OF PIERS.
THE piers and abutments of a bridge must be so con strucced, that each arch may stand independent of its neighbours. For though, by the mutual abutment of arch against arch, the whole may rest upon very slender piers, if once the structure is erected ; yet, as they must be formed singly, and are exposed to many acci dents, it will be best to contrive them, that the destruc tion of one arch may not involve in it that of the whole.
Some of the writers, on the principles of bridges, in treating this department of their subject, have found it necessary, by the help of the higher calculus, to find the centre of gravity of the semi-arch. The solution of the problem, we are convinced, so far as it is useful in prac tice, lies much nearer the surface.
The reader has already frequently seen, that the ul timate pressure may, in every case, be reuuced to two others, viz. the weight of the semi-arch above, and the horizontal thrust. In the equilibrated arch, this pres sure is directed perpendicularly to the joints of the sec tions ; and these being usually drawn at right angles to the curve, the pressure is in the direction of the tan gent to the arch. Hence, we have often called it the tangential pressure. Upon this principle, however, when the curve springs at right angles to the horizon, an in finite pressure is required in the vertical direction,— a supposition which cannot have place in practice. We must accordingly call in the assistance of friction in that case ; a force which may be set in opposition to the ho rizontal thrust, and which, increasing with the superin cumbent weight, very fortunately keeps pace also with what it is intended to oppose.
Granting, then, that the friction is so contrived, upon the principles already explained, that there is no danger of any slide at the horizontal or springing joint ; it will he readily admitted, that no slide is likely to take place in any horizontal course below that, till we arrive at the foundation ; for the disturbing force is constant, but the friction increases as we descend. Our principal care
then must be, that the pier does not overset, by turning on the farther joint E, Plate LXXX!. Fig. 6, of its base, as a fulcrum. Take a in the horizontal Joint, A a as the centre of pressure. Draw uN" to represent the weight of the semi-arch, and N'T the horizontal thrust ; then 'la is the ultimate pressure : and if, when produced, it falls nr ithin the base of the pier, it is perfectly obvious that it can never overturn it. And this k altogether in dependent of the weight of the pier ; for if that were a mass of ice, immersed to the springing in water, the case would he exactly the same.
But the pier itself has a considerable stability, arising from its own weight ; and even though the direction of the ultimate pressure of the arch alone pass out of the base, the tendency to overturn the pier may be balanced by its weight. This weight may be supposed concen trated in the centre of gravity of the pier, and of course to act in the vertical line which bisects it.
Its effect will be nearly found by laying off in that line from the point g, where the direction of the ulti mate pressure of the arch intersects it, qr= to the weight of the pier, and taking q.s= the ultimate pres sure =aT. and completing the parallelogram, the dia gonal drawn from g will represent the direction and magnitude of the united pressure of the arch and pier. This is not strictly accurate ; it would be so if a and g coincided, which is the case with a single arch standing on a pillar : but in general, the ultimate pressure is still more favourable than this. Its direction at any point is in the tangent of a curve, which approaches the vertical as we descend, since the proportion ari sing from the weight of the pier increases with its height.
In order to find analytical expressions for these forces, let the horizontal thrust of the arch =t. The weight of the half arch =a, and that of the pier =/1, the height of the pier to the springing of the arch =h, the breadth at the base =b.