192

RAILWAY.

Although it does not properly fall within the present limits to describe the nume-rous varieties in the modes of construction of bridges to which the rapid extension ofrailways has given rise, it does appear desirable to state the effect which velocity hason bridges subject to deflection in increasing the strain due to a given weight.

Effects of Velocity on Bridges subject to Deflection. —If a weight be laid upon a beamsupported at each end by props, it will produce a deflection ; this deflection will hegreatest at or near the centre, but the greatest curvature will occur at the pointwhere the weight is suspended. If the weight be made to traverse the beam withrapidity, it may be conceived that an additional effect due to the motion of the bodyin a curve will be produced, and that the re-action of the beam w’ould be the resultantof this additional force, w T hich would act in the direction of the radius of curvatureand of the force of gravity. This re-action of the beam may be resolved into twoforces, one acting in the direction of the tangent to the beam, the other at rightangles to it; and this latter force would represent the tendency of the bar to break.It will, upon consideration, be evident that the additional effect above mentionedwould vary directly with the square of the velocity and the deflection, and inverselywith the length of the beam. The mathematical investigation of this question isextremely complicated. The following is a short sketch of the mode of solutionadopted by Professor Willis and Professor Stokes, and given at full length in theAppendix to the Report of the Commissioners for inquiring into the Application ofIron to Railway Structures.

It may be deduced, from what previous writers on the strength of materials havesaid, that if a weight be suspended from a beam (the inertia of the beam being neglected,and the weight considered as a heavy moving point), and if the deflection at the pointof suspension = y , and the distance of the point from the end of the bar = x, the half-length of the bar = a, and the weight suspended = W; then y = c W (2 ax — x 2 ) 2 ,when c is a constant quantity dependent upon the elasticity and transverse section ofthe bar. Now if M be the mass of the weight, S the central statical deflection, ordeflection produced by the weight when at rest, and R the re-action between the bodyand the bar (which, as the deflection is small, may be supposed to act vertically), theelastic re-action of the bar at any point (its inertia being neglected) is equal to theweight which, when suspended at that point, would depress it to the same distance

v 1

below the horizontal line. Therefore, R = W = — • —-r-„; and if x = l, or

c (2 ax —

half the length of the bar, y becomes equal to S; and if R = M g (when g represents

the force of gravity), c = :

Now if the weight be supposed to move with a

Mg. a*

velocity = V, the forces which act on the body are its gravity and the re-action of thebar. Whence is obtained the equation of motion,

dryd t 2

= 9 ~

ga 4

S

(2 ax — x 2 ) 2

which, since V

d x ,

——» becomesa t

d 2 y = g_ y _

dx 2 V 2 V 2 s'(2 ax-x 2 ) 2

The integration of this equation would give the form of the path of the body.But it cannot apparently he integrated in finite terms, except for an infinite numberof particular values of a certain constant involved in it. This constant has been

termed /), and $ = ^ j a small value therefore corresponds to a short bridge and