The spacing of stiffeners is more a matter of experimental determination and judgment than of mere calculation - the fact which our foregoing discussion of the web stresses might suggest to any one. Mr. Jos. M. Wilson* has given a safe rule, which says that "stiffeners in girders over 3 feet in depth shall be placed at distances apart (center to, center) generally not exceeding the depth of the full web plate with the maximum limit of 5 feet. In girders under 3 feet depth they may be 3 feet apart, and in some special cases, where there is little or no shearing stress, at greater distances." This refers, of course, to those parts of the girder where there are no concentrated loadings, as in floor beams supporting stringers, or as in through plate-girders with floor beams, in which cases stiffeners are invariably to be riveted at these points of connections, as shown in Fig. 4, pl. I.

* Compare Penna. B. R. Co. Bridge Specification.

There are several other rules for spacing stiffeners given by different engineers.*

Fig. 7.

There is still another point to be considered in the strength of webs, those of girders in which the maximum bending moment and great shearing stress exist at.

*As one of those worthy of notice, Mr. O. Ghaunte, in the N. Y., Lake Erie & Western K. R. Specification, says that when the least thickness of web is less than 1-80 of the depth of the girder the web shall be stiffened at intervals not exceeding twice the depth of the girder.

The weight of girders can be best determined by the use of a simple empirical formula:

W=10 (s3+a)*

In which s = span in feet (effective length), and W= weight in pounds of entire iron work of a single track deck plate-girder when s is between 20 and 80 feet, and a is the constant depending on the kind of moving load, and the specifications according to which the parts are calculated. (See the appendix.) For the loading shown on next page the value of a in the formula is to be taken at 330.

The approximate weight of a through girder, provided with floor beams and stringers is obtained as follows:

W'=W + 300S in which W' =required weight.

W=wt. of deck span of the same effective length. S=effec. length as before.

* A formula by Mr. Pegram, giving somewhat more accurate results will be found in the Trans. A. S. C. *., 1886. The formula here given, however, being much simpler, is easily remembered and worked.

Some of the results obtained by the use of these formulas are given in the appendix.

As to the rolling load, the diagram given on page 30, showing a load system consisting of two 80 ton consolidation engines, coupled and followed by a uniform load of 3,000 lbs. per lineal foot, represents about the heaviest rolling load system in the United States, although these figures have in few cases been exceeded.

Some passenger engines may for small spans give more unfavorable loading than this, and every case must be investigated for itself. A bridge engineer should have before him all the possible rolling loads of the road for which the bridge is to be designed.

The bending moment due to uniformly distributed loading is at once obtained by the well-known formula of simple beams:

M=« w x (l - x)------------(12)

in which M is the bending moment at any point distant x from one end of the beam; w, the uniform load per unit of length, and I the length of span.

Fig. 9.

The word span will throughout be used to denote the effective length on the distance between centers of bearing3.

For the bending moment at the center (x=« l) equation (12) becomes:

M=⅛ w l2------------(13)

The graphical representation of the bending moment due to a uniform load is a parabola with its apex at the middle of the span.

The maximum bending moment due to the rolling load can be determined either analytically or graphically, the latter method being, however, far more preferable, on account of the rapidity with which the work can be performed. With the analytical method one has to solve the equation of bending moment at different sections of the beam under the position of the moving load giving the maximum moment. Thus, with loading like Fig. 10, we obtain for the reaction "R" at one end:

R=(w"d"+w"d" +w'd') / l.

Fig. 10.

And consequently for the bending moment at any section, distant x from one end, we get:

Repetitions of such a work require, even when accurate results are not needed, considerable labor.

On the other hand, with the graphical method one has merely to construct an equilibrium polygon, and by placing the given span in different positions under the rolling load (which amounts to the same thing as letting the rolling load pass from one end of the girder to the other), the maximum moments at different sections are more readily obtained.

We know from statics that in a simple girder with loads distributed over it either uniformly or arbitrarily, the maximum bending moment at any section is found when the loads on each side of this point are to each other as the parts into which the point divides the span. But for a single load traveling over the girder, it is evident that the maximum moment at any point is found when the load lies at that point. Consequently the maximum bending moment at any point of the girder will be found when one load lies at the point, and the rest are so located as to fulfill the condition already mentioned or expressed in the formula:

Moment will be maximum at any point x (Fig 11) when:

To find the load wx, which when resting on a section, gives the. maximum moment for that point, requires some work, as some trials must be made for every load and point; but it is usually sufficient for all practical purposes, for short spans, to place the second driver of the second locomotive over the section at which we want to determine the moment, and obtain the same by the method to be mentioned in the following pages. Of this one can be easily convinced by a few trials.