the point of contact, its horizontal thrust equals the weight sustained, multiplied by the horizontal, and di vided by the vertical reach of the brace. But the horizontal and vertical reaches are respectively as the sine and cosine of the angle made by the tangent with the vertical; that is, as ab and bd, Fig. 17, while the weight is also as the sine ab, of the angle adb. Hence, the weight by the horizontal reach, is as ab2, or as the square of the sine of adb; and the constant horizontal thrust of the arch at all points, is as

α

FIG 17.

d

Now this condition is answered by the parabola, in

which becd=

bd, and

ab2 bd

ab2

1

2 cb

=constant C, whence ab2cb X constant 2C, which is the equation of the parabola.

=

This quality of the arch truss, allowing nearly all of the compressive action to be concentrated upon almost the least possible length, and consequently, enabling the thrust material to work at better advantage than in plars where this action is more distributed, and acts upon a greater number and length of thrust members, enables it to maintain a more successful competition with other plans than we might be led to expect, in view of the greater amount of action upon materials in the arch truss, than what is shown in trusses with parallel chords. Hence, we should not too hastily come to a conclusion unfavorable to the arch truss, on account of the apparent disadvantage it labors under, as to amount of action upon material. These apparent disadvantages are frequently overbalanced by advan

tages of a practical character, which can not readily be reduced to measurement and calculation.

The preceding general comparisons are to be regarded only as approximations, and should not be taken as conclusive evidence of the superiority or otherwise, of any plan, except in case of very considerable difference. in amount of action, with little or no probable advantage in regard to efficient action of material.

EFFECTS OF WEIGHT OF STRUCTURE.

LIII. In preceding analyses, and estimates of stresses upon the various members in bridge trusses, regard has only been had to the effects of movable load, which may be placed upon, or removed from the structure, producing more or less varying strains upon its several parts.

But the materials composing the structure, evidently act in a similar manner with the movable load, in producing stress upon its members; the only difference being, that the weight of structure is constant, always exerting or tending to exert the same influence upon the members, instead of a varying action, such as that produced by the movable load. In order, therefore, to know the absolute stress to which any member is liable, and thereby to be able to give it the required strength and proportions, we have to add the stresses due to constant and occasional loads together.

The weight of structure evidently acts upon the truss in the same manner as if it were concentrated at the nodes along the upper and lower chords, and of the arch, in case of the arch truss. And, since much the larger proportion of it acts at the points where the

movable load is applied, if we regard the whole as acting at those points, the results obtained as to stresses produced by it, will be sufficiently accurate for ordinary practice. Still, more closely approximating results may be obtained by assigning to both upper and lower nodes, their appropriate shares of weight sustained, as may easily be done when deemed expedient.

If we divide the whole weight of superstructure supported by a single truss, by the number of panels, the quotient, which we may represent by w', will show the weight to be assumed for each supporting point, on account of structure; and the stresses produced by such weights, added to the maximum stresses of the several members, due to the movable load, will represent the true absolute stresses the respective members are liable to bear.

Now, as far as relates to parts suffering their maximum stresses under the full load, such as chords, arches, king braces, and verticals in the arch truss, as to their tension strain, we have only to substitute W, (w+w'), in place of w, in expressions obtained for stresses due to movable load. In other cases, w and w' will have each its peculiar and appropriate co-efficient.

The diagonals of the arch truss, are obviously not affected by weight of structure, as they are not so under full and uniform movable load. Moreover, the weight of structure acts in constant opposition to the compressive action of movable load upon verticals. Hence, in truss Fig. 11, where we find the varying movable load gives a maximum compression upon the longest, equal to 3w", and upon the next shorter, equal to 2w", the weight of structure diminishes those quantities to 3w"-w', and 2w"-w' respectively. Or, if we

would be more exact, we may add in both cases, the weight of a segment of the arch, which has no tendency to produce tension upon the verticals; or we may subtract only or of w'; thus, 3w"-w', and 2w′′—zw', may be taken to represent the compressive action upon the verticals in Fig. 11.

LIV. In the case of truss Fig. 12, the only diagonals acting under uniform load, are oc, fj, nd and ek; the two latter sustaining, of weight of structure, 1w', and the two former, 2w'. And, the maximum movable weight borne by those members, being [XL] 10w" and 15w", the absolute maximum will be 10w"+w' for nd and ek, and 15w"+2w' for oc and fj.

Now, if we place the figure 1 under d and e, (Fig. 12 A), and the figure 2 under c and ƒ, and so on, in case of a greater number of panels, to the foot of the last diagonal cach way, inclining outward from the lower nodes, these figures are obviously, the co-efficients of w' to express the weights contributed by the material of the stru ture, to the stresses of diagonals extending upward and outward from the points to which the figures respect'vely refer.

FIG. 12 A.

Again, we have seen [XL], that a certain condition of the movable load, tends to throw 1w" upon bn, and another condition of such load, tends to throw 3" upon

ст. But, since, as we now see, the weight of structure tends to throw a constant weight of 2w' upon oc, which is antagonistic to bn, the actual maximum weight upon bn, is 1w"-2w', which will always be a negative quantity, in practice; whence bn must always be inactive, and may be dispensed with.

The maximum weight upon cm, as modified by weight of structure, is in like manner reduced to 3w" -w', which will in practice, be either negative, or of quite small amount. Hence, we have the following rule: For the absolute maximum stresses of diagonals (in case of parallel-chord trusses with verticals), we add the effects of weight of structure to the maximum effects due to variable load, where both fall upon the same, and subtract the former, in cases where the two forces fall upon counter, or antagonistic diagonals.

In case of parallel-chord trusses without verticals, we add the effects of constant and variable load upon each diagonal, when alike, i. e., when both tensile or both compressive, and subtract the former when the effects are alike.

DOUBLE CANCELATED TRUSSES.

LV. The use of chords in a truss being to sustain the horizontal action (whether of thrust or tension) of the oblique members, it follows that the aggregate stress of chords, is equal to the aggregate horizontal action of all the diagonals acting in either direction. And, the horizontal action being obviously as the number and horizontal reach directly, and as the vertical reach inversely; also, the length of truss being as