If we strike out the diagonals cm and me, and also ll and If, all the determinate forces necessary to sustain uniform weights at the nodes of the lower chord, would be exerted by remaining members, although we have assigned to those members, cach, the sustaining of weight equal to 2w" under the full load, and twice that weight under certain conditions of partial load ; and it is quite certain that they are necessary to the stability of the truss when partially loaded. But with both halves loaded uniformly, the weight upon each half could be transferred to the nearest abutment, producing equal thrust in both directions upon the central portion of the upper, and equal tension in opposite directions upon the lower chord; whereas, with one-half loaded, there is no means by which the pressure due at the farther abutment could be transferred past the centre, without oblique members in the centre panel. Still, which mode of action takes place under the uniform load, when the diagonals are in place, is a matter involved in a degree of uncertainty. If the centre diagonals do not act, under the uniform load, then ek and fj must sustain each 7w", instead of 6w" for the former, and 9w" for the latter, as above estimated. Also, kg would sustain 7w" by thrust, and different results would be produced as to stresses of various parts of upper and lower chords.

The maximum stress for ek and dn, and for nb kg, would be 7w" instead of 6w", as found above, and would occur under the full, instead of the partial load. The tension of gj and ob, also, would be increased to 14w". The weight sustained by fj, would be only 7w" under the full load, though liable to the same maximum weight of 9w", under a partial load.

For the lower chord, 'we should have the same coefficient, (21) of w" to express the tension of ab and ig, 28 for bc, 35 (a decrease), for cd, and 42 for de.

For upper chord, the co-efficients of wh would be 28 for on and kj, and 42 for the three middle sections; no action being imparted by diagonals at m and l.

XLVIII. This uncertainty of action has no place in trusses of an even number of panels, as in such cases, no transfer of the action of weight can be supposed to take place past the centre, under a uniform load, without involving the absurdity of supposing the same member to carry weight by tension and compression at the same time; except, however, that in case of diagonals crossing two panels, or having a horizontal reach equal to twice the space between nodes of the chords, there will be diagonals filling the same condition of crossing in the centre of the truss, both vertically and longitudinally, as in Fig. 13.

We may obviate mostly, any mischief liable to result in cases of the kind under consideration, by estimating the stresses upon the several parts under both hypotheses, and taking for each member the highest estimate, which will mostly meet all contingencies. Estimating action upon truss 13 in this manner, we obtain the following representative expressions for material :

Making h=v=1. These expressions give, Compression material = 46.855M+ Tension do, 47.714м = total 94,571M.

This shows an aggregate amount of compression and tension action, identical with that of truss Fig. 12, [XLIII.]

DECUSSATION AND NON-DECUSSATION.

XLIX. The elasticity of materials affords a means of answering the question as to decussation of forces through diagonals crossing in the centre of the truss, vertically and longitudinally (as in Fig. 13), in specific cases. But the results will vary in trusses of different numbers of panels, and different inclinations of diagonals.

=

Suppose the truss Fig. 13 to be so proportioned that the maximum stresses of the several parts and members, will produce change of length equal to E, multiplied by the lengths of parts respectively; the vertical ob, v, being the unit of length. Then, the truss being uniformly and fully loaded, and the chords being under their maximum stress, the upper chord is contracted, and the lower one extended at a uniform degree; and, if the diagonals be unchanged in length, their vertical and horizontal reaches have not been changed by the change in length of chords. Hence, the distance between chords is not altered by change in their length. But the diagonals being under stress, by which some are extended and others contracted, according to the stress they are under, as compared with their maximum stresses respectively, the nodes of the chords are allowed to settle to positions below what they are brought to by the mere change in lengths of chords.

Hence, the panels are (generally) thrown into more or less obliquity of form, in consequence of inequality in length of diagonals in the same panel. But the centre panel can not assume obliquity, because any tendency of forces to change the length of one diagonal, is attended by a like tendency of equal forces to produce exactly the same change in the other; so that the vertical reaches of both must suffer the same change, if any, and both must be under tension or compression, according as the acting forces tend to bring the chords at the centre, nigher together or farther apart.

Now, the forces produced by the load being all concentrated at the points o and j (Fig. 13), the point d is depressed with respect to o, by the extension of ob and nd, and by the compression of bn. Hence, assuming decussation to have place, giving tension to the diagonals dl and me, equal to what is due to a weight of 2w", ob is under maximum tension and gives depression equal to E, to the point b, bn and nd are under § maximum stress, and give depression, each equal to Ex d2 * = (1.666/2 + 1.666) F, for the two (D representing length of diagonals, =✔2+1). Then, adding 1E for effect of ob, we obtain (1.666h2 + 2.666) E, depression of point d.

=

=

The point m is depressed by extension of oc under a maximum stress, giving an amount equal to D2E (h2+1) E. Also, by compression of cm under one-half maximum stress, to the extent of (h2 + 1) E. Hence, depression of point m = (11⁄2h2 + 11⁄2) E.

This shows the point d to be depressea more than m, by (1.666h2 + 2.666) E(1.5h2 + 1.5)E(0.166h2 +

* Let the diagonals bd and Bd, of two rectangular panels ac and Ac, Fig. 14 (c and d, being fixed points), be exposed to tension in propor tion to their respective cross-sections, receiving each thereby, extension

1.166) E, and the spaces md and le to be increased to that extent; of course producing tension upon dl and me.

Now, by hypothesis, these diagonals are under the weight of 2w", giving half maximum stress, and requiring an increase of vertical reach, equal to (h2 + 1) E. If then, we give such a value to h, as will make the last co-efficient of E equal to the one above, it will show that the chords have receded just enough to give the assumed tension to dl and me, and the decussation is a demonstrated fact. To find the value of h, producing this result, make,.5h2 + .5 = .166h2 + 1,166, and we deduce .333h2.666; whence h2.

But this requires too great an inclination of diagonals, and a less value of h, gives a space from d to m too great for the supposed tension of dl and me. Making h=1=v, we have increase of distance from d to m = 1.333E, requiring a weight of 2.666w", to stretch dl down to the point d. But as no weight or stress can be added to the 2w" assigned to dl and me, without af

equal to b'e and B'E, respectively. This will cause the points b and B to drop to b' and B', in ab and AB produced. Join be and BE. Then, the infinitesimal triangles bb'e and BB'E, right-angled at e and E, are essentially similar, respectively to the triangles db a and dBA. Hence, the following relations: (1). bb': b'e :: bd: ab, (2). BB': B'E :: Bd: ab.

and

whence, bb' x ab= b'e X bd, and

BB'Xab B'EX Bd,

From these two equations we derive

(3). bb'Xab: BB'Xab:: b'e X bd B'EX Bd.

FIG. 14.

a

A

But, by the law of elasticity

(4).

b'e: B'E :: bd: Bd; whence,

E

B'

(5). b'e x bd : B'Ex Bd :: bd2: Bd2.

Hence dividing the first ratio of proportion (3) by ab, and substituting for the last ratio of (3), its equivalent found in (5), we have, (6).

bb': BB':: bd2: Bd2.

Hence the depression due to the extension of a diagonal retaining the same vertical reach, is as the stress (per square inch), sustained, multiplied by the square of the length of diagonal.