that the rule above described and thus
applied, although possibly it may be considered empirical, results in the
correct solution of the question, and that the stresses shown are
actually those which the girder would have to sustain under the given
position of the live load. Figs. 2 to 10 inclusive show stresses arrived
at in this manner for every position of the live load. An inspection of
these diagrams shows: a. That there is no single instance of compression
in a vertical member of the bowstring girder, b. That every one of the
diagonals is subjected to compression at some point or other in the
passage of the live load over the bridge, c. That the maximum horizontal
component of the stresses in each of the diagonals is a constant
quantity, not only for tension and compression, but for all the
diagonals. The diagrams also show the following facts, which are,
however, recognized in the common formulae: d. The maximum stress in any
vertical is equal to the sum of the amounts of the live and dead loads
per bay of the girder. e. The maximum horizontal component of the
stresses in any bay of the top flange is the same for each bay, and is
equal to the maximum stress in the bottom flange. Having taken out the
stresses in several forms of bowstring girders, differing from each other
in the proportion of depth to span, the number of bays in the girder, and
the amounts and ratios of the live and dead loads, similar results were
invariably found, and a consideration of the various sets of calculations
resulted in the following empirical rule for the stresses in the
diagonals: "The horizontal component of the greatest stress in any
diagonal, which will be both compressive and tensile, and is the same for
every diagonal brace in the girder, is equal to the amount of the live
load per bay multiplied by the span of the girder, and divided by sixteen
times the depth of girder at center." The following formulae will give all
the stresses in the bowstring girder, without the necessity of any
diagrams, or basing any calculations on the assumed action of any of the
members of the girders:

Let S = span of girder.
D = depth at center.
B = length of one bay.
N = number of bays.
L = length of any bay of top flange.
l = length of any diagonal.
w = dead load per bay of girder.
w