The load upon the central support is

65.4 +381 = 103.5 tons

The maximum bending moment occurs at B, viz. 118.6 foottons.

The beam is constructed of two Dorman rolled steel girders 12 inches deep, with flanges 6 inches × 1 inch, and webs & inch thick. The flanges are connected together on the top by means of a plate running the whole length. The web is stiffened by means of channel steel stiffeners. The working stress is 9 tons per square inch, the tensile strength of the steel being 32 tons per square inch.

In consequence of the change in the position of the points of contra-flexure during the passage of a rolling load, the shearing stresses cannot be accurately determined without considerable labour.

In the first example, when the uniform live load of 0.7 ton per foot run advances from one abutment and gradually covers the three spans, the point of contra-flexure in the first span moves through a distance of 135·8 124·3 = 11.5 feet.

If we draw the diagrams of shearing stresses for the extreme positions of the point of contra-flexure, we see that the nearer the point is to the pier the smaller the stress on the pier and the greater the stress on the abutment, so that if we use the point nearer the pier for determining the shearing stresses on the half of the effective span nearest the abutment, and the point farther from the pier for determining the shearing stresses for the remainder of the effective span and cantilever portions, we shall obtain stresses which do not differ much from the true maxima. We can deal with the central span in a similar manner. If the girder is of the lattice type, we first find the extreme positions of the points of contra-flexure, then apply the methods explained in Chapter VIII.

Concentrated rolling loads may be dealt with by calculating the bending moments and shearing stresses for various positions of the wheel loads in the manner illustrated in the foregoing example of the bressummer beam. The positions of the wheel loads which produce maxima stresses could, of course, be found if the points of contra-flexure are known; but this is not the most convenient method, excepting in the case of a traction engine with only two axles, as shown in Figs. 323 and 324.

In this case the points of contra-flexure for the dead load will not be moved very much with the traction engine in any position, so that, if we make two calculations for the extreme positions of the point of contra-flexure, we generally cover the maxima stresses in this case also.

Economy of the Continuous Girder.-In a three-span bridge of the best proportions, i.e. in which the effective spans are about equal in length under the loads which produce maxima stresses, or in which the side spans are about four-fifths of the central span, the economy of material due to continuity is about 50 per cent. for the dead load, and 16 per cent. for the live load, over three independent spans; hence the advantage of continuous girder road bridges where the dead load is considerable. This advantage will vanish if the piers settle unequally.

Mechanical fixing of the Points of Contra-flexure and Cantilever Bridges. We have seen that the points of contra-flexure are continually changing in position during the passage of the live load, and that in a bridge of three spans there will generally be four points of contra-flexure, viz. two in the central span and one in each of the side spans. The points of contraflexure may be fixed mechanically by uniting the girder at the proposed points by some form of hinged connection. The two points may be fixed either in the central or side spans, but not in both, or the stability of the bridge will be destroyed. The fixing of the points of contra-flexure for any span considerably simplifies the calculations for that span.

In the Kentucky bridge, United States, America, which consists of three equal spans, each 375 feet long, there is a hinge in each of the side spans, situated at a distance of 75 feet from the pier, thus reducing the effective length of the side spans to 300 feet. If the two hinges are located in the centre span, we have a form of the cantilever bridge in which the central span consists of two cantilevers projecting from the piers, with an independent girder resting upon their extremities and completing the span. The side spans in this case should be sufficiently long to balance the central span under all conditions of loading, otherwise the extremities must be anchored down.

The Ploughkeepsie bridge over the Hudson River, United States, America, consists of cantilever and rigid spans of almost equal length arranged alternately, so that there is no necessity for anchorages, as the rigid span balances the cantilevers on each side of it.

In the more usual form of cantilever bridge the cantilevers projecting from each side are generally of equal length, so that they balance each other for the dead load; the independent girder rests on the extremities of the cantilevers as before, while the extremities of the side cantilevers must be anchored down to balance the central portion under all conditions of loading.

The ratio of the length of the independent girder to that of the cantilevers varies in different bridges. Examples of this type occur in the bridges over the St. John's, Niagara, and the Frazer rivers, and also the Red Rock Cantilever Bridge, United States, America.

The celebrated bridge over the Forth consists of two cantilever spans, each 1700 feet long; the cantilevers project 680 feet on each side of the piers, with a maximum depth of 343 feet; the independent central girders are each 340 feet span. The Forth Bridge has been fully illustrated in the various engineering journals, and its detailed description will not be attempted here. It is in every respect the greatest constructional achievement in the world.

The cantilever bridge is one of the most economical types for long spans. The calculations may be made by first considering the independent girder, the reactions of which upon the extremities of the cantilevers must be combined with the panel loads of the cantilever.

It will be generally most convenient to consider each panel, proceeding from the extremity of the cantilevers step by step to the pier. Owing to the varying depth, the panel loads due to the weight of the panels are not equal.

Cantilever bridges possess the advantage of being easily erected by building outwards from the piers, and are especially applicable for long spans over deep gorges or rivers, where ordinary scaffolding would be too expensive or subject to great risks. Cantilever bridges should not generally be less than 500 feet span.

N

CHAPTER XII.

STRENGTH OF COLUMNS.

Short Columns.-Fig. 219 represents a rectangular prism subjected to compressive stress between the flat plates of a testingmachine. If the compressive force is applied along the axis of the prism, and if P denote the total load, A the area of the prism, and p the intensity of stress, then

The stress p1 is uniformly distributed over the area of the section.

If the load is applied at some point other than the centre of

The maximum compressive stress p2 due to the couple Py will occur at E, and will be

where a = BE, and I is the moment of inertia of the cross section of the prism.

Hence the maximum intensity of compressive stress at E is

Long Columns. In a long column the ratio of length to the least radius of gyration may be sufficient to cause the column to fail by lateral flexure rather than by direct crushing, in which case we have a compound stress somewhat similar to that existing in the short columns with eccentric loading. The tendency to lateral flexure in a long column may be increased by the eccentricity of the loading caused by the direction of the load. not coinciding with the true axis of the column. The strength of long columns has been investigated mathematically by Euler and Rankine, Professor R. Smith, Mr. Claxton Fidler, and others, and experimentally by Hodgkinson, Christie, and at the Watertown Arsenal, United States, America. Euler's Formula for Columns.-Euler's theory of the strength of columns may be stated as follows:Fig. 220 represents a column fixed in direction at one end only, which bends as shown in the figure.

If the column fails by direct crushing, then

Let p = radius of curvature, M = bending moment at xy, then

FIG. 220.

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