S T O

456

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propagated from above, and slide off ? It was' only theirfriction that could retain them. It is greatly increased bythis very pressure. And it is unquestionable, that a ring ofpolished blocks in that situation would not have hung togetherfor a moment. The force of friction, therefore, makes soimportant a part of our subject, that it deserves a separateinquiry. Let us see how it may be estimated.

“ When a mass of matter is moved along other matter ofthe same kind, the resistance produced by friction has beenusually stated at ^ of the weight. That of freestone, indeed,is supposed to be greater than perhaps it is And inthe case to which we are going now to apply it, there can belittle doubt, that, aided by the inertia of the stones, and thecohesion of the cement, the friction is even much more. Butthis force is inert ; and we are at present inquiring, how farwe are benefited by it in promoting the stability of our struc-i ture. It will, therefore, be proper to underrate it, at leastI until we discover how far we are warranted to say it mustbe beneficial.

“ Figure 12.—Let L, M, N, exhibit the three sections (10°each) of an arch, which we may conceive equilibrated abovethe section L, or 60° from the crown. Draw L T, expressingthe direction and magnitude of the ultimate pressure, perpen-dicular to the upper surface of L. In like manner, tv is thehorizontal thrust, and v L the weight of matter over L to thevertex. Draw the perpendicular t yb ; t l is the directionof the ultimate pressure when propagated to the lower sur-face of L ; y L is its tendency to makeL slide upwards alongthe joint. Now it is evident, that, if yL has to y t a lessratio than the friction has to the pressure, L will not move.Nay, what is more, L will itself have some weight. Take l ato represent it, which, in the case of equal sections, = thetangent x z. Draw t a for the ultimate pressure in the lowersurface of Jj, and a b for the force to be resisted by friction,in this case equal to .1343, or about ft of the pressure, andof course less than the friction, which will at least be one-third of the same.

“ Since L does not move upon the section M, they are tobe considered as one solid mass, and we pursue the pressurethrough the section M. For this purpose, lay off a c for theweight of M, draw the perpendicular t d, and the parallel c dto the joint mo, c d is the force opposed to friction in thatjoint, and still is less than one-third of t d, the pressurebeing, in the case of equal sections, == .2796, or about &Lastly, lay off 1 c e for the weight of the lowest section, N, anddraw as before. It is evident, that e f the force opposed byfriction here, is just equal to tv, the horizontal thrust, asmight have been concluded without any investigation. Inthe case of equal sections, its proportion to rf or v e, theweight of the semi-arch or perpendicular pressure, is as .4425,,or about 7 6 , which is probably more than the friction willoppose without other assistance.

“ If, therefore, the friction on the horizontal bed at theI springing be not equal to the thrust of the arch, we mustincrease it, as by dowelling it, for example, into the lower| stones, or by backing it with other masonry, or by increasing| the pressure on that joint, without altering the thrust of thearch, which may be done by thickening, or loading the archjust over the springing. And here the theorems for thei extrados of equilibration come to our aid j for we see, thatJ i any quantity of matter may be laid over the springing-! i courses, and, far from disturbing the arch, it will tend toj increase its stability.

I “ It may not be improper to inquire, what are the condi-1 tions for equilibrating an arch by means of the friction of itsi segments alone ? that is to say, what are the alterationspracticable in the position of the joints, or in the weights

over the several sections, until the tendency of eachto slide is just balanced by the friction at its lower su t |,e“ Whether we inquire into the position of the joints, theweight that may be applied, there are two cases ; g jj ( jingfriction being an inert force, will resist the stone meither upwards or downwards. . . 0 f the

“ I. Let it be required to determine the position ^joints in an arch, when each section is just prevensliding outwards by the friction at its lower surface- . 0 | n t,“ Figure 13.—Let the arch spring from a horizon j 0 gt

as n n, where, of course, the friction acting in T ?’, e gaveequal to the horizontal thrust, and must thereto ^jghto t n or vn the weight of the semi-arch, the ra the

O ^ fti WJ J D

friction has to the incumbent pressure, say -j- . ff n-

direction of the absolute pressure at the abutm re onTake n m the weight of the section N, t m is the P re ^ ^ v yiHthe joint of M, and making m t m similar to N t n, ^ itsalso represent the extreme friction in that joint, an ^ c> beload, and so on successively. Wherefore, if T f; ilhtang'e 9found, the joints of the arch may be drawn at r ° xaC tly *to these lines respectively, and every stone will be e j^ornthe predicament of N, that is, just kept by its * rlCsliding away. t e( j;

“ The positions of t m, t / may be readily ^' s ?° V e f 0 re> ' v ' ethe angle n' t m must be equal to n t m. If, “ ie , ^akinomake t a equal to t n, draw the tangent a u>, an ^ ina b~ n m, and joining t b, we have a t b = N T gU cce®'this manner, taking ab, b c, &c. for the weights ot . gBJ -r, th esive sections from the scale, and drawing lines ^ tl> uSjoints may be formed perpendicular to the

drawn. traction P er ' f

“ Figure 14.—But a more convenient const nt jty ®haps would be, to take the horizontal thrust, uTof these^friction in the vertical line C d, lay off the weight ^ oil

■ - - ‘ - also ** ® vigi ,

arch d a, draw C a, make C x equal to »wmodi the div>s*:the weight of the sections along a; z, and thiou D n^ ^ pai®

draw lines from the centreto these lines.

red

oi>s

lie 1

the joints requir

“ II. Let it be required, in the next place, to d« ter *”' ll

- ■ ■ that in w "

nine

the other limit to the position of the joints, A“‘ti.e"f r * ti0 °

each section is just prevented from sliding in, hy

on its lower bed. ..

acts

precise'^

“ Here it is evident, that as the friction . a ^ lO : 0 jiitsopposite to its direction in the former case, t l ^ ee 0 f ob 'have, on the opposite side, exactly the same ., e theref° re ’ j a y

Draw,

quity to the position of equilibrium. Draw, 1 ‘"" a c,tangent vy parallel to«c, cut it with coequal fro^ 1off the weights of the sections along vy, ^! IC 1

““ -.vi sciiu.., - j, ioints>

c ; these lines will exhibit the positions ot j ia ve n** rC b

, .>s

of course may be drawn parallel to them- vv Jj ]g half' ar g( jthese two limits of position in three joints o^ one .(jiird,above the same figure, assuming the friction a . a notaking the first section of 30° as equal to the t | ie ci 1-c °_ a9 .other arch might have been introduced as^we ^ t be co r

r tll #t

ual

IS

ponding Jine^frVy t -' ,an ^ Ie cda >makes«, when the ’ °F * n Czx > an angle eq

18° 26'• and ‘’ ? t,on ,s one-third of the pressure,

26° 34'. ’ pi le n \ en friction is one-half, thisformer oa> S1 , 10 ! 1 °f any joint, therefore, mayof the posit; ’ Z ln . I the . 'alter case 26» 34', onplace anions the e 4 u d‘bvium before any sliding 0‘ f . ee .

- , . e f lan one-iialfj perhaps it is two to

.. --- aa- 4'. And it is P'ftoiies,

- -rr.h-SW 1 M

sid®

eitl* r >

the pressure, which would give 33° 4.observe, that this is not confined to the an

nulus

m»y.

b»

of ar cl *

tions

oqserve, mat uiis is not uommeu iu —• gect iui ‘“ *\

but holds equally with whatever weight t t be P°

loaded. We may observe, then, that, in any a