portions of the chords from the ends to the centre of the truss.

In the case of a deck bridge, the weights sustaine-l by thrust uprights, are respectively indicated by the figures over the diagram on the right hand half of the truss, prefixed to w", for movable load, and the figures under the diagram prefixed to w', for weight of structure, being the same weight which gives the maximum stress to the diagonal running upward and outward from the foot of the upright. Tension verticals at the ends sustain no weight.

TRUSSES WITHOUT VERTICALS.

It will be seen upon a general view of the action of the different parts of a truss with parallel chords, that the diagonals (and verticals when used), form media through which weight acting upon the truss, is reflected back and forth between the upper and lower chords, antil it comes finally to bear upon the abutment.

A weight applied at one of the nodes of the lower chord, of course, cannot be sustained by the tension. of that chord, which acts only in horizontal directions; but is suspended by a tension piece, whether oblique or vertical, from a node in the upper chord. But the upper chord acting also horizontally, cannot sustain the weight. Consequently, a thrust member, either oblique or vertical, must meet the force at that point, to prevent the weight from pulling down the upper chord, and destroying the structure.

Hence, we see, that in all the cases we have considered, of trusses with parallel chords, the weight, whether applied at the upper or lower chord, acts alter

nately upon thrust and tension pieces, extending directly or obliquely from chord to chord.

With reference to Fig. 18, we have regarded the weight as transferred from tension diagonals to thrust verticals, and the contrary. But if we conceive the verticals to be removed, except the endmost, we have only to insert a thrust brace from the abutment to the second node (or the first from the angle), of the upper chord, and to so form and connect the other diagonals as to enable them to act by either tension of thrust, and we have a truss capable of sustaining weights applied at all, or any of the nodes of the upper and lower chords, in the same manner as the truss with verticals, represented in Fig. 18. In this condition, the truss will act upon the principles discussed with reference to Fig. 13. For this modification of the truss, see Fig. 19.

To estimate the strains upon the several parts of such a truss, due to weights w, w, etc., at the nodes of the lower chord; we may place the figures 1, 2, 3, etc., over the nodes of the upper chord, as was done in the case of Fig. 18. But, instead of adding alternate figures to form the second series, to be used as co-efficients of w", for expressing the weights sustained by diagonals, we add every fourth figure; because it is only the weights at every fourth node, that act upon the same set of diagonals.

For instance; the weights at 1, 5, 9 and 13, act upon their peculiar set of 8 pieces (excluding the end braces, but including the tension vertical at 1), and none of the weights at the other nodes have any action upon those pieces; as is made obvious by an inspection of Fig. 19.

Again, the weight at 2, 6, 10 and 14, have their peculiar and independent set; and so of those at 3, 7, 11 and 15, and those at 4, 8 and 12. Therefore, in form

ing our second series of numbers, we place under each figure of the first series, the sum of that figure, added to every 4th figure preceding; that is, under 12, place the sum of 12, 8 and 4 24. Under 5, the sum of 5+ 16. The four first figures, having no 4th preceding figures,

are simply transferred, without addition or alteration.

These numbers in the, second series, are the co-efficients of w" (=w divided by the number of panels in the truss, being 16 in this case), to express the greatest weights acting by tension on each diagonal having its lower end under the number used, and the upper end under a higher number. Also the weight act

ing by thrust upon the diagonal meeting the former at the upper chord The last, or highest

number, determines the weight sustained by the tension vertical under the number, the vertical being a member of one of the four sets of alternate thrust and tension pieces connecting the two chords.

A third series of figure, formed by reversing the order of the second-placing the low

est number of the third under the highest of the se cond series, and vice versa, prefixed as before to w", will show the weights sustained by thrust and tension of diagonals in the reversed order; i. e., whereas one series shows the amount of tension a particular diagonal is liable to, the reversed series shows the thrust the same piece must exert in a different condition of the load.

Thus we ascertain, as in the case of truss Fig. 13 [XLV], that nearly all of the diagonals are exposed to two kinds of action, thrust and tension; and it is only the preponderance of the larger over the smaller of these forces, which has place when the truss is fully loaded, and it is only this preponderance which is to be used as co-efficient to (w+w') in estimating the stresses upon the different portions of the chords, and as co-efficient to w', in modifying the effects of the variable load upon diagonals, as affected by weight of structure. But it is to be remembered that the numbers over the diagram are to be divided by the number of panels, before being used before w and w', in the expression of stresses of members. Thus, we have, as the effect of variable. load upon the diagonal 2/4 ..., 2w" (w), as the

16

18

2

16

greatest weight acting by tension, and 16", the greatest acting by thrust. Hence the weight upon this piece, due to weight of structure, is (18—2)w',=w', and it produces thrust or compression, because the thrust tendency is the greater. This weight (w'), added to the greatest effect of variable load shows the maxi6W, mum weight which can act by thrust upon that diagonal, to be w+w'. We have, also, for the greatest weight acting by tension as modified by weight of struc

18

16

18

16

2

16

ture, w-w', which is a negative quantity when w is less than 8w', as will usually be the case in practice; consequently that diagonal can seldom or never be exposed to the force of tension.

16,

16

h

Again (w+w'), (h and v representing horizontal and vertical reaches of the diagonal, as in previous discussions), is the amount contributed toward the maximum tension of the lower chord by the diagonal in question, not affecting, of course, that portion of the chord outside of the connection therewith, or a like portion at the opposite end.

VIII. It is to be remembered that the tension or thrust of a diagonal, is always equal to the weight sustained, multiplied by the length, and divided by the vertical reach of the diagonal.

The method here under discussion for estimating stresses, seems to need no further illustration. But the question as to decussation, affects the case of Fig. 19, as well as that of Fig. 13. The two sets of diagonals which meet the upper and lower chords in the centre, have symmetrical halves on each side of the centre, and no action can pass the centre upon either, when they are uniformly loaded; whereas, the two sets to which 7/9 and 79 belong, have the half of one on either side of the centre, a counter part to the half of the other set on the opposite side; and the diagonals 7/9 and 79, will act or not, according as their opposite points of connection with upper and lower chords, are carried farther apart, or the contrary. Now, as the points 9 and 7, upper chord, are depressed by the change in one vertical and 3 diagonals, while the opposite points at the lower chord are depressed by the