Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

ARCH.
411
Arch, distance below the springing of the spire. This part, be-
ing loaded with the great mass of perpendicular wall, is
fully able to withstand the horizontal thrust from the legs
of those arches. In many spires these thrusts are still
farther resisted by iron bars which cross the tower, and
are hooked into pieces of brass firmly bedded in the ma¬
sonry of the sides.
38. There is much nice balancing of this kind to be
observed in the highly ornamented open spires; such as
those of Brussels, Mecklin, Antwerp, &c. We have not
many of this sort in Britain. In those of great magni¬
tude, the judicious eye will discover, that parts, which a
common spectator would consider as mere ornaments, are
necessary for completing the balance of the whole. Tall
pinnacles, nay even pillars carrying entablatures and pin¬
nacles, are to be seen standing on the middle of the slen¬
der leg of an arch. On examination we find that this is
necessary, to prevent the arch from springing upwards in
that place by the pressure at the crown. The steeple of
the cathedral of Mecklin was the most elaborate piece of
architecture in this taste in the world, and was really a
wonder; but it was not calculated to withstand a bom¬
bardment, which destroyed it in 1578.
Such frequent examples of irregular and whimsical
buildings of this kind show that great liberties may be
taken with the principle of equilibration without risk, if
we take care to secure the base from being thrust out¬
wards. This may always be done by hoops, which can
be concealed in the masonry; whereas in common arches
these ties would be visible, and would olfend the eye.
39. It is now time to attend to the principle of equi¬
librium as it operates in a simple circular dome, and to
determine the thickness of the vaulting when the curve
is given, or the curve when the thickness is given. There¬
fore, let B 6 A (Plate XLIX. fig. 2) be the curve which pro¬
duces the dome by revolving round the vertical axis AD.
We shall suppose this curve to be drawn through the
middle of all the arch-stones, and that the coursing or hori¬
zontal joints are everywhere perpendicular to the curve.
We shall suppose (as is always the case) that the thickness
KL, HI, &c. of the arch-stones is very small in comparison
with the dimensions of the arch. If we consider any por¬
tion HA h of the dome, it is plain that it presses on the
course, of which HL is an arch-stone, in a direction b C
perpendicular to the joint HI, or in the direction of the
next superior element /3 6 of the curve. As we proceed
downwards, course after course, we see plainly that this
direction must change, because the weight of each course
is superadded to that of the portion above it, to complete
the pressure on the course below. Through B draw the
vertical line BCG, meeting [3 b, produced in C. We may
take Z> C to express the pressure of all that is above it, pro¬
pagated in this direction to the joint KL. We may also
suppose the weight of the course HL united in b, and
acting on the vertical. Let it be represented by b F. If
we form the parallelogram b FGC, the diagonal b G will
represent the direction and intensity of the whole pressure
on the joint KL. Thus it appears that this pressure is
continually changing its direction, and that the line, which
will always coincide with it, must be a curve concave
downward. If this be precisely the curve of the dome,
it will be an equilibrated vaulting; but so far from being
the strongest form, it is the weakest, and it is the limit
to an infinity of others, which are all stronger than it.
This will appear evident, if we suppose that b G does not
coincide with the curve A 5 B, but passes without it. As
we suppose the arch-stones to be exceedingly thin from
inside to outside, it is plain that this dome cannot stand,
and that the weight of the upper part will press it down,
and spring the vaulting outwards at the joint KL. But
let us suppose, on the other hand, that b G falls within Arch,
the curvilinear element b B. This evidently tends to push
the arch-stone inward towards the axis, and would cause
it to slide in, since the joints are supposed perfectly
smooth and slipping. But since this takes place equally
in every stone of this course, they must all abut on each
other in the vertical joints, squeezing them firmly toge¬
ther. Therefore, resolving the thrust b G into two, one
of which is perpendicular to the joint KL, and the other
parallel to it, we see that this last thrust is withstood by
the vertical joints all around, and there remains only the
thrust in the direction of the cuiwe. Such a dome must
therefore be firmer than an equilibrated dome, and can¬
not be so easily broken by overloading the upper part.
When the curve is concave upwards, as in the lower part
of the figure, the line b C always falls below b B, and the
point C below B. When the curve is concave downwards,
as in the upper part of the figure, b' C' passes above, or with¬
out b B. The curvature may be so abrupt, that even b’ G'
shall pass without b' B', and the point G' is above B'. It is
also evident that the force which thus binds the stones of a
horizontal course together, by pushing them towards the
axis, will be greater in flat domes than in those that are
more convex; that it will be still greater in a cone, and
greater still in a curve whose convexity is turned inwards;
for in this last case the line Z>G will deviate most remarkably
from the curve. Such a dome will stand (having polished
joints) if the curve springs from the base with any eleva¬
tion, however small; nay, since the friction of two pieces
of stone is not less than half of their mutual pressure,
such a dome will stand although the tangent to the curve
at the bottom should be horizontal, provided that the ho¬
rizontal thrust be double the weight of the dome, which
may easily be the case if it do not rise high.
40. Thus we see that the stability of a dome depends Stability of
on very different principles from that of a common arch,a dome de-
and is in general much greater. It differs also in another Pe.hds on
very important circumstance, viz. that it may be open in jj;ffergnt
the middle ; for the uppermost course, by tending equally from that
in every part to slide in toward the axis, presses all to-ofacorn-
gether in the vertical joints, and acts on the next course mon arch,
like the keystone of a common arch. Therefore an arch
of equilibration, which is the weakest of all, may be open
in the middle, and carry at top another building, such as
a lanthorn, if its weight do not exceed that of the circular
segment of the dome that is omitted. A greater load than
this would indeed break the dome, by causing it to spring
up in some of the lower courses; but this load may be in¬
creased if the curve is flatter than the curve of equilibra¬
tion : and any load whatever, which will not crush the
stones to powder, may be set on a truncate cone, or on a
dome formed by a curve that is convex toward the axis;
provided always that the foundation be effectually prevent¬
ed from flying out, either by a hoop, or by a sufficient mass
of solid pier on which it is set.
41. We have mentioned the many failures which hap¬
pened to the dome of St Sophia in Constantinople. We
imagine that the thrust of the great dome, bending the
eastern arch outwards as soon as the pier began to yield,
destroyed the half-dome which was leaning on it, and
thus almost in an instant took away the eastern abutment.
We think that this might have been prevented without
any change in the injudicious plan, if the dome had been
hooped with iron, as was practised by Michael Angelo in
the vastly more ponderous dome of St Peter’s at Home,
and by Sir Christopher Wren in the cone and the inner
dome of St Paul’s at London. The weight of the latter
considerably exceeds 3000 tons, and they occasion a hori¬
zontal thrust which is nearly half this quantity, the eleva¬
tion of the cone being about 60°. This being distributed