BRIDGE. 



509 



ly. the actual breadth. Are we to look upon this near 

 coincidence aa the effect of chance, of science, or the 

 habit of the builder ? We rather think of the first. 



When 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 di- 

 minished, 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 4-0, 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 calcu- 

 lation be completed, as in the first example, the thick- 

 ness 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 



^ a * ' r , we have for the thickness of 



the 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 feet, or, in other words, 

 an arch of 120 degrees, let the height ot" the pier 

 and vertical thickness be as before. The radius will 

 be 65-i 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 7.4(>, and a similar increase be- 

 comes necessary in the other cases. 



If 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 exactly 40 feet if a horizontal arch with joints 

 drawn to a radius of 130 feet be introduced in its 

 stead. The enormous thickness of pier which be- 

 comes necessary for these fiat segments, precludes, in 

 a great measure, the possibility of employing them in 

 practice ; and indeed we do know, that a horizon- 

 tal 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 al- 

 ready given some idea of the greatest possible arch 

 of stone or brick ; a segment of that circle may, of 

 course, be employed 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 intimately on the thickness of 

 piers. We ought to k;n>w the dimensions of the 

 largest pier that can be trusted, and this, we con- 

 ceive, depends chiefly on the care of the mason; for 



stone, and especially cement, is a compressible tub- Thory. 

 stance ; and when an arch is very flat, a very small S**" 

 yielding at the springing produces an enormous de- 

 pression at the crown, insomuch that there may be 

 reason to dread, lest the arch pass down below the 

 hori/.ontal lin.~, 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 abut- 

 ments, and appears equally difficult of remedy. 



In great hori/.ontal thrusts, where th-: segment ii 

 flat, the immersion of the pier in water comes to have 

 an important effect. On the weight of the pier, in 

 those cases, bh<.. btahiliiy chiefly depends, nnd a de- 

 duction from th. - of two fifths must be compensated 

 by enlarging the thickness. For example, in the arch 

 of 100 feet span, with '25 feet rise, and piers 20 feet 

 high, the ring of stones of 3 feet at the crown may 

 be set on a pier of l-l feet broad, taking the half 

 arch at 180 feet. But if the pier be set in water to 

 the springing, it will lose J- of its weight ; and its 

 breadth must be increased nearly to 1G| feet ere it 

 has the same degree of stability as before. The 

 truth is, that in this case the stability derived from 

 the pier itself is nearly as much as that derived from 

 the arch, (conceiving this always concentrated in the 

 middle of the half of the pier,) a diminution of f 

 from the pier, therefore is f of the whole, and must 

 be provided for by an increase of breadth, not just 

 equal to f ; for we must observe, that the stability 

 derived from the arch is also increased thereby. 



But indeed the immersion of the pier, if it be very 

 tall, that is, if the depth of water be great in propor- 

 tion to the span, will demand attention, although the 

 arch should not be very Hat. In such a case, the 

 stability arising from the pier is often as great as 

 that which is derived from the weight of the arch. 

 It can seldom be greater, and consequently can sel- 

 dom require an addition of more than one fifth ot 

 that breadth, which would be sufficient were there 

 no immersion. 



We might easily give a theorem for this in rec- 

 tangular piers ; but it is hardly worth while ; tin; 

 effect of any addition is easily determined in the first 

 formula, which we think, on the whole, although 

 only tentative, the most convenient rule for the prac- 

 titioner. 



But although the total immersion, even of a lofty 

 pier, will seldom require any great alteration in the 

 thickness, there is yet another circumstance which 

 well deserves attention. Bridges are often built, espe- 

 cially in a tide-way, with the arches springing below 

 the high waters ; we have in that case a diminution 

 from the weight of the arch itself, but unless the 

 keystone be under water, the horizontal thrust is un- 

 changed ; we must, accordingly, in our calculation, 

 make the same diminution for that part of the arch 

 which is thus immersed, as we did in the above ex- 

 ample for the piers. The result will oblige us still 

 more to increase the thickness of pier. 



On the whole, we may conclude from this investi- 

 gation respecting the piers, that the increase of 

 breadth which may be, and usually is given to the 

 pier, is of much less importance, on account of the 

 weight that is thereby gained, than by its increasing 



