Thursday, 10th October.

P. A. PETERSON, Vice-President, in the Chair.

Paper No. 32.

BRIDGE CALCULATIONS.

BY H. E. VAUTELET, M.CAN. Soc. C. E.

The opinion of many well known authorities is that it would be preferable to use a distributed load, that would be safe for all existing types of locomotives in use on railways, and that would leave a margin for the probable increase of weight in the future. The wheel base of a locomotive as well as the weight on each axle is limited by the radii of the curves and the section of the rail; and although the weight of cars is constantly increasing, there exists a necessary relation between the engine and train weights. In general practice, the train weight is considered as distributed and the engine weight as concentrated. The author thinks that the weight of a wheel is always distributed by the rail and ties (more so in locomotives than in cars, owing to the lesser distance between the wheels), and that both weights should be considered as distributed. It will always be necessary to use two different distributed loads, and the equivalent distributed load will vary with the length of span.

Furthermore the stresses, although calculated with the greatest care, are not the actual stresses in a bridge, and frequently discrepancies, amounting to several thousand pounds, are shown during the erection. The general practice is to have the posts fastened by pins or rivets, allowing them to work as tension members, while the top chord is rigid, instead of having articulations at every panel point. It follows that the strains are not what they would be in an articulated system, where the posts could only take compression, the differences being more especially apparent in bridges with inclined top chords, which act partially as an arc, the posts acting as suspenders. Another cause of error is the use of stringers, rivetted to the floor beams, which act as parts of the top or bottom chords, as the case may be.

A

It must not be supposed that the author is advocating free articulations with bolted stringers and slotted holes, as he believes that the actual practice increases the solidity of the bridges, and prefers to have stiffness in his work, at the cost of some uncertainty in his calculations. He would also follow in the lead of an eminent bridge engineer in the United States, whose trusses are rather light, a large quantity of material being used in the stiffening of the bridges, in the top and bottom laterals, sway bracing, and especially in the portals, the last of which is certainly a very important part of a bridge. Most of the actual specifications seem to be made for the perusal of outsiders more than for actual use, and it seems (to give one instance) that the rivetting foreman and inspector should know, without being told, what the appearance of a rivet must be after it is driven.

In treatises on bridges, written by French authors, it is always said that we must not rely too much on calculations, and the best that can be said about their rules is that bridges built according to them and with a large factor of safety have withstood the test of time.

Experience would appear to show that, usually, the longer the specifications the worse are the bridges. What may be considered to be a standard bridge in the United States is built with a two page specification.

This paper sets forth solutions of the two following problems, which the author believes to be new :—

To find the maximum bending moments and shearing stresses in girders or trusses:-1° taking into account a distributed engine load, followed and preceded by a distributed train load, and 2° taking into account the load on every wheel. A simplification in the calculation of bending moments, in continuous bridges of two spans, will also be referred to.

1° Calculation of the maximum bending moment with a distributed load P, occupying a length V, preceded and followed by a distributed. load p.

Let y be the bending moment at C

y = -x2

(3 - 1) + x (P − p) (1 − ĭ) a − (P − p) [VI - V ]

L2 L

If instead of two weights P and p a weight R is distributed,

The weights P and p can then be replaced by a weight R such that

*This formula has been used, if the writer is not mistaken, by the Key

stone Bridge Co.

For a uniformly distributed load the max, shearing force

Hence, that these two shearing forces may be equal,

=

n2RI 2(N-1)

we have a close approximation on the safe side. The weights P and p can be replaced without material error by one weight R; V being the length occupied by the distributed load P, and n the number of panels which must be fully loaded to give the maximum stress.

2° Graphic calculation of bending moments, taking every wheel into

account.

Fig 3

Aa

N

MM, B
-b-

If a weight P is moving along AB, the bending moment at the point of application will be:

Having drawn this parabola, the bending moment at a point D at a distance a from A will be:

απά

and CD being an ordinate of the parabola for an abscissa x=a

The bending moment at D is then equal to the ordinate of the triangle ACB at the point of application of the weight P'. For another load P1 we should have another triangle AC, B1, and ifb is the distance between the two weights, MN+M,N, will represent the bending moment at D produced by the weights P and P1.

1

By sliding the triangle ACB a distance b to the left, M, comes to M, and the bending moment at D for any position of the two weights is given by the ordinates MN + MN, of the two triangles.

The same reasoning will apply to any number of weights.

To apply the method it is sufficient to draw the parabola

To find the moments at Do draw a series of triangles ABC, A,B,C1 A,B,C, etc., so that

AA, A,A2,...being the distances between the weights...