roduce somewhat severer straining than any probable actual
rolling loads. Now, except for very short bridges and very unequal loads, a
parabola can be found which includes the curve of maximum moments. This
parabola is the curve of maximum moments for a travelling load uniform per
ft. run. Let w_e be the load per ft. run which would produce the maximum
moments represented by this parabola. Then w_e may be termed the uniform
load per ft. equivalent to any assumed set of concentrated loads. Waddell
has calculated tables of such equivalent uniform loads. But it is not
difficult to find w_e, approximately enough for practical purposes, very
simply. Experience shows that (a) a parabola having the same ordinate at
the centre of the span, or (b) a parabola having the same ordinate at
one-quarter span as the curve of maximum moments, agrees with it closely
enough for practical designing. A criterion already given shows the
position of any set of loads which will produce the greatest bending moment
at the centre of the bridge, or at one-quarter span. Let M_c and M_a be
those moments. At a section distant x from the centre of a girder of span
2c, the bending moment due to a uniform load w_e per ft run is

M = 1/2w_e(c-x)(c+x).

Putting x = 0, for the centre section

M_c = 1/2w_ec^2;

and putting x = 1/2c, for section at quarter span

M_a = 3/8w_ec^2.

From these equations a value of w_e can be obtained. Then the bridge is
designed, so far as the direct stresses are concerned, for bending moments
due to a uniform dead load and the uniform equivalent load w_e.

[Illustration: FIG. 52.]

27. _Influence Lines._--In dealing with the action of travelling loads much
assistance may be obtained by using a line termed an _influence line_. Such
a line has for abscissa the distance of a load from one end of a girder,
and for ordinate the bending moment or shear at any given section, or on
any member, due to that load. Generally the influence line is drawn for
unit load. In fig. 52 let A'B' be a girder supported at the ends and let it
be required to investigate the bending moment at C' due to unit load in any
position on the girder. When the load is at F', the reaction at B' is m/l
and the moment at C' is m(l-x)/l, which will be reckoned positive, when it
resists a tendency of the right-hand part of the girder to turn
counter-clockwise. Projecting A'F'C'B' on to the horizontal AB, take Ff =
m(l-x)/l, the moment at C of unit load