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The nine 1-inch bars give a much better distribution of the metal inside of the concrete. The superficial area of the nine 1-inch bars is 18 square inches per linear inch of the concrete beam, while the area of the four 3-inch bars is only 12 square inches per inch of length. But an even greater advantage is furnished by the fact that we have nine bars instead of four, which may be bent upward (and bent more easily than the 3-inch bars) as fast as they can be spared from the bottom of the concrete beam. In this way the shear near the end of the concrete beam may be much more effectually and easily provided for. Since the shear is greatest at the ends of the concrete beam, more bars should be reserved for turning up near the ends. For example, in the above case of the nine bars, one or two bars might be turned up at about the quarter-points of the concrete beam. One or two more might be turned up at a distance equal to, or a little less than, the depth of the concrete beam from the quarter-points toward the abutments. Others would be turned up at intermediate points; at the abutments there should be at least two, or perhaps three, diagonal bars, to take up the maximum shear near the abutments. This is illustrated, although without definite calculations, in Fig. 101. This will be illustrated by a numerical example.

A concrete beam having a span of 18 feet supports one side of a 6-inch concrete slab 8 feet wide which carries a live load of 200 pounds per square foot. In addition, a special piece of machinery, weighing 2,400 pounds, is located on the concrete slab so near the middle of the concrete beam that we shall consider it to be a concentrated load at the center of the concrete beam. The concrete floor area carried by the concrete beam is 18 feet by 4 feet = 72 square feet. Adding 3 inches to the 6 inches thickness of the concrete slab as an allowance for the weight of the concrete beam, we have 9 X 12 = 108 pounds per square foot for the dead weight of the concrete floor. With a factor of 2 for dead load, this equals 216. Using a factor of 4 on the live load (200), we have 800 pounds per square foot. 'Then the ultimate load on the concrete beam, due to these sources, is (216 + 800) the reliability of the whole calculation. Therefore the rules which have been suggested for a prevention of this form of failure are wholly empirical. Mr. E. L. Ransome uses a rule for spacing vertical stirrups, made of wires or i-inch rods, as follows: The first stirrup is placed at a distance from the end of the concrete beam' equal to one-fourth the depth of the concrete beam; the second is at a distance of one-half the depth beyond the first stirrup; the third, three-fourths of the depth beyond the second; and the fourth, a distance equal to the depth of the concrete beam beyond the third (see Fig. 100). This empirical rule agrees with the theory, in the respect that the stirrups are closer at the ends of the concrete beam, where the shear is greatest. The four stirrups extend for a distance from the end equal to 212 times the depth of the concrete beam. Usually this is a sufficient distance; but some "systems" use stirrups throughout the length of the concrete beam.

On very short concrete beams, tIe shear changes so rapidly that at 212 times the depth from the end of the concrete beam the shear is not generally so great as to produce dangerous stresses. With a very long concrete beam, the change in the shear is correspondingly more gradual; and it is possible that stir- nips or some other device must be used for a greater actual distance from the end, although for a less proportional distance. When the diagonal reinforcement is accomplished by bending tip the bars at an angle of about 45°, the bending should be done so that there is at all sections a sufficient area of steel in the lower part of the concrete beam to withstand the transverse moment at that section.

Are You in Pelham New Hampshire? Do You Need Concrete Cutting?

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