the supports, at an angle of 30° to the horizontal, and crosses the central plane of the beam half-way between the fifth-point and the tenth-point, that is, at the (1500)-point, of the span. The BENDING-MOMENT CURVE is plotted by computing the actual moments at the different sections and laying them off vertically, to scale, from the base-line. To the same axes are plotted the moments which the steel at any section is capable of resisting. This latter graph is called the curve of the POTENTIAL RESISTING MOMENTS. The difference between the potential resisting moment and the actual bending moment, at any section, therefore represents the additional bending moment which the reinforcement could carry at this section. The point d in the diagram is projected from s, and the point e from r. The points d and e are then joined by a curve, Fig. 6. Arrangement of Longitudinal Steel Reinforcement to Resist the Bending Moments in a Fully-Continuous Beam under a Uniformly Distributed Load showing the variations in the magnitudes of the potential resisting moments at the sections of the beam between these points. The line cedf is the CURVE OF THE POTENTIAL RESISTING MOMENT. The rectangle, abfg, represents the value of the straight rod or rods and the irregular figure, bced, that of the double-bend rod or rods with respect to the positive moment in the beam. As the value of the potential resisting moment is, at all sections, at least equal to that of the bending moment, and varies with it as closely as practicable, the design is shown to be both safe and economical, as far as bending-moment considerations affect the reinforcement. Fig. 6 illustrates the arrangement of longitudinal steel for a fully-continuous beam under a uniform load. The diagrams are based upon the DESIGN-MOMENTS, WL/12 in-lb, at the middle of the span and at the supports, and the resisting moments are plotted for a design in which 50% of the longitudinal steel is raised at an angle of 30° to the horizontal and crosses the central plane of the beam at the fifth-point of the clear span. The curves enclosing the varying positive and negative bending moments are developed from different positions of the loading and consequently are not continuous. Their values represent the AVERAGE CONDITIONS encountered in practice, and illustrate the reason for the recommended disposition of the steel. The positive resisting moment, WL/12 in-lb, is plotted by laying off to scale the values of the potential resisting moments due to the straight rods (equal to ab), and that due to the bent rods (equal to bc). The point d is projected from s, and the point e from r. The points d and e are then joined by a curve as in the previous case (Fig. 5). For the negative resisting moment, WL/12 in-lb, the values due to the rods are similarly laid off, as pf and fg. The two rods, or sets of rods, which are then available to resist the negative bending moment, are the straight portion ry of the double-bend rod extending from the adjoining span, and the section of the double-bend rod of the beam under design, between the points s and y. Assuming, for purposes of illustration, that each rod is a 1-in round rod, and requires 40 diameters of embedment in order to develop its full strength (see page 63), it is evident that, disregarding the effect of the hook, the value of the rod xy uniformly decreases from to its termination x. The projection of these points to i and h, respectively, in the moment-diagram, results in the polygon, hifp, which represents the value of this rod in resisting the negative bending moment in the beam. The value of the other rod, constant from the support y to the point u, where the bend occurs, is then plotted below the broken line hif, locating the points g, j, and k. From k to its value decreases, the variation being shown by the curve, and the zero-value being at the point 7, where the only resisting moment is that due to the value of the rod xy. Since, in this case it is a fair assumption that both rods develop the same unit stresses, mn is laid off equal to om, and the POTENTIAL RESISTING-MOMENT CURVE gjnlh is determined. In a rectangular beam, where steel is not required in the compression-area, the rods in the bottom of the beam, over the supports, need not ordinarily pass the center line of the columns. In T beams, however, these rods are often lapped or even extended into the adjoining spans a short distance beyond the further face of the support, the distance depending upon the need of supplying steel to assist the concrete in carrying the compressive stress due to the negative moment. (See page 41.) Where beams frame into wide supports it should be remembered that the EFFECTIVE SPAN can always be considered, even in simply-supported members, as the clear distance between the supports, plus one-half the depth of the beam or girder, instead of the distance between the center lines of supports. Furthermore, where a 30° angle brings the top of the bend, as at the point u, Fig. 6, too near the face of a support, it is advisable to use a 40° bend, as the distance tu should never be less than 6 in. The general rule of carrying the double-bend rods to the quarter-points of adjoining spans, and anchoring them by a hook as shown, satisfies the conditions usually encountered in standard practice; but where a beam is subjected to a very definite live load, it may be necessary to carry these rods to the one-third points. If reasonable doubt exists, the BENDING-MOMENT CURVE should be plotted for the particular case in question, and the steel so designed that there is no cross-section of the beam at which the curve falls appreciably outside of that representing the potential resisting moment. Fig. 7 illustrates the longitudinal reinforcement for a semicontinuous beam under a uniform load, a typical arrangement for the end-span in a series of continuous members. The diagrams are based on maximum positive and negative Fig. 7. Arrangement of Longitudinal Steel Reinforcement to Resist the Bending Moments in a Semicontinuous Beam under a Uniformly Distributed Load design-moments, WL/10 in-lb, at the middle of the span and at the interior support, and the potential resisting moments are plotted for a design in which 50% of the longitudinal steel is raised over the supports, the raised portion making an angle of 30° with the horizontal. The section at the first interior support, in a series of continuous members, is usually a CRITICAL POINT in the design, and it is recommended to carry the double-bend rod to the one-third point of the first interior span, as the negative moments on that side of the column should be considered to extend to the quarter-point. (See Fig. 1, Diagram 2.) By reference to Fig. 7 it is seen that the arrangement of steel shown at the wall-termination is not quite adequate to resist the bending moment as plotted in the diagram. In this case, however, it must be remembered that the moment curve is plotted on an ASSUMPTION OF ABSOLUTELY NO RESTRAINT at the support, although it is certain that there is some restraint, which is provided for by raising 50% of the steel to the tension-side. Furthermore, the value WL/10 in-lb is used for the maximum positive design-moment, while the corresponding theoretical moment is, from Fig. 1, Diagram 2, only WL/12.9 in-lb. Because of these two considerations a slight overlapping of the curves may be disregarded, and the design considered safe as far as bending-moment considerations are concerned. The method of plotting the curves of the potential resisting moments for continuous beams, explained in the preceding paragraphs, applies also in this case. 4. Typical Design as Governed by the Bending Moments. The following three examples illustrate the method of providing for the bending moments. Specification-data for three examples: Example 1. Typical Design of a Single-Span Reinforced-Concrete Beam with a Uniformly Distributed Load. Distributed Load Fig. 8. Typical Design of a Reinforced-Concrete Beam, under a Uniformly Load = 20 000 lb, uniformly distributed, including weight of beam End-conditions: as shown in Fig. 8. A trial value of 9 in is taken for the beam-width b. If this value is not governed by architectural considerations, the beam is made only wide enough to accommodate the reinforcement with proper space between bars and adequate fireproofing-allowances. The width, however, should, in no case, be less than about one-third the effective depth, or 1/24 the clear span. Total beam-depth 24.9 +0.5+1.5 26.9 in, or 27 in = In this equation, 0.5 in is the approximate half-diameter of the rods, and 1.5 in the allowance for protection. As it is desirable to use an even number of rods of moderate size for the main longitudinal reinforcement, four 3/4-in round rods are chosen. The width of 9 in is then sufficient to allow 1 in between rods and 11⁄2 in fireproofing on each side (see page 13). Two of the 4-in rods are run straight the entire length of beam, and two 3/4-in rods are raised so as to cross the central plane of the beam at a section half-way between the one-fifth point and the one-tenth point of the span, at an angle of 40° to the horizontal; and all rods are anchored by hooks at each end. Fig. 8 shows the completed design, including the stirrups placed to resist the DIAGONAL TENSION, as described on page 53. In all cases the bond-stress should be checked as described on page 61. For a complete design of a rectangular beam, see page 65. Example 2. Typical Design of a Reinforced-Concrete Girder, with a Uniformly Distributed Load Due to its Own Weight, a Concentrated Load at each ThirdPoint of the Span, and with the Ends Fully Continuous. (See Fig. 9.) Fig. 9. Typical Design of a Reinforced-Concrete Girder, under a Uniformly Distributed Load Due to its Own Weight, a Concentrated Load at each Third-Point of the Span, and with the Ends Fully Continuous = 15 000 lb Load P at each third-point Uniformly distributed load due to weight of girder A trial value of 10 in is taken as the girder-width b. M = = PL/3 = (15 000 × 21 × 12)/3 = 1 260 000 in-lb. Since the member is continuous, this moment may be reduced in the ratio of 12 to 8. Maximum moment due to concentrations = = 1 260 000 X 3 840 000 in-lb. = (6 000 X 21 X 12)/12 = 126 000 in-lb. Maximum design-moment = 966 000 in-lb. |