Let s he the length of beam for which the steel at a is to carry the diagonal tension, and Td the total tension in the steel at a.
v = I is the unit diagonal tension on the concrete, and bs cos 45° is the section normal to the tension over which this unit diagonal tension is distributed. The total tension to be carried by the steel at a is The limits within which reinforcement is necessary, and its proper spacing, may be determined in the same manner as for vertical stir rups. Where diagonal reinforcement is used, the spacing should not exceed three-fourths of the effective depth of beans or s= d.
Bending up Horizontal Steel.—Diagonal reinforcement is com monly provided by bending up a portion of the horizontal steel near the supports where it is not needed for horizontal tension. In a simple beam uniformly loaded, the moment diagram is a parabola (see Fig. 51) and the diminution of the moment from the middle toward the ends is proportional to the square of the distance from the middle of the beam. Thus if ill is the moment at the middle AI=, the moment at a point distant x from the middle, and 1/2, the distance from the middle of the beam to the support.
The area of horizontal steel needed at any point varies directly with the moment at the point. If A is the area required at the middle and the area needed at any point distant x from the middle, A is the area of steel that it is allowable to turn up at distance x from the middle of the beam.
The Joint Committee does not consider diagonal tension reinforce ment to be fully effective unless it is firmly attached to the longi tudinal tension bars. When stirrups are looped about the longi steel, the Committee recommends that the allowable unit shear be made 42 per cent of the ultimate compressive strength, while when fully reinforced with bars firmly attached 6 per cent may he allowed.
109. Bond Resistance and Lateral stress carried by the steel in a reinforced concrete beam is transmitted to the steel through the bond existing between the concrete and steel.
Horizontal Tension Rars.—The amount of stress that may be transmitted to the horizontal steel at any cross-section of the beam is equal to the horizontal shear at the section.
If a does not exceed the safe unit bond stress between the steel and concrete at the section of maximum shear, the horizontal shear unay be communicated to the steel without danger of the bars slipping.
The Joint Committee recommend that the safe bond stress between concrete and plain reinforcing bars be limited to 4 per cent of the compressive strength of the concrete, and for good deformed bars not to exceed 5 per cent. For ordinary concrete (compression this would give a value for plain bars, u =SO lb./ and for the best deformed bars, u=100 lb. in 2 In selecting sizes of bars for horizontal tension steel, care should be taken that the bars are not too large to give sufficient surface area to provide properly for bond stress. Thus, suppose a beam; in which b = 0 inches, d = 10 inches, and j = 0.S5, requires for tension steel, A =0.60 in? If the maximum value of shear V=3200 lb. and allowable unit bond stress u=SO the required surface area of steel per inch of length, The b-inch bars are too large for the bond stress; the 44-inch bars are just sufficient; the bars are still better and would probably be selected.
Length of Bar to Prevent Slipping.—The stress carried by any reinforcing bar roust be transmitted to the concrete between the point at which the stress exists and the end of the bar, which trust be accomplished either by having a sufficient length of bar to develop bond stress equal to the maximum tension or by anchoring the bar by other means.
Let stress per square inch in the bar; diameter of bar in inches; u=allowable bond stress per square inch; length required for bond.
For round bar, the total If lb.'in.' and u=80 lb.'in.°, or for safety, the length between the point where the stress of 16,000 lb., in exists and the end of the bar must be 50 diameters.
Anchoring Bar by Bending.—When it is not feasible to secure the length of bar necessary for bond, the end of the bar may be anchored in the concrete by bending to a semicircle. Experiments indicate that, in general, the full strength of a bar in tension may be developed by bending the end to a semicircle, the diameter of which is four tunes the diameter of the bar. Short right-angled bends are found to be much less effective than curves through 180°.