Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

406
Arch.
ARCH.
dz*
Then, by the general theorem, cC = -^3, r being the
radius of curvature. This, by the common rules, is
_ ‘fe* - This gives us
- dydl x — dxd2y & df
dyd^x
or —
arch, the span of which is 100 feet, and the height 40, Arch,
which are nearly the dimensions of the middle arch
Blackfiiars Bridge in London.
dxdhy
s zz C X L
quently C z=
a h % a h -\r It1
)>
L^a h 2
: h +A2)‘
the general value of 3/ zz s x
(ct, h V 2 a h
~^+A+a/(2«^ + 42)
a + ^ + V/ 2 a x d2
X C ; where C is a constant quan-
dy2
tity, found by taking the real value of cC in V, the vertex
of the curve. But it is evident that it is also zza + x—u.
dyd2x — dxdhf ^ ,, _ C
Therefore a x — u zz ^3 X u _ ^ x
fluxion of
dy
If we now substitute the true value of u (which is
given because the extrados is supposed to be of a known
form), expressed in terms of y, the resulting equation will
contain nothing but x and y, with their first and second
fluxions, and known quantities. From this equation the
relation of x and y must be found by such methods as
seem best adapted to the equation of the extrados. _
Fortunately the process is more simple and easy in the
most common and useful case than we should expect from
this general rule; we mean the case where the extrados
is a straight line, especially when this is horizontal. In
this case u is equal to 0.
PI. XLIX. Ex. To find the form of the balanced arch AVL, hav-
tfg. 6. ing the horizontal line cv for its extrados.
Keeping the same notation, we have wzzo, and therefore
C a . cdx
a + * = ^ X fluxKmof^.
d dx C
Assume dy — — \ then — zz v, and X fluxion of
dx_ _ Cvdv that ■ a + x = Therefore adx
dy dx ^ dx
+ xdx zz Cvdv; and by taking the fluents, we have
, ,2 ax x? „ .
2 aa? + a? = O2; and w = */ £; * Consequent¬
ly, zz 'd^'dx — /'being —Taking the flu-
V2 ax + x2' v '
ent of this, we have y — \/ C x L (2 a 2 x
+ 2 \^2 ax -{- x2). But at the vertex, where x zz o,
we have y zz y'C X L (2 a). The corrected fluent is
^a-\-x + V 2 ax + x?
therefore y zz \/ C X L ^ •
It only remains to find the constant quantity C. This
we readily obtain by selecting some point of the extrados
where the values of x and y are given by particular cir¬
cumstances of the case. Thus, when the span 2 s and
height h of the arch are given, we have
0
2
4
6
8
10
12
13
14
15
16
17
18
19
20
6,000
6,035
6,144
6,324
6,580
6,914
7,330
7,571
7,834
8,120
8,430
8,766
9,168
9,517
9,934
y
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
10,381
10,858
11,368
11,911
12,489
13,106
13,761
14,457
15,196
15,980
16,811
17,693
18,627
19,617
20,665
y
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
21,774
22,948
24,190
25,505
26,894
28,364
29,919
31,563
33,299
35,135
37,075
39,126
41,293
43,581
46,000
and conse-
Therefore
r (a x V 2 a x + x?\
L\ a )
zz s X -
)
26. As an example of the use of this formula, we sub¬
join a table calculated by Dr Hutton of Woolwich, for an
The figure for this proposition is exactly drawn accord¬
ing to these dimensions, that the reader may judge of it
as an object of sight. It is by no means deficient in
gracefulness, and is abundantly roomy for the passage of
craft; so that no objection can be offered against its be¬
ing adapted on account of its mechanical excellency.
"The reader will perhaps be surprised that we have Defects n|
made no mention of the celebrated Catenarean curve, the Cate-
which is commonly said to be the best form for an arch;nareatl
but a little reflection will convince him, that althoughcurve'
it is the only form for an arch consisting of stones of equal
weight, and touching each other only in single points, it
cannot suit an arch which must be filled up in the
haunches, in order to form a road-way. He will be more
surprised to hear, after this, that there is a certain thick¬
ness at the crown, which will put the Catenarea in equi-
librio, even with a horizontal road-way; but this thick¬
ness is so great as to make it unfit for a bridge, being such
that the pressure at the vertex is equal to the horizontal
thrust. This would have been about 37 feet in the
middle arch of Blackfriars Bridge. The only situation,
therefore, in which the Catenarean form would be proper,
is an arcade carrying a height of dead wall; but in this
situation it would be very ungraceful. Without troubling
the reader with the investigation, it is sufficient to inform
him, that in a Catenarean arch of equilibration the abscissa
VH is to the abscissa v h in the constant ratio of the ho¬
rizontal thrust to its excess above the pressure on the
vertex
27. Thus much will serve, we hope, to give the reader a
clear notion of this celebrated theory of the equilibrium
of arches, one of the most delicate and important appli¬
cations of mathematical science. Volumes have been
written on the subject, and it still occupies the attention
of mechanicians. But we beg leave to say, with great
deference to the eminent persons who have prosecuted
this theory, that their speculations have been of little
service, and are little attended to by the practitioner.
Nay, we may add, that Sir Christopher Wren, perhaps
the most accomplished architect that Europe has seen,
seems to have thought it of little value; for, among the
fragments which have been preserved of his studies, there
are to be seen some imperfect dissertations on this very
subject, in which he takes no notice of this theory, and
considers the balance of arches in quite another way.
These are collected by the author of the account of Sir
Christopher Wren’s family. This man’s great sagacit),