the member [IX] (disregarding extras in connections), and the same is true of thrust members of similar formed cross-sections, sustaining stress proportional to the square of the length of pieces respectively.

(6). The respective stresses of two thrust members, divided by the squares of respective lengths, give quotients indicative of, though not proportional to, the re lative efficiency of material in the two members, — the greater quotient showing the greater efficiency, or greater power of resistance to the square inch of crosssection [XIII].

With these rules or principles in view, we may proceed advantageously with general analyses and comparisons of different plans, or systems of bridge trussing, adapted to different lengths of span.

THREE PANEL TRUSSES.

XVII. In structures exceeding 25 or 30 feet in length, the length of joists from the centre to the ends, would require cross-sections so great, to give them the requisite stiffness, that their weight and cost would become objectionable. It becomes expedient, then, in such cases, to provide support for more than one principal point, or transverse beam, or bearer. A superstructure from 30 to 40 feet long, may be constructed with two cross beams, supported by two trusses with two pairs of braces each, with the feet connected by a horizontal tie or chord, as seen in Fig. 6.

The cross beams, may be at b b', or suspended at c and d, at equal horizontal distances from a a', and from one another; which latter position they will be re

garded as occupying in this instance. Or, the figure may be inverted, thus reversing the action of the several thrust and tension members.

XVIII. Another, and a more common form of truss for two beams, is shown in Fig. 7. These may be called three panel trusses.

To compare these two trusses, suppose the two to have the same length and depth, and to be loaded with uniform weights, w, at the two points c and d in each. Then, since we know from the principle of the lever, that each weight produces upon each abutment, a pressure inversely as its horizontal distance from them respectively, and that the pressure upon the two abutments is equal to the weight producing it, it follows that ab, Fig. 6, sustains 3w, and compression. Hence making ab = — D,... be➡v, and ach, ... w, becomes Zw, and multiplying this stress by length,

= D,

and

changing w to M*, we have for material in ab,... M. But

2h

D2=h2 + v2, whence 3D = 3 (1⁄2 +v) x = (372 + 2) M.

M

30

Again, ab' sustains w, with length = ✔4h2+v2, and by multiplying and changing as in case of ab, we obtain material in ab', =(12+), which added to amount for ab, gives(6+ v) for the two braces, and (4h+2v)M for the four.

=

The horizontal thrust of abw while that of ab'

w2w. Hence the horizontal thrust of ab and

=

ab=w=tension of aa', and material for chord aaʼ,

4h

equals 3x=4M. Tension of be and b'd, each, equals w, and material for the two-2 v M, which added to amount in aa', makes the whole tension

material equal to (4+2) M, being the same co-efficient

of M as was obtained for compression.

In truss Fig. 7, ... ab and a'b' (= D =✔h3+v3), evidently sustain each a weight equal to w, and a stress ✔k2+v2w. Whence, material — (3+v) м for each,

v

and (213 + 2v) M for both, while bb', equal to h, sustains compression equal to the horizontal thrust of ab, equal to ow, and requires material equal to M, making, with amount in braces ab a'b', (3+2v) M.

v

Now we have just seen that the horizontal thrust of ab, equal to the tension of chord aa', equals Zw, and the

* When is used in the co-efficient of M, then M represents the pro duct of the stress, in terms of w, by length according to any assumed unit, which may be equal to v or not.

length being 3h, the material consequently equals 3h M, to which add 2vм for verticals, and we have

(312 + 2 v)м = whole amount of tension material in

M

truss 7, which is less by M, than in the case of truss Fig. 6.

XIX. With regard to thrust material, it is clear that while in truss 6, the weight on either pair of braces, is transferred in due proportion to both abutments, independently of the other braces, whether one or both pair be loaded; on the contrary, truss 7, when c only is loaded, must transfer w to a', which can only be done through a'b', the only oblique member acting at the point a'. Moreover, the weight must be communicated to a'b', at the point b', through the thrust of bd, and the tension of db', assuming bd and cb' to be thrust members. Now, as either c, or d, is liable to be loaded with weight equal to w, while the other is unloaded, it follows that both bd and cb' are liable to sustain weight equal to fw, and require thrust material equal to (22 + v) μ for each,

2h

or (2 + 2) for the two. The whole amount of

30

thrust material for truss 7, then, equals(34 + 2 v) M,

(the amount found above) + (3 + 3v), equal to

20

(331o + 2 v) м, against (4+2v) for truss 6; the 233 difference being ( — v) M. If this be a positive quantity, the balance is in favor of truss 7, and if negative, in favor of tress 6, as regards amount of action on thrust material; while, if th — v ➡ zero, the

อ

=

amount of thrust action is the same in both trusses.

Either of these suppositions may be true, according to the relative values of h and v. If h = v√2, then th

= 0.

v

ૐ vo. If h, be greater than v✔2... 14-v is 1 zv positive, and if h be less than v2, the value is nega tive. But the amount is trifling in any probable rela tion of h and v, and may be disregarded in this general comparison.

Calling then, the amount of action upon thrust material in the two plans equal, there is a probable advantage in favor of truss 7, as to efficiency of thrust material, while the latter truss, shows a positive advantage over truss 6 in amount of tension material, equal to ( (4+2)-(3+2v) m = = 12 M. This is equal to

อ

4м, when h = 2v2; which is in tolerable proportion for the trusses under discussion, and, substituting these values of h and v in the expressions of tension material in the trusses respectively, we have (16 + 2)m for truss 6, and (12 + 2)м for truss 7, being about 28 per cent more for 6 than for 7.

The same difference would appear with Fig. 6 inverted, the thrust and tension action being the same in amount of each, only sustained by different members, thrust members in one case becoming tension members in the other.

FIVE PANEL TRUSSES.

XX. Truss Fig. 7, may be increased in length and number of panels, by introducing additional panels between the end triangular panels, and the rectangular centre one either of an oblique form, as in Fig. 8, which represents an arch-truss, or of a rectangular form as in