One of the loftiest bridges with which we are ac quainted is that of Alcantara, over the Tagus, in Spain. It is stated by Don Antonio Ponz, in his Viage di Espana, to consist of six arches, the two largest 110 feet in span, the water at the lowest is 42 feet deep ; from the sur face of which, to the beginning of the springing of the middle arches, 87 ; and from thence to the upper sur face, 76 ; which, with the 4 feet and a half of parapet, make the whole 205 feet and a half, (more correctly, 209i). Taking then the thickness at crown as equiva lent to 16 feet, and the diameter 110, the thickness for an infinite height should be 42 feet. They are 38 in thickness, and 129 feet high. Let us now try this thick ness by the general formula given in the earlier part of this section.
The lower or immersed part 42 feet high, and 38 broad, is 1596; but of this are to be deducted on ac count of the immersion, leaving for that part 958. The pier from thence to the springing is 87 by 38, or 3306. We must suppose such a pier built up between the arches to at least Is of the height, or about 20 feet ; but on account of a set off which appears in the design, we shall suppose the breadth still 38 on an average, which makes 760, and the whole pier 5024, and its half is 2512. To this add of the semi-arch ; say x55 x16=660, and we have 3172. By this number let us divide the product of the horizontal thrust and height of pier, that is, 16x55 x129=113520, and we find about 36 Feet, very near the actual breadth. Are we to look upon this near coincidence as the effect of chance, of science, or the habit of the builder ? We rather think of the first.
Vhen the arch is a segment less than a semicircle, a greater thickness of pier becomes necessary. For the span continuing the same, we must either make the arch a part of a circle of greater radius, which would increase the horizontal thrust, or we must, in order to obviate that, diminish the thickness at the crown. In either case the weight of the arch is diminished, and with it the assistance which it gives to the stability of the pier.
Take a segment of 100 feet span and the versed sine 40, and suppose the pier 18 feet high, and the arch 6 feet thick in the crown, as in last example. The radius of this arch will be 51.25, and the thrust 307.5. The weight of this arch will be less than the former; let us take it at 110.0, and if the calculation be completed, as in the first example, the thickness of pier will be found =5.35 feet.
But suppose the pier carried no higher than the spring and ring of archstones, six feet thick, firmly bonded into it. The half arch will be 443 cubic feet ;
the thrust will remain as before ; and from the formula b—igt 2 have for the thickness of the 4h 4h pier 13.35 feet.
And for a ring of stones 2 feet thick, we have 9.35 feet only.
As another example, take a segment of 100 feet span with a rise of only 25 legit, or, in other words, an arch of 120 degrees, let the height of the pier and vertical thickness he as before. The radius will be 651. feet, and the thrust, where the crown is 6 feet thick, will be 393, taking the half arch at 775 ; we have for the pier 7.46, and a similar increase becomes necessary in the other cases.
the versed sine of the same arch be reduced to 10 feet, the radius is then 130 feet, and thrust =780, the arch being taken as every where 6 feet, we find very nearly 40 feet as the thickness of pier : it will be exact ly 40 feet if a horizontal arch with joints drawn to a ra dius of 130 feet be introduced in its stead. The enor mous thickness of pier which becomes necessary for these flat segments, precludes, in a great measure, the possibility of employing them in practice ; and indeed we do know, that a horizontal arch of 100 feet must be, in a great measure, a visionary structure.
There is an interesting subject of enquiry, which might not be unappropriately noticed here, we mean the lowest versed sine that can be used for arches in pro portion to the span. We conceive this, however, as in a great measure a practical question. We have already given some idea of the greatest possible arch of stone or brick ; a segment of that circle may, of course, be em ployed in any situation, but the piers (if the arch be of considerable span and height to the springing) must be made very great. Indeed the investigation depends in timately on the thickness of piers. We ought to know the dimensions of the largest pier that can be trusted, and this, we conceive, depends chiefly on the care of the mason ; for stone, and especially cement, is a com pressible substance ; and when an arch is very flat, a very small yielding at the springing produces an enor mous depression at the crown, insomuch that there may be reason to dread, lest the arch pass down below the horizontal line, and fall to pieces before the stability of the abutments can be acted upon. A compression in the joints is equivalent to a yielding at the abutments, and appears equally difficult of remedy.