The criterion, equation (27), for the greatest bending moments in a
bridge is applicable to any truss whatever, whether the chords are
parallel or inclined, but it is not so with equation (33). If the
chords of the trusses are parallel, the quantity _i_ in equation (33)
becomes infinitely great, and the equation takes the following form:
_l_
_W₁ + W₂ + ... + Wₙ_ = ---- (_W₃ + W₄_ + etc.) (34)
_p_
Ordinarily the span _l_ divided by the panel length _p_ is equal to the
number of panels in the span. Hence equation (34) shows, in the case
of parallel or horizontal chords, that when the moving load is placed
for the greatest web stress in any panel, the total load on the bridge
is equal to the load in that panel multiplied by the total number of
panels.
=99. Influence Lines.=—A graphical method, known as that of “influence
lines,” is used for determining the greatest shears and bending moments
caused by a train of concentrated weights passing along a beam or
bridge-truss. Obviously it must express in essence that which has
already been shown by the formulæ which determine positions of moving
loads for the greatest shears and bending moments. In reality it is the
application of graphical methods which have become so popular to the
determination of the greatest stresses in beams and bridges.
=100. Influence Lines for Moments both for Beams and Trusses.=—It is
convenient to construct these influence lines for an arbitrary load
which may be considered a unit load; the effect of any other load will
then be in proportion to its magnitude. The results determined from
influence lines drawn for a load which may be considered a unit can,
therefore, be made available for other loads by multiplying the former
by the ratio between any desired load and that for which the influence
lines are found.
[Illustration: FIG. 25.—Bending Moment in a Simple Beam.]
_AB_ in Fig. 25 represents a beam simply supported at each end, so
that any load _g_ resting upon it will be divided between the points
of support, according to the law of the lever. Let it be desired to
determine the bending moment at the section _X_ produced by the load
_g_ in all of its positions as it passes across the span from _A_ to
_B_. Two expressions for the bending moment must be written, one for
the load _g_ at any point in _AX_, and the other for the load at any
point in _BX_. The expression for the first bending moment is
_z_
_M = g_ ----(_l-x_), (_a_)
_l_
and that for the latter
_l-z_
_M′ = g_ ---- _x_. (_b_)
_l_
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