By Table II the required core-diameter is 21 in (gross diameter 25 in), with a corresponding effective section-area of 346 sq in. For the spiral a 16-in wire with a 23-in pitch is chosen (Table VIII, percentage 1.99). The section-area of the vertical steel is Fourteen 1-in round rods are used. Example 5. Design based upon the New York City Code. minimum ratio of vertical steel p = axial load = = 2%, 1%, maximum ratio of spiral steel p' 500 000 lb, unstayed height 12 ft, and fireproofing 2 in; then, with 1% vertical and 2% spiral reinforcement, the effective cross-sectional area of the column is By Table II the nearest larger size is a core-diameter of 21 in (gross diameter 25 in), with an effective section-area of 346 sq in. In Tables IV to IX a spiral is selected which makes the percentage 2% for a column of this size, and which has a pitch of not more than 3 in. For this spiral the choice is a 16-in wire with a 23%-in pitch (Table VIII, percentage 1.99). For the vertical reinforcement Ao = 346 X 0.01 = Six -in round rods are used. = 3.46 sq in The smallest column that can be employed under this code to carry the required load, with fe 600 lb per sq in, 2% of spiral, and 4% of vertical steel is, by Formula (9), fel1+ (n-1)p]+2p'fs1 600[1 + (11 × 0.04)] + (2 × 0.02 × 20 000) 300 sq in By Table II, the required core-diameter is 20 in (gross diameter 24 in), with a corresponding effective section-area of 314 sq in. The choice for the spiral is a 16-in wire with a 22-in pitch (Table VIII, percentage 1.98). The sectionarea of the vertical steel is Tables X to XV are adapted to the New York City Code, and are used as follows in determining the most economical design for a spirally reinforced concrete column to carry the 500 000-lb load of Example 5. From Table III, second table-column, a column with a 21-in core-diameter (gross diameter 25 in) and with 1% of vertical and 2% of spiral steel will carry the load (L =507 C00 lb). From the fourth column of the same table the vertical steel consists of eight 34-in round rods. Deducting the load carried on the concrete and steel, as given in the seventh table-column, from the total load (500 000 231 000 lb), there remains 269 000 lb to be provided for by the spiral steel. Table XV, thirteenth column, shows that for a 21-in core, a %-in round wire with a 3-in pitch corresponds to a load of 270 000 lb, which is satisfactory. This completes the design, but if desired, the percentage of spiral steel may be checked by reference to Table IX, thirteenth column, which shows that for a column with a 21-in core-diameter, and with this size and pitch of spiral, the ratio is 1.94%. This system of design is based upon maintaining the percentage of vertical steel, wherever possible, at the 1% minimum, using a value of fe = 600 lb per sq in, and crowding the amount of spiral steel as close as possible to the 2% maximum. When this is not desirable, and it becomes necessary to add more vertical steel, the value of each additional percentage, for the size of column-core employed, can be estimated at a glance from Table III, sixth column. 8. Formulas for Bending-Stresses in Columns. In the preceding articles of this chapter the load on a column has been considered as CONCENTRIC with the column-section and only DIRECT AXIAL COMPRESSION has been considered. When the resultant of all the loads does not pass through the center of gravity of the cross-section, however, a column is subject to FLEXURAL STRESSES as well as to direct compression, and the value of fe, the EXTREME FIBER-STRESS in the concrete, is the algebraic sum of these two separate stresses, or Mc In this equation of the resultant of the forces acting upon the column-section, multiplied by the distance from the median or gravity axis to the face of the column, and divided by the moment of inertia of the cross-section with reference to that axis of the section about which the section tends to rotate. is the value of the moment of the normal component This is the same equation that formed the basis of the formulas developed in Chapter VII, on Bending-Stresses combined with Direct Axial Stresses, all of which, as illustrated in the examples of that chapter, apply to column-sections when the moment, M Ne, is known. To facilitate reference these formulas and the formula-numbers of Chapter VII are repeated here. 9. Notation. The following notation is used: A = area of cross-section; A = bt for rectangular sections; A, = area of cross-section of steel near face of member, least stressed in com A's pression; = area of cross-section of steel near face of member, most highly stressed in compression; A = area of cross-section of transformed section; A. = total area of cross-section of steel in symmetrically-placed reinforce = moment of inertia of cross-section of concrete, about the gravity axis; = moment of inertia of cross-section of steel, about the gravity axis; = moment of inertia of transformed cross-section, about the gravity axis, II + nIs; M = bending moment; M = Ne; R = resultant of all forces acting on or all stresses acting in the section; component of R normal to the section; N = a = distance from median axis of cross-section to steel, in sections with b = C = C1 C2 = symmetrical reinforcement; breadth of cross-section of concrete; distance from median axis to face of section, in plain concrete; c1 = distance from gravity axis, to face most highly stressed in compression; distance from gravity axis, to face least stressed in compression; distance from centroid of cross-section of steel, A's, to adjacent face of section; di d2 = = distance from centroid of cross-section of steel, A,, to adjacent face of section; e = eccentricity, or distance from median or gravity axis of cross-section to point of application of N; f. = unit fiber-stress in steel near face of member, least stressed in compression; f's fc = = unit fiber-stress in steel near face of member, most highly stressed in compression; maximum compressive fiber-stress in concrete; f'c = minimum compressive fiber-stress in concrete; p = percentage of steel in sections with symmetrical reinforcement; Ao/bt; distance from the compressive surface to the neutral surface; t = radius of effective concrete section, in circular sections; total thickness or depth of cross-section of concrete. 1. Plain Concrete sections (rectangular) Tension exists when the value of e is greater than one sixth the value of t. 2. Reinforced-concrete sections (rectangular). (1) Compression over entire section. (a) Unsymmetrical reinforcement. Chapter VII, Formula (4), Abt+n(A', + A,) |