275. Table for Computation of Simple Beams. In Table XVII has been computed for convenience the ultimate total load on rec tangular beams made of average concrete (1:3:5) and with a width of 1 inch. For other widths, multiply by the width of the beam. Since = s II7; and since by Equation 23, for this grade of con crete, = 397 b and since for a computation of beams 1 inch wide, b = 1, we may write = 397 For 1 we shall substitute 12 L. Making this substitution and solving for we have = 265 d' L. Since b = 1, A, the area of steel per inch of width of the beam = .0084 d.
Example. What is the ultimate total load on a simple beam having a depth of 16 inches to the reinforcement, 12 inches.wide, and having a span of 20 feet? Answer. Looking in Table XVII, under L = 20, and opposite d = 16, we find that a beam 1 inch wide will sustain a total load of 3,392 pounds. For a width of 12 inches, the total ultimate load will be 12 X 3,392 = 40,704 pounds. At 144 pounds per cubic foot, the beam will weigh 3,840 pounds. Using a factor of 2 on this, we shall have 7,680 pounds, which, subtracted from 40,704, gives 33,024. Dividing this by 4, we have 8,256 lbs. as the allowable live load on such a beam.
276. Resistance to the Slipping of the Steel in the Concrete. The previous discussion has considered merely the tension and com pression in the upper and lower sides of the beam. A plain, simple beam resting freely on two end supports, has neither tension nor compression in the fibres at the ends of the beam. The horizontal tension and compression, found at or near the center of the beam, entirely disappear by the time the end of the beam is reached. This is done by transferring the tensile stress in the steel at the bottom of the beam, to the compression fibres in the top of the beam, by means of the intermediate concrete. This is, in fact, the main use of the concrete in the lower part of the beam.
It is therefore necessary that the bond between the concrete and the steel shall be sufficiently great to withstand the tendency to slip. The required strength of this bond is evidently equal to the difference in the tension in the steel per unit of length. For example, suppose that we are considering a bar 1 inch square in the middle of the length of a beam. Suppose that the bar is under an actual tension of 15,000 pounds per square inch. Since the bar is 1 inch square, the
actual total tension is 15,000 pounds. Suppose that, at a point 1 inch beyond, the moment in the beam is so reduced that the tension in the bar is 14,900 pounds instead of 15,000 pounds. This means that the difference of pull (100 pounds) has been taken up by the concrete. The surface of the bar for that length of one inch, is four square inches. This will require an adhesion of 25 pounds per square inch between the steel and the concrete, in order to take up this difference of tension. The adhesion between concrete and plain bars is usually considerably greater than this, and there is therefore but little question about the bond in the center of the beam. But near the ends of the beam, the change in tension in the bar is far more rapid, and it then becomes questionable whether the bond is sufficient.
Although there is no intention to argue the merits of any form of patented bar, this discussion would not be complete without a statement of the arguments in favor of deformed bars, or bars with a mechanical bond, instead of plain bars. The deformed bars have a variety of shapes; and since they are not prismatic, it is evident that, apart from adhesion, they cannot be drawn through the concrete without splitting or crushing the concrete immediately around the bars. The choice of form is chiefly a matter of d6igning a form which will furnish the greatest resistance, and which at the same time is not unduly expensive to manufacture. Of course, the deformed bars are necessarily somewhat more expensive than the plain bars. The main line of argument of those engineers who defend the use of plain bars, may be summed up in the assertion that the plain bars are "good enough," and that, since they are less expensive than deformed bars, the added expense is useless. The arguments in favor of a mechanical bond, and against the use of plain bars, are based on three assertions : First: It is claimed that tests have apparently verified the assertion that the mere soaking of the concrete in water for several months is sufficient to reduce the adhesion from t to 3. If this con tention is true, the adhesion of bars in concrete which is likely to be perpetually soaked in water, is unreliable.