43
TRACT II.
Examples. — Let there be allowed in the calculation for insurance weight upon a foot superficial at the vertex of anarch, fifteen times the real incumbent weight.
Let the real incumbent weight per foot superficial at the vertex of an arch be taken at €50lbs. avoirdupois. Then thereal weight + the insurance weight — 250 -f- (250 x 15) = 4<000lbs., a constant quantity being nearly the same as it will bededuced, the bridge of Neuilli is capable of sustaining on a foot superficial at the vertex. In timber bridges, thereal incumbent weight may be taken at 150. Then 150 + (150 x 15) = 2400lbs. But in these examples 4000lbs.are taken, (a)
Height of the key of
MATERIALS.
Weight of one cubicfoot of the materialin lbs. avoirdupois.
= g
Height of a prismaticcolumn one footsquare base, weigh-ing 4O0Olbs. avoir-dupois.
w 4000 ,
Modulus of fracture=/' or radius of cur-vature of an arch atthe vertex, support-ing only its ownweight in
Radius of curvatureof an arch at thevertex, when theheight of the key isone foot, and theincumbent weight4000lbs.
f
an arch, when theradius of curvatureat the vertex is 50feet, or the diame-ter of the circle ofcurvature is 100 feetand the incumbentweight 4000lbs.
w'
Versed sine or heightof a circular arch,when the height ofthe key is one foot,the semi-pan 28.11feet, and incumbentweight 4000lbs.
Semi-span of a cir-cular arch when theheight of the key isone foot, and theversed sine or heightis 28.11 feet,and theincumbent weight is4000lbs.
1+ —n
f-<
= X " C -V c 2 - yS
— y = v \2c — x)x
FEET.
FEET.
FEET.
FEET.
FEET.
FEET.
Brick.
135.5 #
29.52f
858$
28. 1 1 §
1.827||
28.11
28.11
Portland stone. .
151.75
26.36
4337
158.51
.307
2.513H
90.118 **
Cornish granite .
166.375
24.04
5502
219.72
.220
1.806
107.529
Aberdeen granite
164.06
24.38
9581
377.50
.127
1.049
142.943
Fir .
37.5
106.67
7680
71.33
.699
5.774
56.740
Oak.
52.81
75.74
10906
142.11
.348
2.808
86.847
Cast iron . . . .
439.56
9.1
51229
5072.17
.009
.078
558.996
(a) The heights of the keys obtained in the column of the heights of the key of an arch will not be found by a comparison with those in the column of the specificheight of the key in the Tables of Bridges in brick to vary more than might have been anticipated. In respect to the other materials, the same conclusion may be drawn,upon the supposition, that arches with such materials should be constructed with ribs or arcs doubleaux, after the practice of the Gothic architects : for example, supposea bridge were erected of Cornish granite with ribs five feet from the middle of one to the middle of another, and a foot wide, then the height would be 1.1 feet. In castiron, and the ribs 4 of an inch wide, the height would be 0.72 feet; and the limit of the radius of curvature, which in the preceding column is 5072.17., would become, onthe same supposition, only 70.273 feet. In fir, if the incumbent weight be taken at 2400lbs., and the interval between the ribs double the width of them = 2 feet, thenthe height of the key will be 1.26 feet, and a similar increase would take place in the versed sine, and decrease in the semi-span.
The third column of the table in M. Patte’s Edit. Blondel’s Cours D’Archi, vol. vi. 1757. page 194. contains the thickness of vaults at the key, which thickness has beendetermined, after a number of experiments, as he says, by taking the twenty-fourth part of the diameter of a semi-circular arch, to which one foot must be added, andone line per foot taken away from that sum. The same rule will serve for vaults surbaissees, taking the double of the radius of curvature at the vertex for the spanof the arch.
Perronet, Desc. des Proj. &c. des Ponts, page 625, says: “ It is the custom to make the length of the key for great arches which are ‘ surbaissees au tiers’ the twenty-fourth part of their diameter. It is proper to give to keys of small arches a foot in height more than this twenty-fourth part, and to diminish this sum in the ratio of oneline for each foot of the span of the arch, so that the key will be reduced to 27 and to 42 inches for arches of 36 and 72 feet. Semi-circular arches require a key alittle shorter and again, in page 664, he says, “ The thickness of semi-circular arches has been fixed, by a Treatise I have written on the subject, at the twenty-fourth partof their diameter, adding one foot to it, and diminishing the sum in the ratio of one line for every foot of the diameter. For arches surbaissees, the radius of curvatureat the vertex being doubled for the diameter, proceed as with semi-circular arches.
* 2168 4000
= 155.5lbs. See table of the strength of materials, j-
16
135.5
29.52. | See table. § c=-
/'
858
858
ivf 29.52 30.52
1 +— 1 +■-~
n 1
= 28.1 1.
29.52
/'
858
IS
■ = 1.827. f X = c— = 158.51 — 'V / iss.5it._28.11' ~ 2.513. ** y = //( 2e _ 1 (2 X 158.51)—28.11 128.11. — 90.118.