ECONOMIC DESIGN OF BEAMS. The most economic propor tion of steel depends upon (1) the relative cost of a cubic unit of the steel and the concrete, (2) the ratio of the coefficients of elasticity of the two materials, and (3) the unit working stress for each ma terial. The problem of determining the most economical proportion of steel is capable of a mathematical solution; but different results are obtained according to which one of the two following initial assump tions is made: (A) the stress in the concrete is constant or (B) the stress in the steel is constant; and also according to which one of the three following limiting conditions of design is assumed; (a) breadth of beam constant, (b) depth of beam constant, (c) ratio of breadth to depth constant. Finally the results differ still further according to the beam formula employed (§ 445).
The only practicable method of solving the problem is to make a solution for both of the conditions A and B for each of the condr tions a, b, and c, using the values for the ratios 1, 2, and 3 above that fit the case in hand; and then by inspection select the most economical result. For an example of such a solution, see
Engineering News, Vol. l vii, pages 686-88. The following conclusions are from that article.
1 " When the depth is unlimited, a girder will come under condi tion c, that is, breadth divided by depth constant; and a floor slab will come under condition b, that is, breadth constant.
2. " When the depth of the beam is limited, the cheapest beam is very probably an over-reinforced beam.
3. " Whenever an over-reinforced beam is the cheapest, there is a fairly wide range of percentages of reinforcement within which the cost will vary only slightly." Turneaure and Maurer's Principles of Reinforced Concrete Construction, pages 175-84, gives a method of solution along the above lines, and also several diagrams to facilitate its application in practice.