This section is from the book "Cement And Concrete", by Louis Carlton Sabin. Also available from Amazon: Cement and Concrete.

Let us consider a concrete-steel beam twelve inches wide, twelve inches deep and of ten foot span, supported at the ends; reinforcement, one square inch of metal properly distributed in a plane two inches above the bottom of the beam. Let us suppose this beam carries a uniform load of 600 pounds per foot, giving a maximum bending moment of 90,000 inch-lbs., and a stress in steel of 10,000 pounds at the center. The ends of the steel bars are of course without stress. Since the bending moment at any section of such a beam is proportional to the product of the segments into which the section divides the span, the bending moment one foot from the ends will be.

Let us consider the neutral axis in the same position at the end of the beams as near the center. (This is not strictly true, because of the lighter stress near the ends of the beam, but the error made by such an assumption will be unimportant for our present purpose.) Then the tension in the steel will have the same proportion, or, tension in steel one foot from the end = X 10,000 = 3,600 pounds.

The stress in steel, then, which is zero at the end, has increased to 3,600 lbs. in one foot of length. To provide against poor contact near the end, consider two-thirds of this length, or eight inches, to be operative. If the reinforcement consists of four one-half-inch square bars, the necessary adhesion per square inch is = 57 lbs. per sq. in.; but if only one bar is used one inch square, the required adhesion is 114 lbs. per sq. in. The latter would not be good practice, not only because of high adhesion required, but because the steel is not properly distributed.

Where the stress in adhesion is greater than can be safely relied upon for plain rods, it is necessary to use some kind of deformed bar, or to anchor the bar securely at the end. This may be done by passing the end of the tension bar around a rod transverse to the beam near the end. Care should be taken that the safe value of adhesion is not assumed too high.

1 X 9.

5x5.

X 90,000 = 32,400 inch-pounds.

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