same horizontal action as i would do under 3w". Hence these two diagonals, under their maximum stresses, which neither suffers, in the present case, would only compensate of the loss on stress of arch and chord, due to removal of weight from the truss.

It may, therefore, be regarded as a matter of extreme probability, if not a rigidly demonstrated fact, that the arch and chord of an arch truss, undergo their maximum stress in all parts, under the full uniform load of the truss.

It is hoped and believed that the foregoing illustra tions of the manner of determining the strains of the several parts and members of an arch truss of seven panels, will be sufficient to enable the same to be done in the case of trusses of any desired number of panels.

THE TRAPEZOIDAL TRUSS.

So designated from the figure of its outline. XXXIX. This truss may be constructed with diagonals and verticals, as in Fig. 12, or without verticals, except at b and g, as seen in Fig. 13. To explain the operation of these trusses, and determine the maximum, stresses of their various parts, we may use the same notation, generally, as heretofore; that is, let h represent the horizontal, and v, the vertical reach of the diagonal or oblique members, and D, the length of diagonals. Also, let w represent the greatest movable load for at panel length, supposed to be concentrated at the nodes b, c, d etc., of the lower chord; and, let w" be equal to w, divided by the number of panels in the truss (7, in this case), i.e., let w = 7w".

Then, supposing the diagonals (Fig. 12), not including the king braces, ao and i at the ends; the verticals

ob and jg, and the lower chord, to act by tension; and the upper chord, or boom, the king braces, and the four intermediate verticals, to act by thrust, or compression if a weight (w) be applied at 6, it will obviously cause a downward action equal to w" at i, and one equal to 6w" at a.

Now, from what has already been seen, in the discus sion in relation to Fig. 10, the weight acting at i, can only do so by acting successively, or simultaneously, upon bn, and each diagonal parallel with bn on the right, by tension, and upon each compression upright and the king brace j, by thrust; causing upon each of these 10 members, a stress equal to w" upon verticals, and equal to w" upon obliques; D representing the length of obliques, or diagonals.

A weight (w) at c, in like manner, causes a pressure of 2w" at i, through cm, and other diagonals inclining to the right, on the right hand of c. Also, a pressure of 5w" at a. But co being the only member that can transfer weight from c to the left, and, co and bn being antagonistic-stress upon the one tending to relax the other, the result must be, that both can not act at the same time, from the effects of weight at b and c, and only that one can act, to which the greater weight is applied; and that, only with the excess of weight acting upon it, over what is acting, or tending to act Now, as the load at c, tends to throw

upon the other.

a weight of 5w" upon co, while the load at b tends to throw lw" on bn, the former tendency must preponderate co must sustain 4w", while bn is relaxed, and the whole weight at b, is sustained by the tension of ob. In reality then, cm sustains of the weight at c, and none of that at b.

Still, the result is the same, as to pressure at a and i, the former point supporting 7w", the whole of the load at b, plus 4w" of that at c, making 11w", = pressure due at a, from the weights at b and c, while the point i supports 3w", all out of the weight at c. Thus, cm, dl etc., sustain the same proportion of the aggregate weights at b and c, as if each weight acted separately, and independently of the other.

=

Again, applying a weight (w), at d, 3w" tends to bear at i, and 4w" at a, through dn and co. But as we have 3w" tending to act on em, as already seen, this is neutralized by the tendency to action upon dn, and only a surplus of 1w", really acts upon dn in this case, while the 6w", pressure due at i, from the weights at b, c and d, is all made up out of the single weight at d, and the whole of the weights at b and c, together with 1w" from that at d, comes to bear at a, giving a pressure of 15w", at that point; still the same as if each weight acted independently of the others. And, in general, each diagonal, at all times, sustains the preponderance of weight tending to act upon it, over that which tends to act at the same time upon its antagonist. Hence, the greatest weight sustained by any diagonal, is when all the weight tending to act upon it, is upon the truss, and none of the weight tending to produce action upon its antagonist, or counter.

Thus, when b alone is loaded, 1w" is sustained by bn, but when any point on the right of b is loaded, there is

tendency to action by co, and the action of bn is de stroyed or diminished. Therefore 1w" is the maximum weight sustained by bn. When b and c alone are loaded with the weight (w) at each, cm sustains 3w", as already seen, with no tendency to action in dn. But if d, or any point on the right of c, be loaded, there is tendency to action in dn, which must diminish or destroy the action of cm. Hence, cm sustains its maximum weight (= 3w"), when the points b and c alone are under their full load. And, it must be obvious. that the maximum weight is sustained by each diagonal inclining to the right, when the point at its lower end, and all the nodes at the left are fully loaded, and all those at the right are without load. Hence we establish the following easy and expeditious practical method of determining the maximum weights and stresses upon this class of members, in trusses with any number of panels.

XL. Having made a rough diagram of the truss, as Fig. 12, for instance, place over the nodes o, n, m, &c., the numbers 1, 2, 3, &c., high enough to admit of a second series under the first, formed by repeating the 1 under itself, adding the 1 and 2 together and placing the sum (3), under the 2 in the upper series.. Then add 1, 2 and 3, and place the sum (6) under 3, and so on, placing under each figure of the upper series, the sum of that figure, and all those at the left, in said upper series.

Then, it will be seen that each figure in the upper line, prefixed to w", shows the pressure caused at the right hand abutment, by the weight (w) directly under the figure, e. g., the upper figure 3 over d, indicates that 3w" is the bearing at i, produced by the weight (w) at d, and so of the other figures in the upper line.

In the mean time, the figures in the lower line, show the accumulation of the effects of the different weights upon successive diagonals from left to right. Thus, the figure 6 over the point d, shows that dl sustains 6w", pressure due at i, from weights (w) at b, e and d, when those points only are loaded; in which case, dl sustains its maximum weight, as before seen.

=

In like manner, the figures 10 & 15 over e and f, indicate that 10w" and 15w", are respectively the maximum weights sustained by ek and fg, while 21w" (= 3w), equals the maximum weight sustained by ÿj, (by compression, of course), when the whole truss is loaded.

XLI. Having thus ascertained the greatest weights the several oblique members are liable to sustain (those inclining to the left being obviously exposed to the same stresses as those inclining to the right), we find their maximum stresses by rule 4, [xvr]; i. e., multiply the weight by the length, and divide by the vertical reach of the member. Thus, the maximum compression of ÿj, equals 3wo, = 3w✔h2+ v2, and the repre

v

3h

sentative of required material, is (312 + 3v) M.

The maximum stress of ek equals 10w" 13 w✓ and its representative for material is

v

(1322+10) M. Or, the lengths and inclinations being the same, we may take the aggregate maximum weights sustained by tension diagonals, reduced to terms of w, multiply by the square of the common length, divide by v, and change w to M. The ten tension diagonals sustain maximum weights equal to w" multiplied by twice the sum of all the figures in the lower line over