S T 0

458

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A cubic foot of stone weighs about 1601b. per foot ; andbrick weighs less. Suppose, therefore, the arch to be onefoot thick at the crown, and the keystone one cubic foot, itwill bear a horizontal thrust of 50,0001b. that is, 312^ timesits weight.

“ But, 50,000 : 160 : : r : Tang. IT O'' 3"', which will bethe angle of the keystone in that case. So that an archof 312-i feet radius, or a semicircular arch of 625 feet span,might bear to have a keystone of a foot deep, without riskingits being crushed more than in structures which have alreadystood for many years. And this may be called the limit ofstone-arch building ; for if we double the depth of the stone,we will thereby double the weight also, and its ratio to thehorizontal thrust will still be the same. Indeed this limitdoes not much exceed what has been actually executed. Aconsiderable portion of the bridge of Neuilly is an archof 250 feet radius ; and Gautier mentions a platband in thechurch of the Jesuits at Nismes, the camber of which, aftersettling, would make it a portion of an arch of 280 feet radius.The length or span is 26^ French feet, the rise only 4 inches,and therefore the diameter of its circle would be 560English feet.

“ ‘ This singularly bold platband was made under the con-duct of Pere Mourgues, after the design of Cubisol, an ablearchitect. The stones are 1 foot thick, their depth is 2 feettowards the key, and 2 feet 4 inches at each end. It had acamber given it of about 6 or 7 inches, and descended near3 inches on striking the centres.’ ”— Gautier.

“ We see, that the horizontal pressure does not determinethe vertical thickness of the arch-stone. But as we passdown the arch, it is plain that the butting surfaces mustincrease in proportion to the increasing tangential pressure.

“ At sixty degrees from the vertex, granting that the archis equilibrated, the depth of the arch-stones must be doubled ;and though the equilibration be carried no farther, yet, atthe springing or horizontal joint, a small increase will stillbe necessary. The ratio will soon be found. To the squareof the weight of the semi-arch, add the square of the hori-zontal thrust, the square root of the sum is the pressure atthe springing. If we divide this by the horizontal thrust,it will give the thickness at the springing, compared withthat which is necessary at the crown. Or if we divide itby 312£, it will give the smallest depth of joint which shouldbe used at the springing. The thrust and weight are sup-posed to be given in solid feet. If given in pounds, dividethe above quotient by 160, or divide at once by 50,000.

“ If we calculate upon the same principles, the depth ofarch-stone at the spring-course of a semicircle of 100 feetspan, 10 feet thick at the crown, we shall find it to be 5 feet,and at the crown the depth may be 19 inches. In the greatarches of the bridge of Neuilly , the thickness at the crown isabout 4 feet 8 inches, the span 128.2 feet, and height 32.The horizontal thrust is great, the crown being drawn witha radius of 150 feet; consequently, this arch would require adepth at springing of about 4 feet. But when the centrewas struck, the crown of this arch descended 23 inches,which has rendered it a portion of a much larger circle, andhas greatly increased the horizontal thrust.

“ The piers and abutments of a bridge must be so con-structed, that each arch may stand independent of its neigh-bours. For though, by the mutual abutment of arch againstarch, the whole may rest upon very slender piers, if once thestructure is erected; yet, as they must be formed singly, andare exposed to many accidents, it will be best to contrivethem, so that the destruction of one arch may not involve init that of the whole.

“ Some of the writers on the principles of bridges, in .

treating this department of their subject,

foun fl

ha lVthe cen '

necessary, by the help of the higher calculus, to nnu ^ e tp,tre of gravity of the semi-arch. The solution of theP 1 ^cbwe are convinced, so far as it is useful in practice, hnearer the surface. u ltim ate

“ The reader has already frequently seen, that the u ^ (pressure may, in every case, be reduced to two othe ^ ost>the weight of the semi-arch above, and the horizonta pgn-In the equilibrated arch, this pressure is directed P^ ua ]jydicularly to the joints of the sections ; and these b ein ? g j n thedrawn at right angles to the curve, the pressure 0 ften

direction of the tangent to the arch. Hence, we ' ,a ^ | )0 ^v'called it the tangential pressure. Upon this princip^ r j Z on,ever, when the curve springs at right angles to thean infinite pressure is required in the vertical r*eCsupposition which cannot have place in practice. ^ ase ; aaccordingly call in the assistance of friction in tha t j )rU st,force which maybe set in opposition to the li° r iz° n v ery

and which, increasing with the superincumbent ^ el ° jgd t0

fortunately keeps pace also with what it is inoppose. . up 0l l

“ Granting, then, that the friction is so contri j an ger 0the principles already explained, that there is n ° r e!iddyany slide at the horizontal or springing-joint; it w!l i l0 riz° n 'admitted, that no slide is likely to take place in a ”^ t j on ; f° rtal course below that, till we arrive at the foun a ^ ease a n 9the disturbing force is constant, but the friction mwe descend. . t j,at tb a

“ Figure 15.—Our principal care then must ^ o>

pier does not overset, by turning on the farther J . j nt)its base, as a fulcrum. Take a in the horizonta ^ e j,as the centre of pressure ; draw a v to represen n j aof the semi-arch, and v t the horizontal thrust, g yvitb' 1 ’the ultimate pressure : and if, when produced, ll . tt can neV f(the base of the pier, it is perfectly obvious that i ^ ^ vveig' 1overturn it. And this is altogether independent o ^ t o tbof the pier ; for if that were a mass of ice, im£n e gaine , ^springing in water, the case would be exactly l . ar isis»“ But the pier itself has a considerable sta > 0 f t 1from its own weight; and even though the dire t j, e b ase ’

out

A*

gilt

is

of t*‘ e

ultimate pressure of the arch alone pass °. ul , nce d bythe tendency to overturn the pier may be ba g( j jn

the

its

tb®

er-

11

weight. This weight may be supposed concencentre of gravity of the pier, and of course to actical line which bisects it. . ffin th flt ''g

“ Its effect will be nearly found by laying ? p ie8sU I1( ]from the point q, where the direction of the un® ^ pier, aof the arch intersects it, qr = the weight o _i e tinS . 6taking qs = the ultimate pressure = «t, an .|, C re prese nt ^parallelogram, the diagonal drawn from q W1 g f the a ^ adirection and magnitude of the united pressu ^ p e s0 iand pier. This is not strictly accurate ; ’*'. vv0 j e a rch f ta jand q coincided, which is the case with a sin ®, egsU re i 3 8 j aing on a pillar ; but in general, the ultimate P point 1 fl , emore favourable than this. Its direction a ^ rt j c althe tangent of a curve, which approaches t ie ^ e jgjit 0descend, since the proportion arising from 11pier increases with its height. . ,. r thes e 10 t j, e

“ In order to find analytical expressions ^ ^ e jght 0 ( ],elet the horizontal thrust of the arch = t- j ie jght ® . [ithalf-arch = a , and that of the pier ~P ’ 1 t [, e bre aapier to the springing of the arch = , to

the base = b. in A ft b*

“ 1. Then the horizontal thrust acting j crUI n,overturn the pier, and its force round 4 * ie ,. U u | a rdis taI1represented bymultiplying it by theperpendicviz., A X t.