railway trestles. The economic lengths will be greater for light live loads and timber decks than for heavy live loads and concrete decks.

The economics of column spacing for bents when cantilever brackets are employed is an interesting little problem, but the final determination must be in accordance with good judgment as well as economy; for if the spacing be too small, rigidity is likely to be sacrificed. Upon certain assumptions of approximate correctness the mathematical solution of this problem is a possibility; but the equations involved would be so complicated that it is much better for any particular case to assume two or three spacings, compute the total weight of metal in the bent for each, and find the one which will give approximately the least weight of metal. If the columns are placed at the quarter points of the beam, the dead-load bending-moment at the middle will be approximately zero; and if the effect of stress reversion is ignored, the direct and the reverse bending moments for the central portion of the beam will be equal, and this arrangement would be about the most economical possible. But if reversion is considered, the sectional area of the middle-portion of the beam must be greater than that of the outside portions, hence for economy its length should be somewhat less than one-half of the total, and the columns would then be spaced somewhat closer than when they are located at the quarter points. The fact that the brackets are usually lighter near the outer ends than at the inner ones would, for economy, tend to draw the columns together; but, on the other hand, this would increase the weight of the splices and connecting details. The proper column spacing to adopt will depend upon the length of the columns; for it is easily conceivable that the structure could be so high and so narrow that the quarter-point spacing would be too close for proper resistance to wind pressure. Again, in such a case the wind load might be so great as to necessitate an increase in column section above that required to care for the live and dead load stresses only; and thus the effect of wind pressure would enter the economic study. It will be found in most cases that it is inadvisable to space the columns much less than one-half of the total length of the beam.

Elevated Railroads

In respect to the economics for Elevated Railroads, as long ago as 1896 the author, when designing the Northwestern Elevated and the Union Loop Elevated Railroads of Chicago, determined certain economic functions for such structures and published the results in a paper entitled "A Study in Designing and Construction of Elevated Railroads, with Special Reference to the Northwestern Elevated Railroad and the Union Loop Elevated Railroad of Chicago, Ill.," which paper was presented to the American Society of Civil Engineers and published in its "Transactions" for 1897. From it the following statements concerning the economics of elevated-railroads have been excerpted: