M_c = ½w_ec^2;
and putting x = ½c, 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 at F. If this process is repeated
for all positions of the load, we get the influence line AGB for the
bending moment at C. The area AGB is termed the influence area. The
greatest moment CG at C is x(l-x)/l. To use this line to investigate the
maximum moment at C due to a series of travelling loads at fixed distances,
let P_1, P_2, P_3, ... be the loads which at the moment considered are at
distances m_1, m_2, ... from the left abutment. Set off these distances
along AB and let y_1, y_2, ... be the corresponding ordinates of the
influence curve (y = Ff) on the verticals under the loads. Then the moment
at C due to all the loads is