four-in square bars, as explained in Chapter III, the design proceeds with the computations for web-reinforcement. Using the reactions R1 and R2, obtained from the moment-computations, the shear V is determined at the points a, b, c, d, and e, by one of the conditions of equilibrium of external forces acting on the beam. The vertical shear at any section is equal to the sum of the reactions minus the sum of the loads on the right or left of the section, the reaction or reactions and the load or loads considered being always taken on the same side of the section. Then, having Fig. 5. Typical Design of a Reinforced-concrete T Beam or Girder, under a Uniformly Distributed Load Due to its Own Weight, and Three Unequal, Unsymmetrically Placed Concentrated Loads computed the unit shearing-stresses v by Formula (1), the computations are arranged as follows: At c, to the left of load P2, and V = 14 060 · [8 000 + (250 × 9)] = 3 810 lb, At c, to the right of load P2, in The unit shearing-stresses v, laid off to any convenient scale, locate the points. a', b', c', c', d', and e', which determine the shear diagram. Then, by drawing the lines mn and op, parallel to the base, and at a distance therefrom corresponding to v = 40 lb per sq in, it is evident that the unit shearing-stress exceeds this limit throughout the beam, except between the loads P1 and P2, and it is necessary to design web-reinforcement to assist the double-bend rods in taking these stresses. Using a 3-in round-rod stirrup with two legs, the spacing at the left support is, by Formula (12), As the shearing-stresses diminish very slightly from the support out to the point b, under the load P1, the same spacing is retained throughout this portion of the beam, except for the distance 0.82d 18 in, see Formula (6), covered by the two double-bend rods whose area is found by test, as in the preceding example, to be more than adequate. = The spacing at the right support is, by the same formula, This spacing is retained to the point d, under the load P3, except for the distance covered by the double-bend bars. As this is within the limiting distance of 0.82d, given by Formula (6), this spacing is retained between the loads P3, and P1. The unit shearing stress being less than 40 lbs per sq in no stirrups are required, theoretically, through this portion of the girder, but are recommended when the member is subject to impact or vibration. The complete design is shown in Fig. 5. It is seen that the space theoretically covered by the double-bend rods is somewhat reduced in the actual design in order to facilitate the arrangement of stirrups; and in fact some designers entirely disregard their value when they are raised at only one section of the beam. It should also be noted that it is usually inadvisable to alter the spacing of stirrups to provide for the slight variations in shear due to the weight of the beam, or other uniformly distributed loads, in members subjected to heavy concentrated loads, unless a considerable saving is effected, as small changes complicate the construction. 6. Bond between Steel and Concrete. As the TENSION in the longitudinal reinforcement of a beam changes with the varying bending moments, the increment or decrement in these tensile stresses must be transmitted to the concrete; and there are developed BOND-STRESSES between the two materials. To resist this tendency of the reinforcement to PULL LOOSE from or SLIP THROUGH the encircling concrete, there is, first, the grip upon the steel due to the INITIAL SHRINKAGE of the concrete in hardening, and, secondly, the FRICTIONAL RESISTANCE of the rod, or bar, against slipping. The adhesion between the concrete and steel is apparently lower for squares than for rounds, is increased by a slight rust; and considerably reduced when the bars, or rods, are polished. The actual bond stresses at critical sections of reinforced-concrete members subjected to flexure are not easy to determine, but general practise accepts the following evaluation: Let V = total shear at the section, in pounds; v = unit shearing-stress, in pounds per square inch; u = allowable unit bond-stress, in pounds per square inch of surfacearea of the tension steel; 0 = perimeter of cross-section of rod or bar, in inches; Σο = the sum of the perimeters of all horizontal tension-rods at the section considered, in inches; jd = distance between center of compression in concrete and center of tension in steel in inches; distance from extreme fibers in compression to center of gravity of tensile reinforcement, in inches. For concrete developing by test an ultimate compressive strength of 2 000 lb per sq in at an age of 28 days, the value of u is generally taken at 80 lb per sq in for plain rods (rounds or squares), and at 100 lb per sq in for deformed rods (rounds or squares). When SPECIAL ANCHORAGE is provided, as described on page 63, the Joint Committee, 1924, allows HIGHER BOND-STRESSES which, however, are limited by the equation Σο = = Zo the perimeter of the bar under consideration, in square inches; ratio of the average to the maximum bond-stress, computed by Formula (14), within the distance y; Q u = = permissible bond stress 0.04f', for plain and 0.05f', for deformed bars, in pounds; x = the length of bar added for anchorage, including the hook, if any, in inches; y = distance from the point at which the tension is computed to the point of beginning of anchorage, in inches. The length of bar added for anchorage may be either straight or bent. The BOND STRESS is tested at critical sections which, in continuous or restrained beams, is usually considered to be at the FACE OF THE SUPPORT FOR THE TENSION STEEL NEAR THE TOP OF THE BEAM. The Joint Committee, 1924, requires that the bond also be tested in continuous, or restrained beams, at the POINT OF INFLECTION for both the positive and negative reinforcement, see example, page 62. In simple beams, or freely supported end-spans of continuous beams, the CRITICAL SECTION IS CONSIDERED TO BE AT THE FACE OF THE SUPPORT, and the Zo is the sum of the perimeters of the rods carried straight through into the support in the bottom of the beam, see example on page 87. The Joint Committee, 1924, permits bent-up longitudinal bars, within a distance of d/3 from the horizontal reinforcement under consideration, to be included with the straight bars in computing the 20. For example, using the data of example 2, page 37, for a girder with concentrated loads at the third-points, and with the ends continuous, In this equation the numerator is the total vertical shear at the support; 4, the number of rods at the top of the member over supports, two from each span; 2.75, the perimeter of the cross-section of each rod in inches; 0.875, the approximate value of j used in shear computations; 30, the effective depth of the girder in inches. The bond-stress at the fifth-point of the clear span, assumed as the point of inflection, is, In this equation the value of V' is the vertical shear at the fifth-point of the clear span; 2, the number of rods at both the top and bottom of the beam at the fifth-point of the span; the other factors are the same as in the previous operation. This bond-stress is permissible with anchored reinforcement. In case the stress is excessive, the depth of the beam may be increased, or, preferably, the size of the rods reduced and a larger number of bars employed, retaining the same total sectional area, and thereby increasing the sum of the perimeters of the bars: this often results in increasing the width of the member. It should also be noted that bond requirements may determine the number of bars to be carried straight through over supports in the bottoms of continuous beams. 7. Anchorage and Splicing of Reinforcement. Assuming a permissible bond-stress of 80 lb per sq in or 100 lb per sq in it is evident that for any given tensile stress in the steel a sufficient embedment, or anchorage, in the concrete will cause the reinforcement to fail in tension before pulling loose. The length in inches required to develop the full working stress in the steel is determined by the following formula: in which f, is either the tensile or compressive stress in the steel in lb per sq in, and i the diameter of a round rod or the side of a square bar, in inches. For example, if ƒ、 16 000 lb per sq in, u = 80 lb per sq in, and i 1 in, the length of embedment required to develop the full strength of the rod is: = = = 11 000 lb per sq in, u = 80 lb This length of embedment in the concrete is sufficient to permit a 3-in round steel bar to carry 11 000 lb per sq in in compression without passing the allowable bond-stress of 80 lb per sq in. It is the general practice to insure against FAILURE IN BOND, and a consequent SLipping of the roDS, by extending them the distance given by the above formula beyond the section at which a given stressvalue is needed. Then, if the value of the rod in bond falls off more rapidly than its required stress-value, it is necessary to increase the length of its embed |