RAILWAY.
193
a high velocity, and a large value to a long bridge and a lower velocity. If we put g =32‘2 feet, l = length of bridge in feet, V = velocity in feet per second, and S =central statical deflection in inches, we shall have *
0 = 24 - 15 ^.
When 0 is large, the following series will express the ratio of the central deflectiondue to the moving weight = D, to the central statical deflection = S.
13
+ 0 3
D _ 1 j>_
S ~ + 0 + 202
When 0 is equal to, or greater than 100, the first two terms of the series will be trueto the third place of decimals; substituting therefore the value of 0 we obtain D = S
4 V 2 S 2
+ —— Hence, for a given load, the increment of deflection due to velocity
varies nearly directly as the square of the velocity, and inversely as the square of thelength of the bridge. But this “is on the supposition that the mass of the bridge isinsignificant.
When the inertia is taken into account, the problem becomes so complicated thatits complete solution seems to elude the present powers of analysis. It appears,however, that when 0 is less than about unity, the inertia will diminish the centraldeflection due to the moving weight. When 0 is greater than unity, the inertia willat first increase the deflection, and bring it to a maximum, and then diminish it.An approximate solution of this problem has been obtained by Mr. Stokes, for thetwo extreme cases in which the mass of the body or mass of the bridge areneglected; and although in the cases similar to those in practice, where the massesare very nearly equal, the effects are so mixed up together as not to be as yet fullydeveloped, the following Table, which has been calculated empirically, will serve toshew roughly the increments of statical deflection due to different conditions ofbridges, until further experiments and a perfected theory shall have determined thequestion more exactly.
In the following Table, l = length of bridge in feet, V = velocity of moving weightin feet per second, S = central statical deflection, or deflection due to the weight atrest on the centre.
Values of
I 2
24-15
5
6
8
10
15
20
25
30
40
50
100
200
Increments of Swhen mass of baris neglected.
•30
•23
•18
•14
•10
•06
•05
•04
•03
•02
•01
•005
Increments due tothe inertia.
•25
•22
•19
•17
•14
•12
•11
•10
•09
•08
•05
•04
Total incrementsof statical deflec-tion.
•55
•45
■37
•31
■24
•18
•16
•14
•12
•10
•06
•045
To apply this Table to any given bridge, the statical deflection due to the greatestload which is liable to pass over it must be ascertained, and also the greatest probablevelocity: from these data, and from the length of the bridge, the value of
must be calculated. The increment of statical deflection which corresponds
VOL. III.
N