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or height = 50 feet. Let us also assume A m the thickness of the crown, equal to seven feet, nearly 14th part of the span, being nearly the proportion employed. Dividing each half arch H A and R A, into 9 = parts of 10 degrees each, which will do very well to show the form of the extrados, containing each the extend of two wedges; then from the centre O drawing radii through all the points of division, and continued, the first passing through the middle of the key stone: Then, on these radii produced, set off, from the arch of the semicircle, A m, B n, S Q, &c.,
the lengths formed according to the last method of calculation; then drawing a line with the hand through the extremities of all the exterior lines, it will be the extrados required, exhibiting the form and limit of the wall built of uniform materials, so as to constitute an arch of equilibration.
On the Comparative Strength of Arches.
It may be inferred, that the strength of one part of an arch to that of another, will be proportional to the greatest weights those parts are capable of bearing; that is, as the cube of the secant of the curve's inclination to the horizon at those points, divided by the radii of curvature; and when two arches formed of similar materials have the same span and height, their comparative strengths at any two corresponding points will be also in the same proportion; but since if one part of each fails, the whole will fall to ruin, and as the crown is the weakest part in all arches, it will only be necessary to compare them with each other at that point.
As all curves at their vertices have no inclination to the horizon, the angles thereat will always be 0° 0', and .. the secants become the radii of those angles, and are the same or = in every curve; hence the comparative strength at the crowns, of two arches having the same span and height, is reciprocally as the radii of curvature at those points.
Prob. 2.--To find the height of the superincumbent wall, or extrados, above every point in an arch, so that by its pressure all the parts of the arch may be kept in equilibrio.
Rule. The height of the crown multiplied by the cube of the radius, and the product divided by the vertical height of the required point from the horizontal diameter, gives the height for a circular arch.
For an ellipsis-The height of the crown by the cube of the semiconjugate axis, and the product divided by the cube of the vertical height of the point from the transverse axis. Rules for other curves I will omit here.
From all which it appears, that a whole arch of about 108 or 110 degrees, is the part of a circle which may be used for most bridges with the least impropriety, the thickness at the crown being about the 16th part of the span, with a horizontal wall at top.
It is evident the longer the arch stones are, the more stable and secure the whole arch will be: it is of the greatest importance to have arch stones made as long as may be consistent with economy.
The best and safest way, to give the whole masonry of the wall, over the arch stones, the same position of joints as these stones themselves have, that is, to have the joints in the direction perpendicular to the curve of the arch, up to the top of the road way: by this means, the whole has the effect of arch stones, ensuring strength so complete, as to render a deviation from the theory of no effect.
Of the thickness of the voussoirs at the crown, there have been many differing examples; in general it may be put down from to of the span.
On the horizontal drift or shoot of an arch, and the thickness of the piers.
Prob. 3.-To find the thickness of the piers of an arch,
necessary to keep the arch in equilibrio, or to resist its drift or shoot, independent of any other arches.
Preliminary Observations.-If a bridge shall consist of but one arch between the abutments, and each abutment does not rest against an immovable object, such as the bank of a river, it will be evident, if the materials composing the arch have a tendency to yield to any pressure in the direction of the length of the bridge, that the effect of this lateral thrust will be either to push the piers off horizontally, or to overturn them. In either case the thrust must be counteracted, by giving a proper thickness to the abutments. The same precaution ought to be observed in a series of arches.
No theory, purely mathematical, has yet been discovered for determining the equilibration of an arch in this respect. The principle which has been generally assumed as the basis of the investigation is, that the materials of which each semi-arch is composed, have a freedom of motion and a tendency to slide over the intrados by the action of gravity, and that they are retained by the pier.
Others have supposed them so firmly connected with each other, as to form one solid mass, acting like one rigid body only. The equivalent of all these forces being estimated in some particular direction, is considered as the force which tends to push off or overturn the pier or abutment.
To determine this force, it becomes necessary to find the centre of gravity of any longitudinal section of a semiarch; for in that point, by the laws of mechanics, the mass of material may be considered as collected. Or, if the force is to be estimated in a horizontal direction, it will be necessary to determine the situation of a vertical line passing through the centre of gravity. If the arch be perfectly equilibrated by its loading, and only such a portion of the arch is employed as will permit the extrados to serve
accurately for the intended roadway, then in an easy manner we may shew how to determine the situation of the vertical passing through the centre of gravity, and how from thence to determine the lateral thrust exerted by the arch.
It is advisable if possible to construct the piers of heavier materials than the arch, as by this means it will not require so great a thickness, and consequently greater water way will be obtained.
The piers are supposed equally thick: But, instead of that, it is very common to enlarge them towards the bottom, both to give it a broader base to stand on, and make the lever at the base longer, to oppose a greater resistance to its turning about the point for oversetting.
If the piers be constructed as supposed, that is, independently of the resistance arising from the abutments, or from the neighboring arches, a considerable saving in the centering of a bridge, where many arches are required, will be the result; for, the centre of one arch may be struck before the other is begun, and it will serve for its opposite and corresponding one, when of = sizes, which is commonly the case, unless under particular circumstances.
The thickness of the piers in most cases of practice, may be made about of the span of the arch. From what has been said, it is evident, that in order to estimate the proper thickness, the circumstances of the span, height, form of extrados and intrados, must enter into the consideration.
The best form of the protection for dividing the stream, is the triangle; the more acute the projection of the end against the stream, the better it will divide it, and the less will the force of water be against the pier; but it is sufficient to make the angle a right one, as it will make the work stronger.
In rivers on which large crafts pass the arches, it is better to make the ends semi-circular; for though it does not
divide the stream so well as the triangle, it will better turn off and bear the shock of the craft. Bridges are generally placed in a direction perpendicular to the river in a direct line, to give free passage to the water, strength to the bridge, easy construction, &c. A bridge should not be made in too narrow a part of a navigable river, for the breadth being still more contracted by the piers, will evidently increase the depth, velocity, and fall of water under the arches, which will endanger the bridge and navigation. If convenient, all the arches are made of the same size; which will cause a great saving of centering. Bridges are generally made with an odd number of arches. If the bridge has a rise from the ends to the middle, it may gradually rise from the ends to the largest arch in the middle; having the arches decrease from the middle towards each end, all similar segments.
Bridges should rather be of a few large arches, than of many small ones; this will leave a free passage for the water and navigation, and be a great saving of labor, &c.
Before any plan for a bridge is adopted, the engineer must make a careful survey of the river, its bed, shores, the character of the soil, the velocity of the water at the time of the highest flood, &c. A descriptive memoir, containing all possible information, should accompany the map and section.
After this careful survey is made, the form and dimensions of the abutments, piers, and arches, may be regulated according to local circumstances:—no general rule can be laid down for them.
Of Wing-Walls, &c.
In some cases, it is advisable to turn an inverted arch under the bridge, which will procure a safe and durable foundation and structure. The spread of the foundation will depend on the nature of the soil. The stone courses