For FREELY-SUPPORTED BEAMS and slabs, the SPAN IS MEASURED FROM CENTER TO CENTER OF THE SUPPORTS; but the distance is not to exceed the clear span, plus the depth of the member. For CONTINUOUS OR RESTRAINED BEAMS built integrally with the supports, the span may be taken as the CLEAR DISTANCE BETWEEN THE FACES OF THE SUPPORTS. (1) Beams and slabs of equal spans BUILT TO ACT INTEGRALLY WITH BEAMS, GIRDERS, OR OTHER SLIGHTLY RESTRAINING SUPPORTS, and carrying uniformly distributed loads, shall be designed for the following bending moments at critical sections: Beams and slabs continuous for two spans only: Maximum positive moments near middle of each span, Beams and slabs continuous for more than two spans: Maximum positive moments near middle and negative moments at supports of interior spans, Maximum positive moments near middle points of end-spans, and negative moments at first interior supports, Negative moments at end-supports for each of the above cases, (2) Beams and slabs BUILT INTO BRICK OR MASONRY WALLS IN A MANNER WHICH DEVELOPS PARTIAL END-RESTRAINT shall be designed for a negative bending moment at the support of WL M = not less than in-lb (3) Beams and slabs of EQUAL SPANS FREELY SUPPORTED and assumed to carry uniformly-distributed loads shall be designed for the moments specified in paragraph (1), except that no reinforcement for negative moments need be provided at the end-supports where effective measures are taken to prevent end restraint. The span shall be taken as defined for freely-supported beams. (4) Beams and slabs of equal spans BUILT TO ACT INTEGRALLY WITH COLUMNS, WALLS, OR OTHER RESTRAINING SUPPORTS, and assumed to carry uniformly distributed loads, shall, except as provided in paragraph (1), be designed for the following bending moments at critical sections: Interior spans, Negative moments at interior supports, except the first, Maximum positive moments near middle points of interior spans, End-spans of continuous beams and beams of one span, in which I/L is less than twice the sum of the values of I/h for the exterior columns, above and below, which are built into the beams: Maximum positive moments near middle points of spans, and negative moments at first interior supports, End-spans of continuous beams, and beams of one span, in which I/L is equal to or greater than twice the sum of the values of I/h for the exterior column, above and below, which are built into the beams: Maximum positive moments near middle points of spans and negative moment at first interior support, *Most building codes require (5) CONTINUOUS BEAMS WITH UNEQUAL SPANS, or with other than uniformly distributed loading, whether freely supported or restrained, shall be designed for the actual moments under the conditions of loading and restraint. Provision shall be made, where necessary, for negative bending moments near the middle points of short spans which are adjacent to long spans, and for the negative moment at the end-supports, if restrained. The preceding formulas apply to uniform loads and spans of equal length. When spans vary in length, or the loads are not the same for every span, these moments do not apply. Under such conditions the bending moments should be computed by the THEOREM OF THREE MOMENTS and when it is necessary to consider the degree of restraint exerted by the supporting member, the method of SLOPE-DEFLECTIONS is used (see Chapter VI). The four diagrams of Fig. 1, THEORETICAL BENDING-MOMENT DIAGRAMS, illustrate the theoretical bending moments for beams continuous over two, three, or four spans, and supporting uniformly distributed loads. For members with fixed ends, Diagram 1 applies to any number of spans. In the case of beams with simply-supported ends, Diagram 2, illustrating the bending moments in a beam continuous over four spans, can also be used in the design of members having a greater number of spans; but it should be remembered that, as the number increases, the negative bending moments, also, at the middle supports, are slightly increased. For example, over the middle support, for six spans, the negative bending moment is WL/11.6 in-lb, while the positive bending moments remain practically unchanged. For a single span, with one end simply supported, and the other end continuous or otherwise fixed, the bending moments are as indicated in either half of Diagram 4. The bendingmoment diagrams are drawn to scale, by laying off the computed maximum bending-moment values, as vertical lines or intercepts, to a scale of inch-pounds to a linear inch. The proportional bending moments at any other sections of the beam can therefore be easily scaled and determined, and the steel-area of reinforcement required at any points other than those of maximum bending moments, closely approximated. 2. Bending Moments for Concentrated Loads. The bending moment at any section of a simple beam or girder, in a state of flexure under concentrated loads, can be computed by determining the REACTIONS at the supports, and then taking the algebraic sum of the moments of the external vertical forces on either side of that section. Another method is to determine the bending moment due to the concentrated loads, and add to it the bending moment caused by the weight of the member, and by any other uniformly distributed loads. The cases usually encountered in standard construction are those in which equal concentrations, due to beam-loads, are at the middle, the third-points, or the quarter-points of girders. For such members, simply supported, the maximum bending moments due to the concentrated loads are as follows: For a single concentration at the middle, For equal concentrations at the third-points, (1) (2) Diagram 4 Two-Span Beam, Each Span Semi-Continuous Fig 1. Theoretical Bending-Moment Diagrams, all for Uniform Loads and Equal Spans For equal concentrations at the middle and at the quarter-points, (3) It should be noted that these formulas are for use only in computing moments based upon the center to center span. Having determined the maximum bending moments by either of the preceding methods, they may be reduced for semicontinuous and fully-continuous members by multiplying by a REDUCING FACTOR corresponding to the conditions of end-support. For example, the positive moment near the middle point of the span, as computed above for a simply supported girder, would be multiplied by 12 or 2% if the members were fully continuous. The twelve diagrams of Figs. 2, 3, and 4, THEORETICAL BENDING-MOMENT DIAGRAMS, illustrate the theoretical bending moments in beams and girders, continuous over two, three, and four spans, and supporting symmetrically placed, concentrated loads. For members with fixed ends, Fig. 2, Diagrams 5, 6, and 7 apply to any number of spans. In the case of members with simply supported ends, the diagrams illustrating the moments in those continuous over four spans, can also be used in the design of members having a greater number of spans; for although the moments at different sections vary somewhat as the number of spans is increased, the values of the maximum positive and negative bending moments do not differ appreciably from those given in the four-span diagrams. As the diagrams are drawn to scale, the proportional bending moments, for any section of a girder, can be easily read; but it should be remembered that the lines of these diagrams represent the varying bending moments due to the concentrated loads only, and do not include the effect of the weight of the member itself, which should be considered as a uniformly distributed load, and for which the resulting bending moments should be added to those caused by the concentrated loads. 3. Providing for the Varying Bending Moments. Having determined the maximum bending moments in inch-pounds by the principles and formulas given in the two preceding articles, the depth, d, of the concrete is determined by Formula (10), and the sectional area A, of the longitudinal reinforcement, by Formula (11) or (12), Chapter II. If it is decided to conform to general practice and consider the values of the positive and negative bending moments as the same for equal interior spans of continuous members, uniformly loaded, it follows that 50% of the main longitudinal reinforcement in each beam should be raised over the supports and carried far enough into the adjoining spans to resist the negative moments at those supports. As the negative bending moment decreases rapidly from the support outward, it is usually adequate to carry such bars. to the quarter-point of an adjoining span, or to the point of inflection, using |