In the application of the above formulas it must be remembered that N is the normal component of R, the resultant of all forces acting upon the section; and the value of the lever arm, e, of the resultant is obtained by dividing the moment M by the normal component of the total load on the section. The design procedure for a column subjected simultaneously to a direct load and a bending moment requires the determination of the magnitude, direction, and point of application of R, then the computation of N, its normal component, and the resulting value of e. 10. Bending-Stresses in Independent Columns. In the examples of Chapter VII, members with loads of KNOWN ECCENTRICITY were considered. This is the condition when a column supports a crane-load, or another structural member which merely rests on it. Axis of Column As an example of this case the crane-way load illustrated in Fig. 2 may be considered. Here the magnitude of the bending moment on the column, which moment is always greatest at the section of the column where the load is applied, varies with the distance of the load from the ends of the column, and also with the degree of restraint at the ends of the column. If we assume that the column is either fixed at both ends, or fixed at the base and free at the top, which two conditions include practically all those encountered in building-design, the bending moment is a maximum and equal to Pr, when the load is applied at either end of the column, and a minimum, 1⁄2Px, when the load is at the middle length of the column, for columns considered fixed at both ends, and at certain intermediate positions for columns fixed at the base and free at the top. It must be borne in mind, when solving problems of this character, that x is not the eccentricity e, as the latter is affected by the concentric loads, and in all cases by the weight of the column itself, and must be found by dividing the moment, in this case Pr, by the normal component of the total load on the column (which we have previously designated as N) including P, the weight of the column itself, and any other loads which it may sustain. Fig. 2. Column with Bracket-Load of Known Eccentricity 11. Bending-Stresses in Columns which are Part of a Building-Frame. In MONOLITHIC CONSTRUCTION where a column is one of the members of a rigid frame, there is always a certain amount of RESTRAINT exerted at the joint and this restraint depends upon the relative stiffness of the various members, the number of spans of the beams, and the system of loading. The determination of the BENDING MOMENTS in the columns is consequently more difficult than it is under the simple conditions of the previous article. The two general cases which need consideration in building-design are first, an interior column supporting beams of unequal spans, or beams of equal spans with unbalanced live loads, and secondly, a column in an outside wall, supporting one end of a noncontinuous beam constructed monolithically with the column. In both these cases the column is affected by a BENDING MOMENT which it must resist in addition to the DIRECT STRESSES resulting from the axial loads. These latter, on interior columns, are fortunately reduced, under the condition of partial loading that causes the maximum moments in the columns, and this tends to diminish the magnitude of the combined stress. Where the beams are long and shallow, however, such as in flat-slab construction, and especially with stiff columns, the effect of the bending moment on the column-section caused by an unsymmetrical live load is very appreciable; and even in beam-design and slab-design, when the beams or slabs are of only two or three continuous spans, may cause a bending moment on the middle column or columns of sufficient magnitude to cause an extreme fiber-stress in the concrete approximately double that resulting from axial loads due to uniform loading. Where there are columns both above and below a joint, the bending moment caused by an unbalanced load is resisted by both columns, the proportion resisted by each varying with its relative stiffness. In an outside wall column which is not continuous above a beam framing into it, such as a column supporting a roof, the bending moment of the beam tends to cause tension on the exterior face of the column. When such a column is continuous, as in the succeeding lower stories, the effect is to cause tension on the exterior face of the column below the beam and on the interior face of the column above the beam. Interior columns subject to unbalanced loads are affected in the same way. To provide for these stresses, it is usually advisable in both cases to use symmetrical reinforcements. With the usual conditions of loading critical stresses are likely to occur at the bases of top story columns, as the relatively light concentric loads in these members tend to increase the eccentricity of the resultant of the combined loads. For flat-slab construction city building codes stipulate MINIMUM VALUES to be used for THE BENDING MOMENTS on columns, resulting from UNBALANCED LOADS. For example, New York City Code requires that on interior columns used with flat-slab construction the bending moment resulting from unequally W1L loaded panels be given the value and that it be resisted by the columns 40 In the above formulas W is the total live load 1 immediately above and below the floor-line under consideration, and in direct proportion to the values of their ratios of I/h. Under the same code wallcolumns must be designed to resist a bending moment due to the eccentricity WL of the floor-load, equal to 40 on the panel, W the total live and dead load on the panel, L the average centerto-center dimension of panel supported by the lower column, I the moment of inertia of the cross-section of the column with regard to the median axis of the section; and h the height of the column from top of slab to base of capital. WL be resisted The city of Chicago requires that a bending moment, WL 60 by an exterior column supporting a roof, and a bending moment, by each of two superimposed outside columns in the lower stories. In these moments W is the total live and dead load on the panel, L the average centerto-center dimension of the panel supported by the lower column. This code gives no specific requirement for interior columns other than the general stipulation that the least diameter shall be not less than L/12, or one twelfth the clear height of the column; but it requires that the worst conditions due to unbalanced loads be considered in the design. It should be noted in the application of these formulas that although they take account of conditions of unequal live loads on a floor under consideration, the full live loads, or the permitted reduced loads (see page 165), are assumed on the floor, or floors above. The CRITICAL CONDITION, however, may occur when DEAD LOAD ONLY is considered on the supported floors, which assumption naturally increases the eccentricity. The maximum value of a cOMBINED FIBER-STRESS to be allowed for columnsections is limited by the American Concrete Institute to one-and-one-half times the stress permitted for an axial load. The Joint Committee, 1924, allows, for tied columns, a total maximum compressive unit stress on the concrete equal to three tenths of the ultimate compressive strength of the concrete, and limits the amount of the longitudinal reinforcement that may be added, to 2% of the sectional area of the column. For spirally reinforced columns the increase in the compressive unit stress on the concrete within the core-area is limited by these recommendations to 20% of that permitted for an axial load. Example 6. Let it be required to design, under the New York City Code, a typical interior column supporting a flat-slab roof-construction divided into bays 18 ft square, and according to the following data: Column-height, top of slab to base of capital = 12 ft; in which the live load on alternate bays, only, is considered, 1 500 is the weight in pounds of the column capital; bending moment at top of roof-column is M = -W1l 40 1 Referring to Table II it is seen that a round column with a 12-in effective diameter will sustain a direct load of 67 860 lb on the concrete alone. Although the New York City Code permits, in the case of columns without spiral reinforcement, the gross area to be used in the computations, the minimum diameters for columns supporting flat-slab floors is 16 in, which size is chosen. Four 5%-in round rods constitute the vertical reinforcement. Applying the general formula, and using an effective diameter of 12 in, = 1 018 + 260 -8 In this equation 6 is the radius of the circular cross-section of the column-core, in inches; 113 the area of the cross-section of the column-core, in square inches; 11 the value of (n − 1); 1.2 the area of the cross-section of the vertical reinforcement, in square inches; 1018 the moment of inertia of the area of the concrete-section, taken from the top line of Table XVII; and 260 the momen of inertia of the area of the reinforcement, interpolated for the exact percentage in the values given in the top line of Table XIX. The axial load on the base of the roof-column is 45 360 + 2 500 + 1 500 = 49 360 lb in which 45 360 is the full load on a roof-bay (18 × 18 × 140), and 2 500 is the weight of the column shaft. As above, 1 500 is the weight of the capital. Bending moment at the base of the roof-column is in which 150 is the live load on the top floor, in pounds. As this moment must be divided between the column under design and that immediately below, in the proportion of their relative stiffness, or ratios of I/h, these must be computed. In these equations the values of the moments of inertia are taken from Tables XVII and XIX, assuming for the lower column a 14-in core with five 5%-in round vertical rods and a 3%-in spiral with a 2-in pitch. The denominator of each fraction is the clear story-height from top of slab to bottom of capital expressed in inches. The revised moment is which is an acceptable design for the given conditions.* 12. Determination of Bending Moments in Columns by the Slope-Deflection and Area-Moment Method. In the last example the axial load on the column was found by merely multiplying the area of the bay by a load per square foot. If, however, the column under design had been the first interior column supporting a series of girders simply resting on exterior walls, the load would have been affected by the conditions of support at these exterior terminations, and an AXIAL LOAD determined, as in the example, would be about 15% too low. For this and similar cases of equal spans uniformly loaded, and for the more usual cases of UNBALANCED LOADS on members of only a few spans, the axial loads on columns due to the floor system may be found by adding the VERTICAL SHEARS given in the diagrams of Fig. 1. Again, even for unequal spans and unequal loading, if the RESTRAINTS at the columns are neglected, the negative bending moments in the girders, at the supports, can be determined by the THEOREM OF THREE MOMENTS, and from these moments the VERTICAL SHEARS and the REACTIONS (see page 99) at the columns, due to the weight of the floor-construction. But where the restraint at the columns must be considered, as in the case of comparatively stiff columns and shallow floor-members, either APPROXIMATE MOMENTS must be assumed, as in the New York and Chicago Codes governing flat-slab construction, or more exact moments computed by one of the systems of rational analysis, such as that of SLOPE-DEFLECTIONS and AREA-MOMENTS discussed in Chapter VI. In problems of this character the first step is to compute the axial loads from which a somewhat liberal design is made, based on concentric loading. The sectional area and reinforcement of the column can then be approximately determined, and from the general SLOPE-DEFLECTION FORMULAS an expression for the value of the SLOPE of a girder at its junction with a column. Having determined the slope, the BENDING MOMENT for the column-section is given by Formula (42), Chapter VI, M = EKON In this formula E is the modulus of elasticity of the concrete, K the moment of inertia of the column-section divided by its length, 0 the slope of the girder, and N a factor depending upon the condition of restraint at the opposite end of the column from that for which the moment is being computed. As noted in Chapter VI, page 138, the value of N is 3 if the end of the column is considered hinged, and 4 if the end is considered fixed. In multi-story buildings the most critical condition of loading would require a value of N = 6, but the value 4 is ordinarily used. If it is not desired to find the moment, the value of the extreme fiber*If the live load on the roof is omitted, these stresses for the net sections become = 723 and fe" 147, but if computed on the gross cross-sections they reduce - 60. = to fc 400 and fc" -- = |