[TeX: $W = \frac{ 2240 \times bd^3}{ L^2}$]
2240 x 8 x 1000 W = ---------------, or reducing, W=55308 lbs. 324
[TeX: $W = \frac{ 2240 \times 8 \times 1000}{ 324} = 55308$]
Now, these proportions will give ample strength for both flexure and compression, for if we block the two sticks composing the end brace together, and firmly connect them by bolts, we shall have a built beam
2240 x 24 x 1000 of 24" x 10"--whence W = ---------------- = 165925 lbs., 324
[TeX: $W = \frac{ 2240 \times 24 \times 1000}{ 324} = 165925$]
and as 134400 lbs. was all that the conditions demand, we really have an excess of strength. The next set of braces supports the weight of the rectangle included between the upper ends of the braces and the two chords, and the dimensions of the sticks are calculated in the same manner. We find, as we approach the centre of the bridge, that the strains on the braces become less, and consequently their scantling should be reduced, but in ordinary practice this is seldom done.
=Rods.= The next thing is to ascertain the dimensions of the various tie rods. It is evident that the same weight comes upon the first set of rods, as on the first set of braces--which will give for the rods at one end of one truss, 134400 lbs.; and as there are two of these rods, each will sustain a strain of 67200 lbs.--and, at 15,000 lbs. per square inch, will have an area of 4.48 sq. inches, and, by Vose's Tables, must have a diameter of 2-1/2 inches. The sizes of the rods in each set will decrease towards the centre of the bridge as the weight becomes less.
[Illustration: Pl. II. with Fig. 1., Fig. 2., Fig. 3., Fig. 4.]
=Counterbraces.= Now, as to the necessity of Counterbracing, there are various opinions. The object of it is to stiffen the truss and check vibrations. If a load be placed over any panel point, it causes that portion of the truss to sink, and produces an elevation of the corresponding panel point at the other end of the truss--thus producing a distortion, which change of form is resisted by proper counterbraces. The strain to which this timber is subjected is caused by the moving load on one panel only--and requires only scantling of the size of the middle braces. These counterbraces should not be pinned or bolted to the braces where the cross--as their action is thereby entirely altered--but it is well to so confine them as to prevent vertical or lateral motion.