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pressed more than the point j, by the extension of diagonals, by as much as the square of gl exceeds the square of il, or as 8 to 5; assuming diagonals to incline at 45°. The panel gm, must therefore be oblique, and the distance gm, greater than ni. Again, the point f suffering the same depression from the extension of fm, as the point g suffers from that of gl, and a still further depression from the compression of mi, and the extension of il, it follows that the panel fn must also be oblique, and the distance fn, greater than the distance og. Now, the obliquity of both of the panels gm and fn, manifestly contributes to the excess of distance fm, On the contrary, the centre panel eo has no obliquity due to extension of diagonals, or compression of uprights; since there is no cause for obliquity in one direction, more than the other. It seems to follow that en, crossing one oblique panel, must undergo extension; but not so much as fm, which crosses two.

over oi.

Now, fm and en being equal in length, the weight sustained by each, is manifestly as the cross-section and extension combined; and as the former, fm, should be the larger in the ratio of 9 to 6, or as their maximum stresses; if we allow their extensions to be as 2 to 1, the greater for fm, the relative weights sustained would be as 18 to 6, or as 6 to 2. Our decussation theory gave their relative stresses as 7w" to 2w". This is not a wide discrepancy, seeing that the above computation is based in part upon a mere approximate data.

We may conclude then, that in cases like the one under consideration, decussation does actually take place. Still it obviously depends upon conditions which are not of the most determinate character. For, if en and fq, be relaxed or removed, under a full load of the

truss, decussation can not take place, for the same obliquity of the two panels next to the centre one, whicǹ produces the tendency toward tension of en and fq, on the contrary, tends to relax do and gp, through which latter alone decussation could take place, in the absence of the former.

On the other hand, if en and fq be sufficiently strong, they may be strained to such a pitch as to bear all the weights at e and ƒ, and leave fm and er entirely inactive. Hence, there is an uncertainty as to the action of these diagonals, which may be best obviated by estimating stresses upon both theories, and taking the highest estimates; as recommended with reference to trusses without verticals, and as previously suggested with reference to the case in hand.

In view of preceding facts and principles, it may be advisable to avoid the odd panel in trusses with verticals, when practicable without incurring more import ant disadvantages in other respects.

DECK BRIDGES.

LXI. Are those having the movable load applied at the nodes of the upper, instead of the lower chord, as generally assumed in preceding analyses.

It will readily be seen, on a brief contemplation of Figures 12 and 13, for instance, that weights applied at the upper chord, act directly upon compression mernbera, either erect or oblique, as the case may be ; and are thence transferred to tension members at the lower chord; according to the general principle, that weight applied at the upper end of a member, always acts by compression, and that which is applied at the lower end, by tension.

. In the case of truss Fig. 12, the action of tension di agonals is precisely the same, whether the weight be applied at the upper or lower chord. But the compres sion verticals, in the deck bridge, sustain as their maximum, the weights indicated by the figures immediately above them respectively, from the centre toward the right hand; and these weights, of course, are equal to those acting upon the diagonals respectively meeting the verticals at the lower chord; and consequently, greater than when the weight is applied at the lower chord. For illustration, in Fig. 12, as the truss of a deck bridge, the vertical fk sustains 15w", the same as fj, whereas, in the case of a "Through bridge" (with load applied at the lower chord), fk sustains only 10w" communicated to it through ek.

In the deck bridge also, the tension vertice's he and jg are essentially inactive, merely sustaining a small portion of the lower chord. The chords suffers the same stress in both through, and deck bridges.

LXII. Load applied at the upper chord of truss Fig. 13, acts by thrust directly upon the diagonals meeting at the upper chord, and the maximum weight (from movable load), sustained by diagonals meeting at one of the upper nodes, are indicated by the two figures immediately over the node; the larger figure referring to the diagonal running toward the nearer abutment; e. g., the numbers 4 and 6 over the point m, signify 6w" greatest weight borne by mc, and 4w" = the greatest borne by me.

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It is obvious also, that the maximum thrust of any diagonal, equals the maximum tension of the diagonal meeting the former at the lower chord; that is, inaximum thrust of me, is equal to 6w'D maximum ten

sion of co. The maximum thrust of bn being equal to 9, the maximum tension of bo, equals 9w". This is an extra weight thrown upon the point o, in consequence of the vertical bo, being turned out of its regular direction of a diagonal in the position of bp,* in order to throw its load upon oa, whereby op and pa are rendered unnecessary. The weight borne by oa, therefore, instead of being 12w", as indicated by the figure 12 at o, is 12w+9w", 21w", = 3w.

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The figure 1 over o, denotes the tendency of 1w" to act upon oc, by thrust, by which tendency the tension of oc, under a full load of the truss, is reduced to 520D.

LXIII. If Fig. 12 be assumed to represent a truss with tension verticals and thrust diagonals, the figures over the upper nodes, prefixed to w" indicate the weights tending to act by compression upon the diaRonals descending toward the right from the nodes respectively; which weight is transferred to the vertical meeting the diagonal at the lower chord. This constitutes the maximum load of the vertical, in case of a deck bridge. Otherwise, the maximum stress of verticals is shown by the figures immediately over

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*The point p, not seen in Fig. 13, is assumed to be at the intersec tion of a vertical line through the point a, with the upper chordproduced. The arrangement above alluded to, gives the truss a reciangular, instead of a trapezoidal form of outline, which involves no more action upon material, 'though it increases the number of members in the truss. [See Fig. 13A.]

them, prefixed to w", provided, that in this case, the maximum stress of a vertical can never be less than w. the weight applied immediately at its lower end.

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RATIO OF LENGTH TO DEPTH OF TRUSS.

LXIV. Having explained and iliustrated, it is hoped intelligibly, methods by which may be computed the stresses of the various parts composing most of the combinations of members capable of being used in bridge trusses, with a view to giving to each part its due proportions, it may be proper to give attention tc the general proportions of trusses, and such other considerations as may affect the efficient, and economica! application of materials in bridge construction.

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The ratio of length to depth of truss is susceptible very great range, and it is obvious that some certain medium, in this respect, will generally give more advantageous results, than any considerable deviation toward either extreme. For, it will be observed, that in the expressions we have derived for the amount of action open chords, appears as a factor; v representing the depth of truss, between centres of chords. Hence, the smaller the value of v, the greater the stresses of chords, so that when v=0, these stresses become infinite, and the chords require an infinite amount of material; in other words, the case is impossible. On the other hand, if v be infinitely great, though the stress of chords be reduced to nothing, the verticals and diagonals being infinitely long, and sustait ing a definite weight, also require an infinite amount of material.

Now, between these two impracticable extremes where shall we look for the most advantageous ratio!

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