The quantity h represents in any given case the height which is fixed, such as the height from grade to top of footing, height from grade to bottom of footing, height from underside of girder to top of footing, or height from underside of girder to bottom of footing, as the case may be. There is always a considerable range of lengths for which the quantities remain nearly constant. The formula gives values a trifle greater than those for which the quantities are a minimum, since the use of heavier sections will reduce slightly the unit costs of the concrete.

Reinforced-concrete piles should be used under footings when a suitable foundation is to be found only at a considerable depth, or when a very large footing-area would be required in order to reduce the pressures to a proper amount. A comparison must be made for each case as it arises, allowing properly for the costs of the column shaft, the footing, the piles, and the excavation. This latter item must not be overlooked.

The curves of Figs. 56t to 56y, inclusive, of "Bridge Engineering," will be found of great value in studying the questions of economy of reinforced-concrete-girder bridges, as most of the points involved can be settled directly thereby.

Arch Bridges

Economic Rise with Span-Length Unchanged

The economics of arch bridges are much more complicated than those of girder bridges. The important factors are the costs of the arch ribs and those of the piers or abutments; and the main economic point to determine is that of ratio of rise to span-length. For any fixed span-length, the greater the rise, up to a limit of one-third of the opening, the smaller will be the cost of both arch-ribs and piers. By increasing it further, up to the limit of one-half of the opening, the cost of the rib will be but little augmented, and the cost of the pier above the springing will be increased, while that of the portion thereof below the same will be reduced. If the increase in rise is secured by lowering the springings, the greater the rise the greater the economy of material and cost; but if the increase must be secured by raising the grade, the springing remaining at a fixed elevation, it will rarely be economical to increase the rise above the limit of one-third of the opening. The exact limit in any case will depend upon the distance from the springing to the bottom of the base, and upon the massiveness of the pier-shafts above the springing; also upon the spans of the arch-ribs resting on the pier, and upon the character of the substructure employed. If the springing is but little above the top of the base, a comparatively-low rise will be economic.

If the pier carries two arch-spans of the same length, and if the live loads are small as compared with the dead loads, a low ratio of rise to span-length will be economic. On the other hand, if the distance from the springing to the bottom of the base be great, the live load large as compared with the