Let the moving load extend from A to any point D to the right of C. The two reactions R and Rʹ may be found by the methods already indicated. Let W represent the uniform load resting on the portion CD of the span. The shear S′ existing at C will be

Sʹ = Rʹ - W.(21)

Let R‴ be that part of Rʹ which is due to W, and Rʺ that part due to the load on AC. Evidently R‴ is less than W; then

Sʹ = Rʺ + R‴ - W.(22)

Since the negative quantity W is greater than the positive quantity R‴, S′ will have its greatest value when both W and R‴ are zero. Hence the greatest shear at the point C will exist when

Sʹ = Rʺ.(23)

Obviously the loading must extend at least from A to C in order that Rʺ may have its maximum value. Hence the greatest shear at any section will exist when the uniform load extends from the end of the span to that section, whatever may be the density of the load.

If the segment of the span covered by the moving load is greater than one half the span, the maximum shear is called the main shear; but if that segment is less than one half the span, the maximum shear is called the counter-shear. The reason for these two names will be apparent later in the discussion of bridge-trusses.

This rule for determining the maximum shear at any section of a beam is equally applicable to bridge-trusses under certain conditions, and has an important bearing upon the determination of the greatest stresses in some of the members of bridge-frames, although it has less importance now than it had in the earlier days of bridge-building.

85. Bending Moments and Shears for Cantilever Beams.—The case of a loaded overhanging beam or cantilever bracket, as shown in Fig. 16, is sometimes found. In that figure a single weight W is supposed to be applied at the end, while a uniform load w per unit of length extends over its length l. The bending moment at any point C distant x from the end will obviously be

The greatest value of the bending moment will be found by placing x equal to l in equation (24), and it will have the value

The shear at any point and at the end A respectively will be

S = W + wx and S₁ = W + wl.      (26)

The shear due to W is equal to itself and is constant throughout the whole length of the beam.

The second term of the second member of equation (24) is the equation of a parabola with its vertex at B, Fig. 16. Hence if AF be laid off equal to (wl²) /2, and if the parabola FHB be drawn, any vertical intercept, as HK, between that curve and AB will represent the bending moment at the corresponding point. On the other hand, the first term of the second member of equation (24) shows that the bending moment due to W varies directly as the distance from B. Hence if AG be laid off vertically downward from A equal to Wl to any convenient scale, then any intercept, as KL, between AB and BG will represent the bending moment due to W at the corresponding point of the beam.

86. Greatest Bending Moment with any System of Loading.—One of the most important positions of loading to be established either for simple beams or for bridge-trusses is that at which any given system of loading whatever is to be placed on any span so as to produce the maximum bending moment at any prescribed point in that span. In order to make the case perfectly general a system of arbitrary loads, like that shown in Fig. 17, is assumed and the system is supposed to be a moving one.

The separate loads are placed at fixed distances apart, indicated by the letters a, b, c, d, etc., W₁ being supposed to be at the head of the train, while W₏ is the last load having a variable distance x between it and the end of the span. In Fig. 17 this system of moving loads or train is supposed to pass over the span l from right to left. The problem is to determine the position of the loading, so that the bending moment at the section C of the beam or truss will be a maximum, the section C being at the distance lʹ from the left-hand end of the span. The complete analysis of this problem is comparatively simple and may readily be found, but it is not necessary for the accomplishment of the present purpose to give it here. In order to exhibit the formula which expresses the desired condition, let Wₙʹ be that weight which is really placed at C, but which is assumed to be an indefinitely short distance to the left of that point, for a reason which will presently be explained. The equation of condition or criterion sought will then be the following:

If the loads are so placed as to fulfil the condition expressed in equation (27), the bending moment at section C will be a maximum. If the variation in the train weights is very great, it is possible that there may be more than one position of the train which will satisfy that equation. It is necessary, therefore, frequently to try different positions of the loading by that criterion and then ascertain which of the resulting maximum moments is the greatest. It is not usually necessary to make more than one or two such trials. The application of the equation is therefore simple and involves but little labor.

It will usually happen that Wₙʹ in equation (27) is not to be taken as the whole of that weight, but only so much of it as may be necessary to satisfy the equation. This is simply assuming that any weight, W, may be considered as made up of two separate weights placed indefinitely near to each other, which is permissible.

After having found the position of loading which satisfies equation (27), the resulting maximum bending moment will take the following form:

In this equation x corresponds to the position of loading for maximum bending, while the sign (?) represents the distance between the concentrations W₍ₙʹ₋₁₎ and Wₙʹ. This equation has a very formidable appearance, but its composition is simple and it is constantly used in making computations for the design of railroad bridges. The loads W₁, W₂, W₃, etc., represent the actual weights on the driving-axles and other axles of locomotives, tenders, and cars, and the spacings a, b, c, etc., are the actual spacings found between those axles. In other words, these quantities are the actual weights and dimensions of the different portions of moving railroad trains.

The computations indicated by equation (28) are not made anew in every instance. Concentrated weights of typical locomotives, tenders, and cars are prescribed by different railroad companies for their different classes of trains, ranging from the heaviest freight traffic to the lightest passenger train. A tabulation is then made from equation (28) for each such typical train, and it is used as frequently as is necessary to design a bridge to carry the prescribed traffic. The tabulations thus made are never changed for a given or prescribed loading.

87. Applications to Rolled Beams.—It is to be remembered that these last observations do not limit the use of equations (27) and (28) to railroad-bridge trusses only; they are equally applicable to solid and rolled beams and are frequently used in connection with their design. Great quantities of these beams and various rolled steel shapes are used in the construction of large modern city buildings, as well as in railroad and highway bridge structures. The steel frames of the great office buildings, so many of which are seen in New York and Chicago as well as in other cities, which carry the entire weight of the building, are formed wholly of these steel shapes. The so-called handbooks published by steel-producing companies exhibit the various shapes rolled in each mill. These books also give in tabular statements many numerical values of the moment of inertia, the section modulus, and other elements of all these sections, so that the formulæ which have been established in the preceding pages may be applied in practical work with great convenience and little labor. Tables are also given showing the sizes of rolled beams required to sustain the loads named in them. Such tables are formed for practical use, so that, knowing the distance apart of the beams, their span, and the load per square foot which they carry, the required size of beam may be selected without even computation. Such labor-saving tables are quite common at the present time, and they reduce greatly the labor of numerical computations.