Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

ARCH.
round the circumference, occasions a strain on the hoop
7
= 2 ^ 22 t^ie t^rust’ or near]y 238 tons. A square inch
of the worst iron, if well forged, will carry 24 tons with
perfect safety ; therefore a hoop of 7 inches broad and 1^
inch thick will completely secure this circle from bursting
outwards. It is, however, much more completely secur¬
ed ; for, besides a hoop at the base of very nearly these
dimensions, there are hoops in different courses of the
cone, which bind it into one mass, and cause it to press
on the piers in a direction exactly vertical. The only
thrusts which the piers sustain are those from the arches
of the body of the church and the transepts. These are
most judiciously directed to the entering angles of the
building, and are there resisted with insuperable force
by the whole lengths of the walls, and by four solid masses
of masonry in the corners. Whoever considers with at¬
tention and judgment the plan of this cathedral, will see
that the thrusts of these arches, and of the dome, are in¬
comparably better balanced than in St Peter’s church at
Rome. But to return from this sort of digression,
the friction of the joints and the cohesion of the cement.
An equilibrium, accompanied by some firm stability, pro- '
duced by the mutual pressure of the vertical joints, may
hyVdx? -}- dy2 ^ d2x
be expressed by the ^ or
hy Vdx1 -f- djft _ dx dt_^ t js somg variable
\fhyVdx>+dy2 dxt
positive quantity, which increases when x increases. This
last equation will also express the equilibrated dome, if t
be a constant quantity, because in this case — is = 0.
Since a firm stability requires that
hydx V dx2 + dy2
J' hy s/dx1 + dy1
shall be greater than d2x, and CG must be greater than
CB; hence we learn that figures of too great curva¬
ture, whose sides descend too rapidly, are improper. Also,
since stability requires that we have ^<,j2 + thJ_
d2x
the curves t]ie thrust b C, exerted by all the courses above HILK,
proper for an(j ^g force j or the weight of that course, be every-
domes. w^grg coincident with b B, the element of the curve, we
shall have an equilibrated dome: if it falls within it, we
have a dome which will bear a greater load ; and if it falls
without it, the dome will break at the joint. We must
endeavour to get analytical expressions of these conditions.
Therefore draw the ordinates b d b", BDB", C d C". Let
the tangents at b and b" meet the axis in M, and make
MO, MP, each equal to be, and complete the parallelogram
MONP, and draw OQ, perpendicular to the axis, and pro¬
duce b F, cutting the ordinates in E and e.1 It is plain
that MN is to MO as the weight of the arch HAS to the
thrust bC which it exerts on the joint KL (this thrust
being propagated through the course HILK) ; and that
MQ, or its equal b e, or d d, may represent the weight of
the half HA.
Let AD be called x, and DB be called ?/. Then be=dx,
and eC = dy (because 6 C is in the direction of the ele¬
ment j3 b). It is also plain that if we make dy constant,
BC is the second fluxion of x, or BC = cPx, and be and
6E may be considered as equal, and taken indiscriminate¬
ly for dx. We have also bC — V dx1 + dy2. Let h be the
depth or thickness HI of the arch-stones. Then hVdx? -j- dy2
will represent the trapezium HL; and since the circum¬
ference of each course increases in the proportion of the
radius y, hyVdx2 + dy? will express the whole course.
If f be taken to represent the sum or aggregate of the quan¬
tities annexed to it, the formula will be analogous to the
fluent of a fluxion, and J hyVdx? + dy2 will represent the
whole mass, and also the weight of the vaulting, down
to the joint HI. Therefore we have this proportion,
bV,
f hyVdx2 + dy2 : hyVdx? + dy2 :
— bd: CG, = dx : CG.
be
b e : CG,
Therefore 06 = ^"+^
Jhy V/dx? -f- dy2
If the curvature of the dome be precisely such as puts
it in equilibrium, but without any mutual pressure in the
vertical joints, this value of CG must be equal to CB or
to d?x, the point G coinciding with B. This condition
will be expressed by the equation^-^jj^
hyVdx? -f dy2 <px
or, more conveniently, by jhys/d^^df-di'
form gives only a tottering equilibrium, independent of
Arch.
XVUIIlt;. JJUL l/U ICLU11J1 XAV/111 txno ovxi. 1/ WJ. p
Theory of 42. We have seen that if b G, the thrust compounded of greater than J hy V dx? + dy2, we learn that the upper
part of the dome must not be made very heavy. This,
by diminishing the proportion of i F to & C, diminishes the
angle CiG, and may set the point G above B, which will
infallibly spring the dome in that place. W e see here
also, that the algebraic analysis expresses that peculiarity
of dome-vaulting, viz. that the weight of the upper part
may even be suppressed.
hy V dx? + dy2 _ d?x , dt
The fluent of the equation ^ +T
is most easily found: it is I./Xy v7dr + ihf = L<fr +
L*, where L is the hyperbolic logarithm of the quantity
annexed to it. If we consider dy as constant, and cor¬
rect the fluent so as to make it nothing at the vertex, it
may be expressed thus, L; f hy s/dx1 4~ dy2 L « \>dx
f hy s/dx2 + dy2 T dx
— L % + L *. This gives us L ^ - L dyt
, fhy*Sdx? + dy2 _ dx
and therefore
This last equation will easily give us the depth of
vaulting, or thickness h of the arch, when the cuive is
. hy \/dx? + dy2 _ dtdx + td?x
given. For its fluxion is ^ “ yy >
^ adldx + cttd-x_ ;g ap expressed in known
and h = ydy >Jfaf+~dy2'
quantities; for we may put in place of ^ any power or
function of * or of y, and thus convert the expression
into another, which will be applicable to all sorts of
curves- dt
Instead of the second member ^ + 7> we might em-
ploy where p is some number greater than unity.
This will evidently give a dome having stability, because
the original formula ^^gi^will then be greater
7 paxv~ld?x
than d?x.
This will give h —
Each of
= t-. But this cases
ydyv V dx? + dif
these forms has its advantages when applied to particular
ad?x
Each of them also gives h =
ydy V dx? + dy2
when the curvature is such as is in precise equilibrium.
* The letters e and d are wanting in the plate; e ought to be at the intersection of 6 E and C c", and d at that of AD and C cT.