Chap. VIII.
487
Mr. John Southern stated that the weight of a foot lohg of the road, at any point a, oughtto be to the weight of the same length at d , as the cube of the cosine of the arch c b is tothe cube of the radius, or as the line de is to the line df wherein d is thecentre of the arch c h , b, the summit or crown, a b, a tangent to the curve atby c e parallel to a by af at right angles to da, and ca and b f continuations ofthe radii d c and d h respectively. In the case of the proposed bridge,wherein the point c at the abutment is about 24 £ degrees from b , the weightof a foot long of the road, at the abutment to that at the crown, is as threeto four nearly.
As the weight of the arch will not, in fact, be inconsiderable, this differencewill not be so great as this rule makes it, but unless the real or the relativeweights of the parts of the bridge were given, it is not easy to express theratio required.
The fourth question demanded, What pressure will each part of the bridgereceive, supposing it divided into any given number of equal sections, theweight of the middle section being known; and on what part, and with whatforce, will the whole act upon the abutments ?
Dr. Nevil Maskelyne , upon the supposition that the whole acts as one frame of iron, statedthat there would be no horizontal thrust acting on the abutments, and the bridge will restand press perpendicular to the horizon upon the two abutments, each of which will supporthalf its weight, and if the thickness of the rib frames be made in the proportion mentionedin the answer to the third question, the stress will be alike on every part.
Rev. A. Robertson said, proceeding from the crown to either of the abutments, and sup-posing half the bridge to be divided into sixty-two sections, each of them, except the last,over a portion of the arch 5 feet in length, the weights of the several sections will be as ex-pressed in the second column of the following table. The first column of the table containsthe numbers of the sections of half the bridge, reckoning from the crown. The secondcolumn contains the weights of each of the sections, the weight of each section being on thesame line with its number. The third column contains the whole weight of 123, &c. sections,reckoning from the crown of the bridge : in this table the weight at the crown is supposedto be 1. If, therefore, any weight be determined upon for the crown, the weight of anysection, regulated according to this determination, may be obtained by multiplying theweight so determined upon, by the weight of the same section in the table.
Fig. 483.
15/ Col.
2d Col.
3 dCol.
1st Col.
2 d Col.
3rf Col. \
Is/ Col.
2d Col.
3 d Col.
1
4*9969
4*9969
22
5-0120
110*8500
43
5-4374
221 -5345
2
5*0009
9*9978
23
5*1228
115-9728
44
5-4791
227-0136
3
5*0022
15*0000
24
5*1342
121 *1070
45
5-4990
232*5126
4
5*0031
20*0031
25
5*1459
126*2529
46
5-5311
238*0437
5
5-0052
25-0033
26
5-1585
131*4114
47
5-5524
243-5961
6
5*0075
30*0158
27
5*1713
136*5827
48
5-5743
249*1704
7
5*0105
35*0263
28
5-1847
141*7674
49
5-6047
254*7751
8
5*0150
40-0413
29
5-1987
146*9661
50
5-6319
260*4070
9
5*0164
45*0577
30
5-2134
152*1795
51
5*6600
266 0670
10
5-0220
50-0797
31
5-2247
157*4047
52
5*6887
271-7557
11
5-0231
55*1028
32
5*2437
162*6484 '
53
5-7182
277*4789
12
5-0324
60*1352
33
5*2604
167*9088
54
5-7484
283*2223
13
5*0375
65*1727
34
5*2652
173*1740
55
5-7797
289*0020
14
5*0439
70*2166
35
5-3067
178*4807
56
5*8114
294 *8134
15
5*0509
75*2675
36
5*3127
183*7934
57
5*8399
300*6533
16
5*0580
80*3255
37
5*3314
189*1248
58
5-8778
306*5311
17
5*0656
85-3911
38
5-3504
194-4752
59
5-9120
312*4431
18
5*0740
90*4651
39
5*3704
199*8456
60
5-9400
318*3831
19
5*0827
95-5478
40
5-3909
205-2365
61
5-9908
324*3739
20
5*0937
100*6415
41
5*4082
210-6447
62
5*1764
329*5503
21
5-0965
105-7380
42
5-4524
216-0971
The pressure against each abutment, in a direction parallel to the horizon, as 724*8,the weight at the crown being 1. If, therefore, a weight be determined upon for thecrown, the pressure at each abutment will be found to be 724*8 times this weight. Ifthe weight over a certain portion of the arch adjoining to the abutment be given, theweight at the crown may be found as mentioned above, and the horizontal pressure atthe abutment may then be determined. Thus, if the weight of matter over a portion of thearch 10 feet in length, adjoining to the abutment, be 340 tons, the weight at the crown willbe 28*354, and the horizontal pressure at each abutment will be 20550*9792 tons. In this
i i 4