except in case of long slim pieces which break b lateral deflection, under a comparatively small compressive force. We vill, however, use the rule for the present occasion.

The efficiency of the material then, will be as the power of resistance and the horizontal reach directly, and as the stress produced by a given weight, inversely; which stress is as+h. Whence we have

d'h

v2 + h2

+(

=

d'vh

v

(+) proportional to the efficiency

of material in a thrust brace. Making d3v=1, the last ex

h

pression becomes (v2 + h2), and the value of h which gives

the greatest value to this function, will indicate the inclination at which a thrust brace will act with the greatest efficiency, as it regards the brace alone. Differentiating, and putting the result equal to 0, we

have:

d (

h

· (v2 + h2 ) { '

_d? (va + h2) { —¥h (y2+h2) ‡×2hdh=0; whence,

(v2 + h2) 3

multiplying by the denominator (v2+h2), we obtain

3h2-h2 2h2, and by evolution, v =

= 0.7072v.

h✔2, and h =

h

If we deduce the value of the expression: (which is equal to the horizontal reach divided by the cube of the length of brace), putting hv and h successively, we find the degree of efficiency less than the maximum, as above determined, by about 9 per cent in the former, and 8 per cent in the latter case showing that considerable deviations may be made in the inclinations of thrust braces without much detriment to efficiency of material in braces, when required

by other considerations; which will often be the case, as will be seen hereafter.

EFFECTS OF INCLINATION OF DIAGONALS UPON STRESS OF CHORDS AND VERTICALS.

LXVII. The comparative effects of different posisitions of diagonals upon the chords, may be illustrated with reference to Fig. 21. It is manifest that a given weight w on the centre of this truss, will produce a vertical pressure equal to w at each of the points ≈ aud b, and that each oblique member between a and w, will sustain a weight equal to w; and will exert each a horizontal action upon the upper and lower chords, equal to w2. Hence, the stress of chords in the centre,

will equal wxn, in which n represents the number of oblique members between a and w, or between But n equals whence wn =

a and c.

ac

h

h

h ac X- -P h

The term h having been eliminated from the last expression, it shows that the inclination of diagonals has no effect upon the stress of chords in the centre, produced by weight in the centre of the truss; and by similar reasoning it is shown that the same is true in relation to other parts of the chords, or to weight at any othe⚫ points in the length of the truss; the only difference being that the shorter the panels, or the smaller

the value of h, the shorter the intervals at which the increments in the stress of chords are added, and the less the magnitude of such increments, in the same

oportion. Hence, in general, there is no difference in the stresses of chords, whether the diagonals have cae inclination or another.

With regard to the effect upon verticals, that part of their stress which they receive through diagonals, is equal to the weight sustained by those diagonals, and is the same for a given weight, whatever be their inclination. On the vertical we, the pressure is received directly from the weight. But on the next adjacent vertical, on either side, one-half of the same. pressure is received through to the intervening diago nal, and transmitted to the next, and so on to the end.

Consequently the aggregate action of verticals, produced by the weight w, is equal to w + wn, taking n for the number of verticals receiving their stress through the medium of diagonals, and which is equal to the whole number less 3, when the number is odd, and the verticals act by thrust, as assumed in the case of Fig. 21. If the weight be applied at the lower chord, the whole action of verticais is communicated. through diagonals, the latter acting by tension.

Hence the aggregate action of verticals increases and diminishes with their number, and economy as regards those members, would require the diagonals to incline at a greater angle with the vertical than that which is most favorable as to the diagonals themselves.

We have seen, however [LXVI] that by placing the diagonals at 45° when they act by thrust, we lose about 9 per cent in economy of those members, and we now learn that such an arrangement increases the economy. in verticals to a considerable extent by diminishing

their number; the actual amount depending somewhat upon the number, and not deducible by a general rule..

We shall not, however, err greatly in assuming, that with an inclination of 45°, for thrust diagonals in conjunction with tension verticals, the loss upon the former is quite made up by saving in the latter, and that a less inclination in this case, should be regarded as very questionable practice.

In case of tension diagonals and vertical struts, a зaving in material may undoubtedly be made by mak ing the horizontal greater than the vertical reach of the diagonal, whenever such a course is found consistent with a proper regard to just proportions of the truss in other respects; such as width of panel, depth of truss, etc.

THE WIDTH OF PANEL.

LXVIII. Which we have represented in our formulæ by h, has only been hitherto considered as to its relations to v, representing the depth of truss.

With regard to the best absolute value of h, the question is affected by the relative expense of floor joists, and the extra amount of material and labor in forming connections at the nodes of the chords; as well as, in some cases, the lengths of sections in the upper chord. The latter requires support laterally and vertically at intervals of moderate length, depending upon the absolute stress, which, other things the same, governs the cross section.

The upper chord usually, of whatever material, has a cross-section so large as to exclude all danger of breaking by lateral deflection, in sections of 10 to 14 feet; and, as there will seldom be occasion for exceeding these lengths in cancelated trusses, the increased

expense of joists for wide panels, and the expense of extra connections in narrow ones, are the principal considerations affecting the absolute value of h, as an element of economy.

The transverse beams, supposed to be located at the nodes between adjacent panels, may, of course, be proportioned to the width of panel, so as to require essentially the same material in all cases. But the joists, or track stringers of rail road bridges, the depth being proportional to the length between supports, have a supporting power as their cross sections; and since the load, at a given weight to the lineal foot, is directly as the length, it follows, that to support the same load per foot, as bridge joists are required to do, the cross-section should be as the length. The expense of joists and stringers, therefore, is directly as the width of panel.*

On the contrary, the expense of connections will be as the number of panels, nearly, and consequently, inversely as their width, or inversely as the

*The thickness of joist most economical for a short reach would be liable to buckle with greater length and depth. Hence joists require increase of thickness with increase of length and depth. The thickness should be as the depth, and the cross-section, as the square of the depth (d).

Upon this basis, the required material for joists, increases at a greater ratio than the increase in width of panels. The supporting power of a joist or beam of a given form of section, or a given ratio of depth to thickness, is as the cube of the depth directly, and the length (4) inversely; or, as If there be two joists of depths respectively as d and x, and lengths asl and nl, their supporting powers P, P', for load similarly applied, will be as to nľ But the power should be as the load; in other words, as the length of joists. Hence we have the proportion, ::: nl, whence, nd and r dng. Now n is as the length of joists, and the depth, therefore, is as the power of the length, and the cross-section, and consequently the required material, as the power of the length. Hence, if m represent the material for joists with panels of a given width, the material for panels twice as wide, will be represented by m × 2m 16 m2.52. But this is rather anticipating the subject of lateral, or transverse strength of beams.

n