By CHARLES LEAN, M. Inst. C.E.
_Bowstring Girders._--Having had occasion to get out the stresses in
girders of the bowstring form, the author was not satisfied with the
common formulæ for the diagonal braces, which, owing to the difficulty of
apportioning the stresses amongst five members meeting in one point, were
to a large extent based on an assumption as to the course taken by the
stresses. As far as he could ascertain it, the ordinary method was to
assume that one set of diagonals, or those inclined, say, to the
right-hand, acted at one time, and those inclined in the opposite
direction at another time, and, in making the calculations, the
apportionment of the stresses was effected by omitting one set.
Calculations made in this way give results which would justify the common
method adopted in the construction of bowstring girders, viz., of bracing
the verticals and leaving the diagonal unbraced; but an inspection of
many existing examples of these bridges during the passing of the live
load showed that there was something defective in them. The long unbraced
ties vibrated considerably, and evidently got slack during a part of the
time that the live load was passing over the bridge. In order to get some
definite formulæ for these girders free from any assumed conditions as to
the course taken by the stresses, or their apportionment amongst the
several members meeting at each joint, the author adopted the following
method, which, he believes, has not hitherto been used by engineers:
Let Fig. 1 represent a bowstring girder, the stresses in which it is
desired to ascertain under the loads shown on it by the circles, the
figures in the small circles representing the dead load per bay, and that
in the large circle the total of live and dead load per bay of the main
girders. A girder, Fig. 1A, with parallel flanges, verticals, and
diagonals, and depth equal to the length of one bay, was drawn with the
same loading as the bowstring. The stresses in the flanges were taken
out, as shown in the figure, keeping separate those caused by diagonals
inclined to the left from those caused by diagonals inclined to the
right. The vertical component of the stress in the end bay of the top
flange of the bowstring girder, Fig. 1, was, of course, equal to the
pressure on the abutment, and the stress in the first bay of the bottom
flange and the horizontal component of the stress in the first bay of the
top flange was obtained by multiplying this pressure by the length of the
bay and dividing by the length of the first vertical. The horizontal
component of the stress in any other bay of the top or bottom flange of
the bowstring girder--Fig. 1--was found by adding together the product of
the stress in the parallel flanged girder, caused by diagonals inclining
to the right, divided by the depth of the bowstring girder at the left of
the bay, and multiplied by the depth of the parallel flanged girder; and
the product of the stress caused by diagonals inclining to the left
divided by the depth of the bowstring girder at the right of the bay,
multiplied by the depth of the parallel flanged girder. Thus the
horizontal component of the stress in D=
_ _
| Stress caused by diagonals Length of right Depth of parallel |
| leaning to left. vertical. flanged girder. |
| | +
|_ 15.75 × 1/4.5 × 10 _|
_ _
| Stress caused by diagonals Length of ver- Depth of parallel |
| leaning to right. tical to left. flanged girder. |
| |
|_ 24 × 1/8 × 10 _|
= 65; and the vertical component =
Horizontal component. Length of bay.
65 × 1/10 × (8.0 - 4.5) = 22.75.
In the same way the horizontal and vertical components of the stresses in
each of the other bays of the flanges of the bowstring were found; and
the stresses in the verticals and diagonals were found by addition,
subtraction, and reduction. These calculations are shown on the table,
Fig 1B. The result of this is a complete set of stresses in all the
members of the bowstring girder--see Fig. 2--which produce a state of
equilibrium at each point. The fact that this state of equilibrium is
produced proves conclusively 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 formulæ: 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 formulæ 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.
w¹= live load per bay of girder.
W = total load per bay of girder = w + w¹.
Then: S/B = N.