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change in 3 diagonals only, we might naturally expect to find greater depression in the upper than the lower points; though this does not follow as a matter of necessity, since the less number of members, by being more nearly under a maximum stress, might give greater depression than the greater number, under less stress, as compared with their maximum. Now, the vertical at 1, being under maximum weight, gives depression E; (adopting the notation used with reference to Fig. 13 [XLIX.]). The two diagonals 1/3 and 3\5 being under 3 maximum, give depression equal to 2×4E(making h=v=1),−3.81E; while the diagonal 5/7, under maximum, gives depression= 0.4×2E.

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0.8E, making a total depression of point 7, upper chord, = 5.61E. Again, the diagonal 1\3, under maximum stress, gives depression 2E, while 3/5 and 5\7, under 1 maxm u m stress, give depression #x 4E, 3.2E, making a total for the point 7, lower chord, equal to 5.2E, which is less by 0.41E, than the depression of the opposite point in the upper chord, whereas it should be greater by 0.8E, in order to give to 79 and 7/9, the tension assigned to them by the decussation theory.

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But we must not conclude from this fact, that there is no decussation in this case. For, if we assume that 79 is inactive under the full load, it follows that 5/7 is also inactive, and that 1/3+3\5 sustain only 15 maximum stress, producing § E, = 3.05 E, which added o 1E for the vertical at 1, makes 4.05 E, depression at point 7, upper chord; while the 3 diagonals contribut ing to depression of the opposite point in lower chord, are under maximum stress, producing depression = 6 E. Hence, we see, that upon this hypothesis, the distance between these two points, measuring the vertical reach

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of the diagonal, is increased by (6-4.05) E 1.95 E. This can not be, without producing tension upon diagonala 79 and 7/9. Since then these members can not be entirely without action, and as previously shown, they can not have as much action as the decussation theory assigus to them, it follows, in this case, that they must act, but with less intensity than the theory assigns them.

In this case, as well as in that of Fig. 13, the result would be changed somewhat, by taking into the ca culation the weight of structure, which would chang to a small extent, the relation between the maximur. stresses of diagonals, and the stresses they sustain un der a full load. For the stress due to weight of struct ure, is constant, and that due to variable load, is greater. upon most of the diagonals, under certain conditions of a partial, than under a full load. Hence, while 57 sustains (under full load), only maximum upon that part of the material provided for variable load, it sustains a full maximum upon the part provided to sustain weight of structure. It is easy enough to take these things all into account, in estimating the amount of decussation in special cases. Still, it is doubtft' whether any better practical rule can be adopted, than the one previously given, [XLVIII]; namely, to estimate stresses upon both hypotheses, and take the highest estimate for each part.

DECUSSATION IN TRUSSES WITH VERTICALS.

LIX. In trusses of this class with odd panels, and diagonals crossing two panels, as in Fig. 20, it will be seen, on subjecting them to analysis, such as was explained with reference to Fig. 18 [xvr], that, while in trusses of even panels, the figures in the second line.

over the diagram, indicate the maximum stresses of diagonals, and those under the diagram, the stresses under uniform load (which are generally less than the maximum under partial loads), in case of the truss with odd panels, the bottom figures show, for certain diagonals, greater stresses for the full, than the upper figures give as the maximum for partial loads. Thus, in Fig. 20, the number 16 over m, indicates 16w" (=6w), as the maximum weight for il, while the fig ure 2 under the point i, indicates that il sustains 2.v (= 18w'), under the full load. It should be remarked here, that the figure 1 under the first two nodes on either side of the centre, and the figure 2 under the next, are thus

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placed upon the assumption that all the weight on either side of the centre, is made to act on its nearest abutment. This would necessarily be so, if en and fq were removed or relaxed. But, with those members in place, and properly adjusted, there may be a decussation of forces through them, whereby a portion of the weights at e and ƒ, may be made to bear upon the more remote abutments. Now, as the maximum on en is 6w" and that of its antagonist only 4w", the latter is not sufficient to neutralize the former entirely, but leaves a balance of 2w" which may be transmitted through en to gl, as an offset for a like amount trans

mitted through fq to ds. If this be so, then fm and en do not sustain the full weight of 1w, but only 7w", which, being transmitted to il, makes, with the weight w (=9w"), applied directly at i, 16w", as indicated by the figures over the diagram, instead of 2w (= 18w"), as the figure 2 under the point i would indicate.

Now, whether the two diagonals en and fq, being apparently, in a state of partial antagonism, do in whole or in part neutralize the tendency of each other to transmit weight past the centre each way under a uniform load of the truss, is not quite obvious, and it may be proper to estimate stresses under both hypotheses, and take the highest estimate for each part of the

t.uss.

It will be seen that il and cs are the only diagonals in Fig. 20, which show greater stress with a full than a partial load, upon the non-decussation hypothesis. But all the diagonals undergo different stresses, with the uniform load, as viewed under the different theories, and consequently, their effects upon the chords are different. The end brace as, sustains 4 (w+w) · 4W substituting W for w+w'), under either theory, and the tension of ac equals 4w" (making h=ab, and v=bs). es sustains 2W, or W, whence cd sustains either 6Wh or 57 Wh Again, ds sustains W, or 13W,

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the former without, and the latter with decussation. This diagonal having a horizontal reach of 2h, adds 2W or 24 W to tension of chord, making 8Wh

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83 Wh

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as the tension of de. For er, we have W without decussation, making a tension of 10W for ef; while with decussation, er sustains W, from which we subtract W, for opposite action of e, leaving W

g:ving horizontal pull =13W to be added to 83 W making 9 Wh -tension of ef.

Upon the non-decussation hypothesis, sr and m l, of the upper chord sustain thrust equal to 8W, and the remainder of the chord, 10W. By the other hypothe

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sis, sr and ml sustain 83 W rq and nm sustain 97

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W, and the other 3 sections, 103

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LX. We may derive some more light upon this subject, by considering the conditions resulting from the elasticity of materials. Supposing the upper and lower chords to be so proportioned as to be uniformly contracted or extended under a uniform load of the truss, this does not require or imply any appreciable difference in lengths of diagonals. But the stress upon chords being produced by the action of diagonals, the latter, when, as here supposed, acting by tension, necessarily undergo extension, by which means, the panels (except the centre one), are changed from their original form of rectangles, to that of oblique trapezoids. For instance, the figure gjln becomes longer diagonally from g to l, than from n to j, whence the point g falls lower than it would do, if the diagonal suffered no change.

Suppose then the truss to be fully loaded, and the diagonals il, gl and fm, to be each exposed to the same stress to the square inch of cross-section. In that case, il and gl suffer extension proportionally to their respective lengths, thereby causing depression of the pointe i and g respectively as the squares of those lengths. [See note in section XLIX.] Hence, the point g is de

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