BRIDGES and veiy 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 foot run. Let we be the load per foot run which would produce the maximum moments represented by this parabola. Then we may be termed the uniform load per foot equivalent to any assumed set of concentrated loads. Mr Waddell has calculated tables of such equivalent uniform loads. But it is not difficult to find we, approximately enough for practical purposes, very simply. Experience
Fig. 12. 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 Mc and Ma 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 we per foot run is M = W- (c-x) (c+x). Putting x=0, for the centre section Mt= WeC" 2~ and putting x = ic, for section at quarter span 2 M ='_ 3w7ec T~’ From these equations a value of wt 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 we. 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. 13 let A'B' be a
Fig. 13. girder supported at the ends and let it be required to investigate the bending moment at O' due to unit load in any position on the girder. When the load is at F', the reaction at B' is mjl 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
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counter-clockwise. Projecting A'F'C'B' on to the horizontal AB, take — 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 at1 C due to a series of travelling loads at fixed distances, let I 1) I: I^j ... be the loads which at the moment considered are at distances m-,, m2, . . . from the left abutment. Set off these distances along AB and let iq, y2, . . . 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 M = P1y1 + P2y2+ . . . The position of the loads which gives the greatest moment at C may be settled by the criterion given above. For a uniform travelling load w per foot of span, consider a small interval Fk — Am on which the load is wAm. The moment due to this, at C, is ivm(l-x)Am/l. But m(l-x)Am/l is the area of the strip F/hk, that is yAm. Hence the moment of the load on Am at C is wyAm, and the moment of a uniform load over any portion of the girder is x the area of the influence curve under that portion. If the scales are so chosen that a inch represents one inch ton of moment, and b inch represents one foot of span, and iv is in tons per foot run, then ab is the unit of area in measuring the influence curve. If the load is carried by a rail girder (stringer) with cross girders at the intersections of bracing and boom, its effect is distributed to the bracing intersections D'E' (Fig. 14), and the part of the influence line for that bay (panel) is altered. With unit load in the position shown, the load at D' is (p-n)/p, and that at E' is njp. The moment of the load at C is m{l-x)/l-n(p-n)lp. This is the equation to the dotted line RS (Fig. 13). If the unit load is at F', the reaction at B' and the shear at O' is mjl, positive if the shearing stress resists a tendency of the part of the girder on the right to move upwards ; set up F/'= vijl (Fig. 15) on the vertical under the load. Repeating the process for other positions, we get the influence n H line AGHB, for the x, shear at C due to unit load anywhere on the girder. GG = xjl and Fiv. 14. CH = - (Z - x)jl. The lines AG, HB are parallel. If the load is in the bay D'E' and is carried by a rail girder which distributes it to cross girders at D' E', the part of the influence line under this bay is altered. Let n (Fig. 16) be the distance of the load from D', Xj the
distance of D' from the left abutment, and p the length of a bay. The loads at D', E, due to unit weight on the rail girder are ip - n)!p and n/p. The reaction at B'is {{p-n)x1 + n{x1+p)} jpl. Thex shear at C' is the reaction at B' less the load at E', that is { P{ i + n) - nl)lpi, which is the equation to the line DH (Fig. 15). S. II. — 48
