Granting that this proportion has been decided, and dealing first with the case in which the centre girder is simply attached to the cross-girders, and takes no dead load other than its own weight, then the live load carried by the outside girders, and previously borne wholly by them, will be reduced by the amount it is intended to transfer to the centre girder, and will become
Ll - (c × Lt) = live load on outer girders (2)
Ll being the total live load, and Lt the total dead and live load carried by the bridge. From this the deflection of the outer girders corresponding to this modified live load may be derived.
Fig. 70.
It is next necessary to ascertain the vertical movement, commonly a depression, of the cross-girders at the centre relative to their ends, when subject to the running load only, and supported at the middle and ends, the centre reaction being obtained as before indicated [(1)]. This movement will be the difference (if any) between the deflection on the whole span of the cross-girder due to the live load, and the upward flexure of the girder due to the centre reaction, considered as separate effects. Stress values having been estimated for the two conditions, these results may readily be deduced by simple flexure formulæ, observing that while the curve of moments due to live load sufficiently approximates to that for a distributed load to justify, for this, the use of a distributed load formula as given in the chapter “[Deflections],” the flexure due to the centre reaction will be but 0·80 of that which corresponds to the same stress for distributed loading. Or, the curve assumed by the girder under live load may be plotted by a method to be later explained.
The sum of the movements now determined—that is, the live-load deflection of the outer girders, and depression, as is commonly the case, of the cross-girders—will give the extreme depression (marked m in [Fig. 70]), from the dead-load condition of the middle cross-girders, when supported to the extent desired by a centre girder whose proportions are not yet known, but which, carrying the required percentage of the total load, must, subject to a reservation presently stated, deflect only this amount. The unit stress in the flanges of the new girder, governed by this flexure, will for a plate girder be