S T 0

455

S T 0

Hi at .;-;-;-—

fc ut "® magnitude or weight increases with the length only,bg l , ' s weight must, from the principles already laid down,'he difference of the tangents of its abutments ; the5 n | th > therefore, must be in that ratio. Accordingly, weth e breadth at different distances from the vertex, inat /- a 0 me way with the weights of the sections : the breadthtbat° 01 us ' be double, and at 55° must be about triple oftt ar , a ' ’■be crown, and will increase still more rapidly after-8 **°rt S fl- b’ropcrtions such as these may answer well in thefor flight of steps for a flying staircase, but are quite unfitin Ul ' present purpose. When we recollect, however, that’ *'aun 'be extraordinary expansion towards the

s Ur e p 8 ' s materially corrected by the increased pres-to n °‘ 'be incumbent mass in that part, we are encouraged

8 <?Ci

P r °ceed; °nd

a little farther, and consider the effect of the

p. mode of effecting the equilibrium.

; li,i, 11.—The pressure of matter upon each section

I i Sth / e .ady been stated as proportional to tv x sw ; but tv

1 8 Uof> 6 ^'fibrence of the sines of the angular distances of the’ '-sssiv

><1

j 8 *ye abutments from the vertex, and s w is the mean

'lieT' SIne to the given thickness at the crown, when

K " O ''" 1 - - -

the

- r ° ad way is horizontal. We have therefore the pressure

te 8s difference of the sines X (mean versed sine -f tliick-3s y e *'tex.) But these pressures are also, from the theory,<< j difference of the tangents of these angular distances,^.‘he common mode of building, we must give the archV e "t thickness at the keystone, to resist the horizontal; bpon •’ ensu re stability, and bear the loads likely to come! I 'ea 3 "• We must also cover this part with a certain thick-. K 01 gravel, or other matter, so as to form a road-way.H(. n ® r yj n g pressure of the wheels of a loaded carriage,i 8 q f 48 propagated through this stratum of gravel, willMove • diffused as not to disturb the stone immediatelys. "b nor injure the bridge by splintering away its cor-^bis thickness is made as small as possible, that the01 not be unnecessarily elevated, and the roadwayerv ‘“ 1 -The other courses of arch-

il ri es ~ e * v ed nearly horizontal,tllfi

too > do not often differ much in thickness from th

at

l|| sr e ' l"°Wn. But although these things are pretty constant,

Wg a considerable degree of latitude in filling up the

i dot> 6t '! een back of the arch and the roadway. It may

r r CoV V ' t b su bstances varying in density, from the lightest

' u’i or ° r P um ' ce > °P en shiver, or chalk, to closely rammed

11 'iiri even solid masonry ; and it is not uncommon to make,

ways, open spaces in the mosonry of the spandrel,

. 11 ® 'bem above, so as still to support the roadway.

therefore, be proper for us to inquire, what

tr 'hi ? ns ''y requisite over every section of an arch, where

ar e | ' ess °f the crown is given, the roadway horizontal,

'be a "ulform thickness, and the angles of abutment

Jtie c „ evera l sections constant, that is, all drawn from the

&i ^'itr tFe ’ or > what is the same thing, let us suppose the

t 8i! ’ amt up to the horizontal roadway with parallel

q'^"t'e i t * len ' n 1 u ' r e, what is the proportion between the

V.^orne by each section in this way, and the pressure

s le 'le nK .° riu m ; we shall thereby discover the ratio in which- ■ - -

(if ^

t8 t;r V6 j e arc b, that the advantages of equilibration may be

the backing must, if needful, be diminished ;Quantity of expansion necessary towards the spring-

k ’ even ' n 'bis state of things.

i^Oty ,, e We give a more rigid determination, we should wishbe e * e . Practical builder, that the solution of this problemk i a bl e8 asi y approximated, by the help of the trigonometri-es ti,_' -b or we may suppose the matter of the arch-stones

I Sa "te in specific gravity with that which lies above

‘ ( ' 11 as . a8 'here can be no impropriety in considering the'ygonal, from joint to joint, our mean versed sine

is only half the sum of those at the two joints. The suppo-sition is not strictly accurate, but it is sufficiently near .greater strictness would only serve to render the calculationmore complicated, without making it more useful.

“ It was customary in the construction of bridges, to fill upthe haunch with solid matter, such as gravel, earth, or thelike, until a roadway of a proper slope was procured. Wherethe arches were small, this might not be attended with anyperceptibly bad effect, provided the arch-stones were of agood depth. But the necessity of lightening the haunches,has been forced upon the attention of builders, wheneverlarge arches have been attempted.

“ In all probability, the first inventors of this mode ofbuilding, besides employing it with the view of equilibratingthe arch by lightening the part over the haunches, had alsoan idea of steadying it by the lateral abutment. Theyappear to have considered the spandrel walls as a sort ofhoops, that would keep the parts of the arch together, andhinder any stone from moving, by their great friction, inertia,and mutual abutment. Hence various ingenious modes havebeen employed for locking them into the back of the arch-stones, propagating the pressure through, and securing themfrom sliding away at the bases.

“ They indeed act in this way; nevertheless the equilibra-tion of the arch should be attended to in their construction,that every unnecessary strain may be avoided. The thick-ness of these walls may be varied indefinitely, and the vacantspaces made in any proportion to the solid parts. The wallsought to be near each other, that their effect may be felt overthe whole arch, and perhaps they should spread out towardsthe bottom ; but this is not so very necessary, for the coursesof arch-stones break joint with each other, and the inequalityof pressure in one course is immediately corrected by beingpropagated to the succeeding.

“ We have now determined a method of constructing anequilibrated arch for sixty degrees oh each side of the vertex;and this method, so far from having anything unusual, is evenstrictly analogous to that which is adopted by the practicalbuilder. Why then cannot we keep pace with him through-out, and give a construction for the entire semicircle ? Nodifficulty is felt by the mason in that case. He constructssuch arches every day. Nay, they are not only the mostcommon, but the most ancient of all arches. But the readermust have ere now observed, that our theory is in this par-ticular defective. The enormous expansion of the roadway,or the infinite height of superincumbent matter, which itseems to require when the joints are nearly horizontal, arealtogether preposterous and impracticable. We are sure theyare unnecessary; for many semicircular arches have existedfrom the time of the Romans, and are still in good order.What is more, the failure of such arches near the springing,where they differ farthest from the theory, is a most unusual,and, indeed, unheard-of phenomenon. Is our theory erro-neous, then, or is it only defective ? There is no reason fordistrusting any of the consequences we have hitherto deduced.They are mathematically derived from an unquestionableprinciple, the action of gravity. But we have not yet con-sidered all the causes of stability. The lateral resistance ofthe masonry, or other matter behind the arch, acts powerfullyin preventing any motion among its parts ; and independentlyof that, the friction of the arch-stones, assisted by the cohe-sion of the cement, affords a great security to the structure.We have even seen a semicircular ring of stones, abandonedto itself without any backing, and stand very well; longenough, at least, to admit of the other work being leisurelyapplied to it. Here was no lateral pressure ; no equilibra-tion : why did not the lower courses yield to the pressure

*

I