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and eg are necessarily tension members in all cases, in practice.

Again a weight, w, at c, must cause pressure equal to w at f, through tension of ci and dg, and thrust of id and gf. This, with the fw, from the weight at b, makes fw, acting on ci. But the weight at c, causes pressure equal to w at a, necessarily through tension of cl; and siuce cl and bk are antagonistic, the action upon one tending to produce relaxation of the other, it follows that only one can act at the same time, unless unduly strained in the adjustment of the truss. Hence, the tw, which acts upon bk, when b alone is loaded, is overbalanced by the w, tending to act upon cl, on account of the load at c; and the result is, that bk is relaxed, the whole weight at b, is necessarily sustained by bland la, and the gw, which must by a statical necessity, bear at f, in consequence of the loads at b and c, is all made up from the weight at c, leaving only

w of this weight to bear at a, through cl. Now, since it is obvious that all load at c, d, or e, must contribute to the pressure at a, which can only occur through action upon cl, it follows that bk can only sustain the whole weight of w, when the point b alone is loaded; and consequently, that w is the greatest weight that bk can ever be subjected to.

Then, applying another weight, w, at d, it must add w to the pressure at ƒ, through tension of dg and thrust of gf; which last amount, added to gw, communicated to dg through ci and id, makes gw, as the weight sustained by dg. But the weight at d, also causes pressure at a, equal to gw, which can only be done through action, or tendency to action upon dk, and since dk and ci are antagonists, only one can act at once, and that, only with a force equal to the excess of tendency to

action of the one, over that of the other. Now we have seen that weights at b and c, tend to throw w upon ci, while the weight at d, tends to throw zw upon dk. Hence, in these circumstances, ci only sustains tw, which is transferred to dg through thrust of id, while dk is relaxed, and the whole weight at d, is sustained by dy; making, with the w from ci, just above mentioned, gw, equal to the pressure due upon the abutment at f, on account of weights at b, c and d.

Lastly, a weight, w, at e, tends to give pressure equal to w at f, through eg and gf, and a pressure equal to ★ w at a, through ei, dk, etc. This latter tendency has the effect to diminish by w, the tendency of previously imposed weights, to throw gw upon dg, reducing it to gw, and to neutralize the balance of w acting upon ci, after the imposition of the weight at d, leaving c and dk both inactive, while eg sustains the whole weight applied at c, equal to w.

Now, as we have seen, any weight at d or e, tends to throw action upon dk, thereby diminishing action upon ci, and since weight at b and c, both contribute to the stress of ci, it follows that the maximum action upon ci, occurs when b and c are loaded, and d and e, unloaded. For similar reasons, the maximum action upon dg, occurs when e alone is unloaded.

The maximum weight sustained by lb, and eg, is the weight applied directly at each of the points b and e, equal to w, and the maximum weights sustained by e, dk, and cl, are the same as those sustained by bk, ci and dg, each respectively, as just above determined; while al and gf, both receiving action from weight on any part of the truss, obviously sustain their maximum weight, equal to 2w, under the full load of the truss.

The section ab, of the lower chord, suffers a stress equal to the horizontal thrust of al, which of course, is greatest when al sustains the greatest weight. This has just been seen to be equal to 2w, and occurs under a full load of the truss. Hence the greatest stress upor ab equals 2w, and is communicated without change to be, bk being inactive when the truss is fully loaded. The section cd, suffers stress equal to the combined horizontal action of al and le, which must be greatest when this combined action is greatest. That is also under the full load of the truss. For, though le sustains w more weight when b is unloaded, the same cause relieves al of the amount of w. Consequently, the weight borne by the two, is gw less in this case, than when the truss is fully loaded. The greatest combined weights, then sustained by al and le, being equal to 3w, the greatest stress of ed equals 3. This is 3w also the greatest compression suffered by the upper chord lg, since the latter is also equal to the combined horizontal thrust and pull of al and le. The stress of this chord is the same throughout, because the obliques meeting at k and i, are inactive when the truss is loaded throughout.

The maximum compression upon ck and id, equals the greatest weight sustained by ci and dk, already found to be equal to zw.

XXV. Having thus ascertained the greatest weights sustained by the several oblique members, and the greatest stresses of the horizontals and verticals, we may deduce the required amount of material, or, perhaps more properly, the amount of action upon the material required for the truss, as compared with like

amount of action in trusses 8 and 9, thus: Max. weight on end braces, 2w × length✔h2 + v2 2w✓ h2 + v2.

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Max. weight on 2 verticals zw× length of the two (= 2v), gives................ Max. stress of upper chord =

3 wh

x length (= 3h), gives amount of action =

Making total amount of action on

thrust material

=

Aggregate max. weight on 6 tension diagonals =w=4w. This by the length (

=

=

✔h2 + v2), gives stress

4 20√πi2 + v2 whence amount

of action on material, equals

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stress =

(4h2 + 4v) M.

1v M.

9 21 M.

13h2

(1312 + 5; v) M

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2 tension verticals sustain each, 1w, with length v, giving amount of action for the two =

Stress of middle section, lower chord

=3w, x length (

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SYNOPTICAL STATEMENT IN REGARD TO TRUSSES (Figs. 8, 9 and 10.

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Making h=v=1, the above table will be as

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XXVI. This shows very nearly the relative amount of tension material required in the several plans; while, as previously stated, the amount of compression material is not so nearly indicated by the figures and expressions giving the amount of action (sum of stresses into lengths of pieces), as in case of tension members. The compression material in No. 8 (the arch truss), is undoubtedly more efficient in action than in either of the others, while that in No. 9, is unquestionably the least 80. In fact, this truss will be hardly considered as possessing advantages of any kind, sufficient to induce

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