These statements will be true under the conditions named, within the elastic limit of the material; but some advantage would be derived in the second case, and a more marked benefit in the third, if the load assumed to be a maximum were exceeded, or if the composite bar were tested to destruction; as, however, these effects would be outside the limiting conditions imposed, it must be a matter of judgment as to how far this reserve of strength may be considered of value.
If, instead of simply adding section to the bar, some part of the constant load is put upon the new section by the manner of attachment, the combination will, of course, be more effective.
To apply these considerations and illustrate the way in which the two methods of adding flange section work out when reduced to figures, the case will be supposed of a girder 6 feet deep, carrying a load of which one-third is dead and two-thirds live. To the flanges of this girder are added plates equal to 50 per cent. of the original areas, in order to reduce the stress of 7 tons per square inch to which the girder before strengthening is liable, the depth remaining substantially unaltered. With dead load only the original section would be stressed to 2·3 tons per square inch, the new section being then unstressed. Under full load the new and old material take 3·1 tons per square inch additional, making the modified stress on the original section 5·4 tons per square inch, as against 7 tons; or a reduction of 22 per cent. This compares with 33 per cent., the relief due to 50 per cent. increase of flange area under ordinary conditions of stress distribution.
Let the second method of strengthening the girder now be considered, using, for purposes of comparison, the same total amount of new material to increase the girder depth by an addition to the top flange. This section will be equal to the area of one flange, which, though it may be applied in many different ways, giving a greater or a less increase to the depth, would probably be used in some such manner as that shown in [Fig. 64], increasing the effective depth for live-load stress by nearly 10 inches.
Fig. 64.
The added material will, as in the previous case, leave the dead-load stress unaltered, or 2·3 tons per square inch. The stress in the bottom flange due to live load will, however, now be 4·1 tons per square inch, making a total stress of 6·4 tons per square inch, against 7 tons—the original stress. The reduction here is 8 per cent. only, as compared with 12 per cent., the relief due, under ordinary conditions, to an increase of effective depth from 6 feet to 6 feet 10 inches, and by the use of additional material, equal, as before, to one-half of the total flange areas before the alteration.
The effect on the top flange need not be here gone into in detail, but it may be said that, owing to the increase of gross section and of depth, the ultimate stresses of both the new and old material are greatly less than as given for the bottom flange.