This bridge, executed after the designs of Mr. G. W. Buck, is remarkable for its acute angle, which is 241°. The weight of the iron work employed on the six ribs was 540 tons, and the whole was admirably screwed together; the remainder of the viaduct is formed with brick arches of 45 feet span.

DIMENSIONS OF VIADUCTS IN VARIOUS RAILWAYS.

In the Grand Junction Railway, at Vale Royal, is a viaduct of stone over the river Weaver, in which are five arches, each 63 feet in span and 60 in height; the length of the viaduct is 456 feet. Where this railway crosses the Mersey and Irwell canal there is a viaduct of stone having 12 arches, the two in the centre are 75 feet in span, and the remainder from 40 feet to 121.

In the Newcastle and Carlisle Railway, near Brampton, is a viaduct, which crosses the public road and the river Gelt, at the height of 80 feet above the bed of the river, over which it is carried in an oblique direction. The arches, which are three in number, are 33 feet in direct span, and are built at an angle of 45°.

In the Birmingham and Derby Junction Line, between Kingsbury and Tamworth, over the Anker, is a viaduct of 18 arches, each of 30 feet span, and one oblique arch of 60 feet span; its height above the river is 23 feet, and the cost was £18,000. In the same line, between Tamworth and Burton-on-Trent, is a viaduct mile in length, built upon above 1000 piles, driven 15 feet below the bed of the river.

In the Newcastle and Shields Railway there is a viaduct over the Ouseburn of 9 arches, two of which at the ends are of stone, and the others are of timber, resting on stone piers; the three central arches have each a span of 116 feet, and two others 110; the total length of the viaduct is 750 feet, and its height above the water 180 feet. In the same line, the viaduct at Willingdon Dean has 7 timber arches, five of which span 120 feet each, and the two exterior each 115 feet; the whole length is 1050 feet, and the height 82 feet.

In the Taff Vale Railway, near Quaker's Yard, is a viaduct crossing the Taff, the length of which is 600 feet, and the height above the river 100 feet, having 6 arches. In the same line, at the conflux of the Rhondda and Taff, is a viaduct having an arch of 100 feet span, and 60 feet in height.

In the London and Brighton Railway, across the valley of the Ouse, is a viaduct 1437 feet in length, and the height varies from 40 to 96 feet; it is formed of 37 brick arches of 30 feet span.

In the Stockton and Darlington Railway, where one of its branches crosses the river Tees at Stockton, is a suspension-bridge, 240 feet in length between the piers, and 30 feet above low water mark.

NOTE.

It was on this railway that the first locomotive steam-engines were employed, which excited considerable interest, and led to the great changes that have since taken place in the transit of passengers, and to the establishment of the numerous lines existing and in progress at the present time. Part of this railway was laid out above a quarter of a century ago, which drew the author's attention to preparing his system of laying out railway curves, which is given in Section III. of this part of the work.

The London and Greenwich Railway is a continuous viaduct of more than 1000 brick arches, each 18 feet span, 22 feet in height, and 25 feet in width. It is 33 miles in length, and cost £266,322 per mile.

The extension of the South-Western Railway through the metropolis, from the Nine Elms to Waterloo Bridge, is a continuous viaduct, like the last-mentioned one; in which are now being constructed several strong and elegant oblique iron arches for the purpose of crossing some of the principal streets.

I shall conclude this section by recommending to those who wish for scientific and practical information of the first order on this subject, The Theory, Practice, and Architecture of Bridges, by J. Hann, C.E., of King's College, R. Stevenson, C. E., and others; Practical and Theoretical Essay on Oblique Bridges, by G. W. Buck, M. Inst. C.E.; Treatise on the Equilibrium of Arches, by Joseph Gwilt, Architect, F.S.A.; Cresy's Encyclopædia; and a Practical Treatise on the Construction of Oblique Arches, by J. Hart.

SUPERELEVATION

SECTION VII.

OF EXTERIOR RAIL IN CURVES.

THE superelevation of the exterior rail in curves, the radii of which are within certain limits, is absolutely necessary to counteract the centrifugal force caused by the velocity of the train, since all moving bodies have a tendency to continue their motion in a direct line. From this cause the carriages of a railway train are driven towards the exterior rail, and would finally be thrown off the rails, were it not for the conical inclination of the tire and the flanges of the wheels.

Let W = weight of the moving body or train, V = its velocity per second, R = radius of the curve, and g force of gravity at the earth's surface; then, per Dynamics, the centrifugal force

that is, the centrifugal force that urges the moving body to leave the curve, in this case, is of its weight.

22

This force is in most cases counteracted by the conical inclination of the tire of the wheels, each pair of which are firmly fixed on the axle that turns with them, the inclination of the tire being usually about an inch in the whole breadth of the wheel, which is 31

inches, or about 1 in 7. This inclination of the tire together with the lateral play of the flanges of an inch on each side, and the centrifugal force impelling the carriages of the train, when moving in a curve, towards the exterior rail, enlarge the diameter of the exterior wheel, and diminish that of the interior, thus causing the train to roll on conical surfaces, which necessarily produces a centripetal force, the centre of which force is the vertex of the cone, of which the increased and diminished diameters of each pair of wheels are sections.

Let d be the outer diameter of the wheels, & the increment and consequently the decrement that the diameters of the exterior and interior wheels respectively receive, through the joint action of the centrifugal force and the inclination of the tire: then under these circumstances the respective diameters of the exterior and interior wheels will be

d+8 and d-d;

also, if R' be the radius of a circle which the centre of the carriage would describe in consequence of the inclination of the tire of the wheels, and b the breadth of the road or gauge then R' + 16 and R'b, are the radii which would be respectively described by the exterior and interior wheels; and by similar triangles

= inclination of the tire, and ▲ the deviation of the

Also W and V representing the weight and velocity of the train, as in (1.), and g the force of gravity, the centripetal force corresponding to the radius R' will be

W V2
9 R

(3.)

Since the forces ƒ and f, (1.) and (3.), act in contrary directions, they will hold each other in equilibrium when they become equal, and the train will cease to have a tendency to leave the curve; this takes place when

which is the deviation required to produce the equilibrium between the centripetal and centrifugal forces of the train. Therefore since R = R', i. e. the vertex of the imaginary cone of which each pair of wheels are sections will coincide with the centre of the curve, and there will in consequence be no dragging of either of the wheels on the rail.

3 feet, b = 4 ft. 81⁄2 in. = 4.7 feet = breadth of the

, and ▲ = an inch, there will result for the

least possible radius of curvature, in which the two forces balance each other, supposing the two rails to be exactly level,

R =

b d n 4 A

= 4.7 × 3 × 7÷4 × 24592 feet.

(6.)

But as there might occur an accidental depression of the exterior rail, which would cause the flange of the wheel to rub the rail on that side, it is thought advisable, for the sake of greater safety, to limit the value of R' to not less than 1000 or 1500 feet, or what amounts to the same thing, to take the lateral deviation of the train at about or of that taken in (6.), or at about or of an inch, thus producing a less disagreeable displacement of the carriages of the train.

From what has been already shown it will at once appear that in curves of less than 1000, or 1500 feet radius, a superelevation of the exterior rail will be necessary to counteract the excess of the centrifugal force above the centripetal force.

Let the superelevation of the exterior rail be denoted by x, and since b expresses the width of the way, the plane on which the train moves will be inclined% to radius = 1, and therefore the gravity of

the train will draw it to the interior rail with the force

This force together with the centripetal force, due to the deviation of the carriages of the train on the rails, must hold the centrifugal force in equilibrium: we shall therefore have from equations (1.), (3.), and (7.)

which is the Formula for the superelevation of the exterior rail. In which if R = 900 feet, R' = 1000 feet, b = 4.7 feet, g = 321/ feet, and V = maximum velocity = 60 miles per hour 88 feet per

This highly important Formula is due to the Comte de Pambour; see Chap. XVIII. Sect. III. page 534. of his valuable work on Locomotive Engines; to which I would refer those who wish for extensive information on these subjects; also to Hann on the SteamEngine, with practical Rules for the use of Engineers, and to Tate's Exercises on Mechanics and Theory of the Steam-Engine, &c., which two last works cannot be too highly recommended on account of their important bearing on a subject of real practical utility.

I would also recommend the following Tables.

Dr. Hutton's Mathematical Tables, containing Logarithms of Numbers, Natural and Logarithmic Sines, Tangents, &c.

Professor Barlow's Mathematical Tables, containing the Factors, Squares, Cubes, Square Roots, Cube Roots, and Reciprocals of all Numbers, from 1 to 10,000; with several other useful Tables; well calculated to afford assistance in various arithmetical and mathematical investigations.

A small work by Mr. R. Farley, containing Logarithms of Numbers from 1 to 10,000; and of Sines and Tangents to every minute of the Quadrant.