As is readily seen from Fig. 47, diagonal compression equal to the diagonal tension exists in a direction at right angles to the tension. The diagonal compression is unimportant and need not be considered in beam design, as these stresses are always small in comparison with the compressive resistance of the concrete. It is, however, commonly necessary- to reinforce concrete beams to prevent diagonal tension cracks, as failures of beams frequently occur from this cause.
In a homogeneous beam, the maximum tension at any point on the tension side of the neutral axis is the resultant obtained by com bining the diagonal tension clue to shear with the horizontal tension due to moment at the same point. In a reinforced concrete beam, the steel is supposed to carry all of the horizontal tension, and the con crete none. Some horizontal tension will necessarily be carried by the concrete, but, if sufficient horizontal reinforcement be used, the reinforcement for diagonal tension need provide only for tensions due to shear.
Fig. 48 shows the form of failure likely to occur from diagonal tension, where horizontal reinforcement only is used. The diagonal tension at c becomes greater than the tensile strength of the con crete and the concrete cracks. A horizontal crack above the steel then follows, which separates the concrete from the steel and causes failure.
The safe resistance of concrete, without reinforcement, to diagonal tension is stated by the Joint Committee to be about one-third of the safe resistance to direct shear, or about 2 per cent of the ultimate compressive strength. For ordinary concrete, breaking at 2000 when twenty-eight days old, reinforcement against diagonal tension is necessaiy when v is greater than 40 lb./in 2.
Two methods of placing steel for diagonal tension reinforcement are commonly employed.
(a) Vertical stirrups may be used, designed to carry the vertical component of the diagonal tension, leaving the horizontal component to be taken by the horizontal tension steel.
(b) The steel may be placed at an angle of 45° with the hori zontal and parallel with the tensions due to shear. In this ease the diagonal steel must be rigidly connected with the horizontal steel to prevent slipping horizontally-, which is often accomplished by bend ing up part of the horizontal reinforcement near the end of the beans, where stresses clue to moment are light.
Vertical Stirrups.—Fig. 49 shows a beam reinforced for diagonal tension by the use of vertical stirrups.
Let s= length of beam to be reinforced by one stirrup; V= Total vertical shear in section s; v=unit shearing stress; Ta=Tota1 diagonal tension in distance s; 7;= Total vertical tension in stirrup.
The average unit shear dd. This is also the unit diagonal J tension due to shear, acting at an angle of 45° with the horizontal, and the total tension is in place of that given in Formula (12).
In designing stirrup reinforcement, the spaces s may be assumed and the required area :10 of stirrups computed, or the spacing may be determined for stirrups of given area. The value of v to be used should be the average value for the space s.
In order to avoid danger of cracks between stirrups, the spaces s should not exceed one-half the effective depth of beam, 2d. The shear is a maximum at the support, and the first space should be measured from the middle of the bearing on the support.
Diagonal tension reinforcement is needed only in the portion of the beam in which the shear exceeds the allowable unit shear for plain concrete (where v is greater than 2 per cent of the ultimate compress ive strength of the concrete). In a uniformly loaded beam, the shear is zero at the middle of the beam and increases uniformly with the distance from the middle to a maximum at the support. If r,n is the maximum unit shear at the support and 1 the length of the beam the unit shear (or unit diagonal tension) at any point distant x from the middle of the beam is Reinforcement for diagonal shear is needed from the point where v equals the allowable shear for unreinforced concrete to the end of the beam. For ordinary concrete, in which the allowable unit shear is 40 we have Diagonal Feinforeemenx.—Fig. 50 represents a beam reinforced for diagonal tension by bars inclined at 45° with the horizontal. As before, if the unit shear is more than 2 per cent of the ultimate com pressive strength of the concrete, the beam needs reinforcement against diagonal tension.