Let us consider for simplicity a very narrow portion of the beam, having the full length and depth, but so narrow that it includes only one set of fibres, one above the other, as shown in Fig. 91. In the case of a plain, rectangular, homogeneous beam, the stresses in the fibres would be as given in Fig. 90; the neutral axis would be at the center of the height, and the stress at the bottom and the top would be equal but opposite. If the section were at the center of the beam, with a uniformly distributed load (as indicated in Fig. 89), the she' ar would be zero.
A beam may be constructed of plain concrete; but its strength will be very small, since the tensile strength of concrete is compar atively insignificant. Reinforced concrete utilizes the great tensile strength of steel, in combination with the compressive strength of concrete. It should be realized that the essential qualities are com pression and tension, and that (other things being equal) the cheap est method of obtaining the necessary compression and tension is the most economical.
258. Economy of Concrete for Compression. The ultimate compressive strength of concrete is generally 2,000 pounds or over per square inch. With a factor of safety of four, a working stress of 500 pounds per square inch may be considered allowable. We may estimate that the concrete costs twenty cents per cubic foot, or $5.40 per cubic yard. On the other hand, we may estimate that the steel, placed in the work, costs about three cents per pound. It will weigh 480 pounds per cubic foot; therefore the steel costs $14.40 per cubic foot, or 72 times as much as an equal volume of concrete or an equal cross-section per unit of length. But the steel can safely withstand a compressive stress of 16,000 pounds per square inch, which is 32 times the safe working load on concrete. Since, however, a given volume of steel costs 72 times an equal volume of concrete, the cost of a given compressive resistance in steel is !II (or 2.25) times the cost of that resistance in concrete. Of course, the above assumed unit prices of concrete and steel will vary with circumstances. The advantage of concrete over steel for compression may be somewhat greater or less than the ratio given above, but the advantage is almost invariably with the concrete. There are many other advantages in addition, which will be discussed later.
259. Economy of Steel for Tension. The ultimate tensile strength of ordinary concrete is rarely more than 200 pounds per square inch. With a factor of safety of four, this would allow a working stress of only 50 pounds per square inch. This is generally too small for practical use, and certainly too small for economical use. On the other hand, steel may be used with a working stress of 16,000 pounds per square inch, which is 320 times that allowable for con crete. Using the same unit-values for the cost of steel and concrete
as given in the previous section, even if steel costs 72 times as much as an equal volume of concrete, its real tensile value economically 372to (or 4.44) times as great. Any reasonable variation from the above unit-values cannot alter the essential truths of the economy of steel for tension and of concrete for compression. In a reinforced concrete beam, the steel is placed in the tension side of the beam. Usually it is placed from one to two inches from the outer face, with the double purpose of protecting the steel from corrosion or fire, and also to better insure the union of the concrete and the steel. But the concrete below the steel is not considered in the numerical calcu lations. Even the concrete which is between the steel and the neu tral axis (whose position will be discussed later), is chiefly useful in transmitting the tension in the steel to the concrete. Although such concrete is theoretically subject to tension, and does actually con tribute its share of the tension when the stresses in the beam are small, the proportion of the necessary tension which the concrete can furnish when the beam is heavily loaded, is so very small that it is usually ignored, especially since such a policy is on the side of safety, and also since it greatly simplifies the theoretical calculations and yet makes very little difference in the final result. We may therefore consider that in a unit-section of the beam, as in Fig. 92, the concrete above the neu tral axis is subject to compression, and that the tension is furnished entirely by the steel.
260. Elasticity of Concrete in Com= pression. In computing the transverse stresses in a wooden beam or steel I-beam, it is assumed that the modulus of elastic ity is uniform for all stresses within the elastic limit. Experimental tests have shown this to be so nearly true that it is accepted as a mechanical law. This means that if a force of 1,000 pounds is required to stretch a bar .001 of an inch, it will require 2,000 pounds to stretch it .002 of an inch. Similar tests have been made with concrete, to determine the law of its elasticity. Unfortunately, concrete is not so uniform in its behavior as steel. The results of tests are somewhat contradictory. Many engineers have argued that the elasticity is so nearly uniform that it may be considered to be such within the limits of practical use. But all experimenters who have tested concrete by measuring the propor tional compression produced by various pressures, agree that the additional shortening produced by an additional pressure, say of 100 pounds per square inch, is greater at higher pressures than at low pressures.