hear due to Travelling Loads._--Let a uniform train weighing w per
ft. run advance over a girder of span 2c, from the left abutment. When it
covers the girder to a distance x from the centre (fig. 41) the total load
is w(c+x); the reaction at B is

R_2 = w(c+x)x(c+x)/4c = w/4c(c+x) squared,

[Illustration: FIG. 41.]

[Illustration: FIG. 42.]

which is also the shearing force at C for that position of the load. As the
load travels, the shear at the head of the train will be given by the
ordinates of a parabola having its vertex at A, and a maximum F_{max.} =
-1/2wl at B. If the load travels the reverse way, the shearing force at the
head of the train is given by the ordinates of the dotted parabola. The
greatest shear at C for any position of the load occurs when the head of
the train is at C. For any load p between C and B will increase the
reaction at B and therefore the shear at C by part of p, but at the same
time will diminish the shear at C by the whole of p. The web of a girder
must resist the maximum shear, and, with a travelling load like a railway
train, this is greater for partial than for complete loading. Generally a
girder supports both a dead and a live load. The distribution of total
shear, due to a dead load w_l per ft. run and a travelling load w_l per ft.
run, is shown in fig. 42, arranged so that the dead load shear is added to
the maximum travelling load shear of the same sign.

[Illustration: FIG. 43.]

24. _Counterbracing._--In the case of girders with braced webs, the tension
bars of which are not adapted to resist a thrust, another circumstance due
to the position of the live load must be considered. For a train advancing
from the left, the travelling load shear in the left half of the span is of
a different sign from that due to the dead load. Fig. 43 shows the maximum
shear at vertical sections due to a dead and travelling load, the latter
advancing (fig. 43, a) from the left and (fig. 43, b) from the right
abutment. Comparing the figures it will be seen that over a distance x near
the middle of the girder the shear changes sign, according as the load
advances from the left or the right. The bracing bars, therefore, for this
part of the girder must be adapted to resist either tension or thrust.
Further, the range of stress to which they are subjected is the sum of the
stresses due to the load advancing from the left or the right.

[Illustration: FIG. 44.]

[Illustration: FIG. 45.]