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raising 1 in 264, that would be sufficient to draw the same load along a horizontal plane, the velocities on both planes being the same.
Mr. Tredgold made many calculations on the expense of ascending inclined planes. comparing them with cost of cutting them to a level, and the result made him be of opinion it was better follow the undulations of the surface.
When becomes necessary to conduct a line over a considerable elevation, it becomes a matter of great importance to determine whether it is better to distribute the rise and fall equally along the line, or concentrate them in a few steep planes, which are to be ascended by means of additional power.
It is the opinion, that a certain degree of similarity in the gradients is very necessary. Should the road present an inclination so steep, that the ordinary power, by a reduction of speed, could not surmount it; then additional power must be employed, or the engine must run over other parts of the road without its greatest load; by which an unnecessary expense is incurred. But as long as this is avoided, it is the opinion of eminent engineers, that the degree of inclination on a railway, with equal traffic in both directions, is of little consequence. The amount of power lost in ascending being somewhat equal to what is gained in descending, brings the entire amount of power required to pass over, in both directions, the whole distance, equal to what is required to pass over an equal distance on a level road.
Dr. Lardner brought this under the test of experiments. The engine, and 12 carriages employed, making a gross weight of 80 tons, was run from Liverpool to Birming ham, and back, under as nearly as possible the same circumstances, distance 95 miles. The time passing each quarter of a mile carefully observed. The following table
exhibits the results of his experiments on gradients from a level to 1 in 177, nearly 30 feet per mile.
By comparing the number in the table, we see that although the gradient is 1 in 177, diminished the speed from 30.93 miles per hour, which is the velocity on a level, to 22.25 miles per hour, the ascent, the deficiency was more than compensated for in descending the same plane, the velocity per hour in the descent being 41.32 miles, making the mean velocity 31.78, which exceeds the speed on the level by 85 part of a mile per hour. Comparing the mean velocities per hour in both directions on the different gradients, with the speed per hour on the perfect level, we learn the fact, that a railway, with gradients from 20 to 30 feet per mile, may be worked in both directions by the same expenditure of power as a dead level. But when the traffic is unequal, that is, when heavy goods are to be transmitted from one place to another, while light goods only return; in this case there is no doubt but that the heavy goods or loads ought to descend along the line. Therefore the most level line may not best answer the purpose in this case.
To determine the amount of power requisite to draw a given train over a given railway. M. Navier takes the
following method. He takes the friction per ton 11bb, therefore Then he says, "we conclude from this, that in order to transport, with any velocity whatever, constant or variable, a weight W, to a distance a, on horizontal line, it is necessary to employ the power represent bya; that is, the power necessary to raise the weight to the height.'
In forming the excavations or embankments, it is very important to approximate as near as possible to the inclination at which the ground would naturally stand without slipping; for if the slope be greater than necessary, it will increase the labour, and take up more land than is required The proper slope for each particular soil can best be found from observation. In cutting through a rock it might be perpendicular; in cutting through a mixture of clay and sand, compact clay, and compact stony sol, the slopes will vary, not only with the soil, but with the height so. A slope of 1 to 1, that is of 45° is found to answer for ordinary earth; for clay, 1 to 1, or 33°.41'; in other cases it may require to 1: some clay would slip at 3 to 1. When stone is convenient, it would be well to build a small wall of dry stone at the foot of the slopes. In some cases, the wall that supplies the slopes of excavation should be carried up to the natural surface of the ground; and the wall that supplies the embankment should be built to the surface of the 10adway. In no case should it be neglected to protect the slopes by sods or grass seed.
To determine what angle the natural surface assumes when left to itself, which is the only best possible criterion to judge of the proper slope, there is a small instrument called a batter level, or clinometer, constructed for the purpose. (Fig. B.) This instrument consists of a quadrant CD, attached to a flat bar AB, about six inches long. The quadrant is graduated into degrees and half degrees from D to C. The index Q turns upon the centre
of the quadrant and carries a rit-level by which the in
dex may be set horizontal; at p, is a hing-joint, by which means the bar AB may be folded. To use this little instrument, open the hing-joint, and rest the edge of the bar on the surface of the slope; then move the index Q, round its centre until the bubble is in the centre of its tube, then the angle pointed out by the index on the arc, will be the measure of the inclination. It is evident the longer the bar DB, the more accurate the angle.
Width of ground required for the construction of a Railway.
The width generally allowed for a double line of railway is 14 or 15 yards, allowing 6 feet on each side for drainage and fencing. The quantity of land required for one mile, allowing 15 yards, is 5.455 acres; and if you allow the perpendicular cutting with its proper slope and find the additional area for the sides for one mile, and if multiplied by the number of miles in the line, will give an average area for the whole line. But it is evident this is only an approximation, as you can only use an average
depth of cuttings and height of embankments.
When the natural surface of the ground, both on the length and breadth, is upon the same level as that of the intended works, it is extremely simple to set a stake out the width. Suppose we take the base 36 feet, when prepared for the ballasting; the rates of the batter or slope to the heights, both in cuttings and fillings, to be 2 1; at the outward edge, a piece of land 12 feet for the fences and drains. The centre line is first to be staked out and correctly levelled.
It would be well to drive stakes into the ground at the end of each chain's length, and if their tops be on a level with the surface they will afford stations for the levelling staff to be held upon.
1. Now if the ground at any stake is upon the same level with the base of the intended railway, you have no more to do than to measure at right angles to the centre line half the breadth of the railway, suppose 18 feet, and 12 feet for fences, &c., that is 30 feet on each side of the centre. But if the centre line be a curve, the cross line must be at right angles to a tangent at that point. 2. Case. When the ground is not on the proposed level of the road, but the cross-section horizontal, having the cutting or embankment what it may.
Rule-Multiply the cuttings or filling by the ratio of the slope, to the double of this add the breadth of the railway, the sum will be the breadth of the cross section on the natural surface of the land. The method for finding the width for an embankment when the cross section of the ground is horizontal, is the same as for cutting; it is only inverting the figure under the same circumstances.
In laying out a rail-road, the direct course is often interrupted by a steep hill, a town, &c.; and consequently a knowledge of the method of correctly laying down these curves upon the ground which will connect the straight portions of the line becomes very necessary.
Previously, however, to proceeding to the practical part, it may be desirable to remark that the straight portions of the line are always to be connected at the points of change of direction, by an arc of a circle, to which these straight portions are to be tangents.
Let A D and BC, (Fig. 24,) be two portions of a straight road, whose extremities are to be joined or connected by an arc of a circle, to which both are to be tangents. Having surveyed from A to D, set up the theodolite at the point D, and take the angle ADC: then measure the distance DC; and plot the figure, that is, draw A D of