152 Mr Buchanan’s Report on the Bridge of Suspension 
of strain upon every square inch. Experiments differ in some 
degree as to this particular, some carrying the strength much 
alter the figure of the curve of equilibrium, in every part of which, however, 
the strain still invariably depends on the two circumstances already mentioned, 
namely, the weight which that part sustains, and its obliquity. 
Without entering, however, into the mathematical laws which determine 
the direction of any part of the funicular curve, Avhatever be the disposition 
of the load, the strain on the arch of a suspended bridge may be easily deter- 
mined by two simple, but important, considerations: 1st, In the middle or low- 
est part of the curve, the strain in different arches increases exactly as their 
depth is diminished, and diminishes exactly as their depth is increased, suppos- 
ing the span to remain the same. In other words, this strain is inversely pro- 
portional to the depth, or versed sine of the curve : Secondly, When the 
depth is the eighth part of the span of the arch, then the strain in the centre 
is exactly equal to the whole weight of the bridge. In every arch, then, if 
the depth exceed the eighth part of the span, the strain in the centre will be 
proportionally less than the natural weight of the bridge ; and if the depth be 
less than the eighth part of the span, this strain will be in the same propor- 
tion greater than the natural weight. So that, in general, if the depth be the 
nth part of the span, then in every case the strain in the centre will be equal 
to the whole weight of the chains and roadway, augmented or diminished in 
the ratio of 8 to n. Suppose, for example, the depth to be the 16th part of 
the span, then n — 16, and the ratio of 8 to n or 8 to 16, is the same as that 
of 1 to 2. In the centre, then, the arch will be strained with a force equal to 
double the natural weight of the bridge. Suppose, again, the depth equal t© 
one-fourth of the span, then n = 4 and the ratio of 8 to n or 8 to 4, is the 
same as that of 2 to 1, or 1 to §, so that here the horizontal tension is only 
the half of the weight of the bridge. Let W denote the weight of the bridge 
and its load, and T the strain in the lowest part or the horizontal tension, 
nW 
then T = — g- » 
The horizontal tension, or the strain in the centre, is less than the strain 
on any other point, as this increases gradually towards the points of suspen- 
sion, where it is greatest of all. To find the strain there, however, we have 
only to add to the horizontal tension the nth part of the weight of the bridge ; 
wW W 
so that, if t denote the strain at the point of suspension, t = 1 . 
8 n 
The arch of the chains is usually considered as a catenary curve, and even 
in this vierv the above propositions are sufficiently near the truth for practical 
purposes ; for, when the depth is the eighth of the span, the strain in the centre, 
calculated in this manner, is only about ^th part too little. But they will be 
found much more exact, if we consider that, in suspended bridges, the curve is 
in general much nearer to a parabola than to a catenary, as the level roadway 
tends always to bring it nearer to this latter figure, in proportion as its weight 
exceeds that of the chains. The arch, in fact, is only a catenary, if we sup- 
pose the weight of the roadway to be as nothing, compared with that of the 
