of the main line; B and C the tangent points to be connected by the curve whose radius is 40 chains.

Rule. First reduce the radius and the distance into feet, then square the distance set off on the tangent line, and divide the quotient by twice the given radius, the product will be the first offset in feet and decimals, which multiplied by 12 will reduce it to inches and decimals. (See Table, No. 6.)

Then twice the first offset will give the second offset, and all succeeding offsets to the curve required.

As .825 x 2 = 1.650 × 12 = 19.80 inches, the second offset

Having ascertained the first and second offsets, set out one chain from B to a, in line with A B; then set off at right angles to A B the first offset a 1 equal to 9.90 inches, and fix a mark accurately on the point.

Then range out the line from B through 1 to b equal to one chain, and from b at right angles to B 1 b set off the second offset equal to 19.80 inches.

With the last dimension 19.80, set off from every chain produced in like manner every offset to the end of the curve.

The objection to the above method, by the numerous points hanging on to so delicate an operation, may be easily remedied, and be a perfect check on the whole, by adopting the same system upon a larger scale, taking at least four times the distance at each operation, and filling up the intermediate spaces afterwards.

In order, therefore, to arrive at perfect accuracy, it will be necessary in the first instance to calculate the lengths of the lines B F and B 3 F by the rule before given, which upon a larger scale would be considerable, whereas on the smaller scale it is but trifling, but will make considerable difference when repeated frequently.

Problem 24.

Fig. 2, Plate 31. To lay out a curve (with the chain only) by offsets from the tangent.

Range out the main lines A B and C D until the tangents intersect each other at E, and there fix a flag; let B and C be the two tangent points, and the radius 40 chains.

The first offset in this example is obtained by the same rule as in the former example, and all the offsets are set off one chain apart from the lines B E and CE at right angles as before; the succeeding offsets, 2, 3, 4, &c., are to be respectively the square, or 4, 9, 16 times the first offset; as, for example, the radius given is 40 chains:

The offsets on the tangent from E to C are precisely the

same, only reversed.