As the length of this column is only about eight times its diameter, and the load concentric, the minimum allowance of steel will not be increased; and the section-area of the steel is, therefore, Five-in round rods are used, and also 1⁄44-in round hoops spaced 5% in X 15 = 9 in on centers. (See page 169.) It is to be noted that splices are lapped 18 in, and wired. (See page 168.) The steel is placed 11⁄2 in inside of the surface. Example 2. Design of a column according to the same specification as in Example 1, but to sustain an axial load of 175 000 lb (including an assumed weight of the column). The section-area, with 0.5% vertical steel, is The section-area of a 19-in (net greater than the required area. diameter) round column is 283.5 sq in, slightly But let it be assumed that for the purpose of uniformity it is desired to use a column 18-in in diameter, the section-area of which is, as in Example 1, 254.4 sq in. determined by Formula (2), and is [175 000 1 The section-area of the steel is then Ao = [PƒcA]/[ƒc(n − 1)] = (600 X 254.4)]/[600(121)] = 3.38 sq in The column-size or diameter, between outside dimensions, is = Five-in and one 3/4-in round rods are used, and also 4-in round hoops, spaced 12 in on centers (see page 169). It is to be noted that splices are lapped X24 21 in and wired (see page 168). The steel is placed 11⁄2 in inside of the surface. 6. Design-Procedure for Columns with Spiral Reinforcement. The same notation is used as for columns without lateral reinforcement. The TOTAL EFFECTIVE VOLUME of a column of this class is the part on the inside of the spiral reinforcement, which must be surrounded by a shell of concrete 11⁄2 or 2 in thick, the thickness depending upon the fire-resistance required. The PERCENTAGES of both the vertical and lateral reinforcements are based upon the amount of concrete inside of the spiral, and the percentage of the latter is determined by computing the volume of spiral steel in a UNIT LENGTH OF COLUMN and dividing it by the volume of concrete in the same unit length. Columns should never be designed with spirals alone, a certain amount of vertical steel being necessary to resist any tendency to bend. The AMOUNT OF VERTICAL REINFORCEMENT is usually limited by code, both as to minimum and maximum amounts. For example, both the New York City Code, 1926, and the recommendations of the American Concrete Institute, 1920, place the limits at 1 and 4% respectively, while the Joint Committee, 1924, allows from 1 to 6%. As a general rule 1% of vertical steel should be the very minimum allowable amount, and a minimum of 12% is preferable for an approximate design standard. The vertical rods are wired directly to the spiral, no other hoops being used, and each column is supplied with three or four spiral-spacers. In any particular design the MAXIMUM PERCENTAGE OF VERTICAL STEEL is also limited by good practice to the number of rods which may be placed along the perimeter of the spiral, with a clear spacing of 11⁄2 in. The AMOUNT OF SPIRAL REINFORCEMENT, and the method of computing its value, vary considerably in the recommendations of different authorities and in the requirements of various building codes. This variation is due to the fact that although it has been established by test that the ULTIMATE STRENGTH of a column is considerably increased by lateral reinforcement, there is a difference of opinion in regard to the increase to be allowed in the UNIT WORKING STRESSES on that account. The Final Report of the Joint Committee, 1916, recommends that spirally reinforced columns be designed by the formula used for columns with vertical reinforcement only, but with an increase in the UNIT STRESS in the concrete of 55% over that allowed for columns without lateral reinforcement. In order to permit this increased unit stress, 1% of spiral reinforcement is required. The height, also, is limited to ten times the CORE-DIAMETER and the clear spacing of the spiral to one sixth of this diameter, that is, of the diameter of the enclosed column; but in no case is it to be more than 21⁄2 in. The American Concrete Institute (1920) recommends that the SPIRAL REINFORCEMENT, within the limits of 1⁄2 and 2%, be considered four times as effective as LONGITUDINAL REINFORCING STEEL of equal volume; and that the stresses in the concrete and vertical steel be computed by the methods employed for columns without lateral reinforcement. The amount of vertical reinforcement, under this specification, as noted above, can not be less than 1% nor more than 4% of the section-area of the column, and its percentage can not be less than that of the spiral reinforcement. The clear spacing of the latter is limited to one sixth its outside diameter, and in no case is it to be more than 3 in. At least four vertical spacer-bars are required. In the Chicago Building Code, 1924, a value of the lateral reinforcement 21⁄2 times that of an equivalent section of vertical steel is used. The New York City Code, 1922, on the other hand, permits for spiral reinforced columns an increase in the unit compressive stress on the effective section-area of the column equal to twice the percentage of lateral reinforcement, multiplied by the permissible tensile stress in the lateral reinforcement. This is added to the allowable compressive stresses of the concrete and steel as computed for columns with vertical reinforcement only. The amount of spiral is limited to between 1⁄2 and 2%, and there is no requirement in regard to its ratio to the vertical reinforcement. The spacing of the spiral is the same as in the specification of the American Concrete Institute. At least three vertical spacer-bars are required. The Joint Committee, 1924, recommends that the total safe axial load on spirally reinforced columns be computed by the formula The allowable value of the safe working stress, fc, in this formula is determined by the formula fc = 300+ (0.10 + 4p)ƒ'c In this formula f'e is the ultimate compressive strength of the concrete at an age of 28 days, determined in accordance with the specifications of appendices XIII and XIV of the Report. It is specified that the amount of spiral reinforcement shall not be less than one fourth the volume of the longitudinal reinforcement, and the maximum spacing not more than one sixth the diameter of the core or, in any case, not more than 3 in. At least three vertical spaces are required. The simplest procedure in the DESIGN of SPIRALLY REINFORCED-CONCRETE COLUMNS is to compute the required section-area, based upon the minimum percentage of vertical steel and the maximum percentage of spiral steel permitted by the specification which it has been decided to adopt. This combination of reinforcements with a rich mixture of concrete usually makes the MOST ECONOMICAL COLUMN. Knowing the required section-area, a column with a suitable section is chosen and its strength checked; or the reinforcement is adjusted, as in the case of columns without spirals. In no case should the pitch of the spiral be less than 1% in, as this spacing is required to permit the flow of concrete to the outer shell. The AVERAGE COMPRESSIVE UNIT STRESS, f, from which is obtained the ECONOMICAL AREA, is determined as explained in the following paragraphs. Following the recommendations of the Report of the Joint Committee, 1916, = f = P/A = 0.35f'c[1 + (n - 1)p] C (7) If f. 16 000 per sq in, ultimate crushing strength f' 2 500 lb per sq in, fs 12, minimum ratio of vertical steel p n = = p' = 1%; then, using these minimum ratios, 1%, and fixed ratio of spiral steel Following the recommendations of the American Concrete Institute, 1920, If fs = ƒ = P/A = ƒc[1 + (n − 1)p + 4p'n] = (8) 16 000 lb per sq in, fc = 625 lb per sq in, n = 12, minimum ratio of vertical steel p 1%, and maximum ratio of spiral steel p' 2%; then, determining ƒ on the basis of 2% spiral and 2% vertical reinforcement, since the latter under this specification can not be less than the • former, by Formula (8), = f = 625[1 + (11 × 0.02) + (0.08 × 12)] = 1 362 lb per sq in Following the requirements of the New York City Code, 1926, f = P/A = fd[1 + (n − 1)p] + 2p' fs1 (9) in which fs, is the allowable unit tensile stress in the spiral reinforcement. If 16 000 lb per sq in, fe = 600 lb per sq in, fs1 minimum ratio of vertical steel, p = 1%, and maximum ratio of spiral steel p' = 2%; then determining ƒ on the basis of 1% vertical and 2% spiral reinforcement, which makes the most economical column under this code, f = 600[1 + (11 × 0.01)] + (2 × 0.02 × 20 000) = 1 466 lb per sq in Having found the value of ƒ for the particular code governing the design, determined as above for the most economical ratios of vertical and lateral reinforcement, the resulting cross-sections of the column are found from the formula, A P/f. = These three methods of designing SPIRALLY REINFORCED COLUMNS illustrate only too clearly the wide divergence of present-day practice as applied to this particular type of reinforced-concrete column-design. It has been thoroughly established by tests that spiral reinforcement increases the ULTIMATE STRENGTH of a column, but has little effect when the unit stresses are less than the ELASTIC LIMIT; and as the DEFORMATION very quickly becomes excessive after the YIELD-POINT is reached, it does not seem logical to base the ALLOWABLE WORKING STRESS upon the ultimate strength. Neither is it possible, under working loads, to utilize any large amount of lateral reinforcement. In short, the value of the spiral is to increase the FACTOR OF SAFETY, and it is a question of judgment on the part of the designer to what extent this added security may warrant an increase in the working stress. = = 12, minimum ratio 7. Typical Design for Columns with Spiral Reinforcement. The following problems illustrate the method of design outlined in the previous article. Example 3. Design based upon the Report of the Joint Committee, 1916. If fs 16 000 lb per sq in, f'c 2 500 lb per sq in, n = of vertical steel p 1%, fixed ratio of spiral steel p' 1%, axial load 500 000 lb, unstayed height = 12 ft, and fireproofing 2 in; then, with 1% vertical and 1% spiral reinforcement, the effective cross-sectional area of the column is = = = By Table II the next largest effective area is 530 sq in, corresponding to a corediameter of 26 in (gross diameter 30 in). Although this section is a little larger than necessary, neither the vertical nor spiral steel can be reduced, since the percentage of each is already a minimum. In Tables IV to IX a spiral is selected which makes the percentage 1% for a column of this size and which has a pitch of not more than 21⁄2 in. For the spiral reinforcement the choice is a 16-in wire with a 24-in pitch (Table VI., percentage 1.02), and for the vertical reinforcement nine -in round rods. Again, if a somewhat smaller section is desired, a column which has a 25-in core-diameter (gross diameter 29 in), has an effective section-area of 490 sq in. In Table VI, a 16-in wire with a 23%-in pitch corresponds to 1% of spiral steel for a column of this diameter, and is satisfactory under this code. The vertical steel is found by Formula (3), and p = [P — fcA]/[ƒ.(n − 1)A] = [500 000 (875 X 490)]/[(875 × 11 × 490)] = 0.015 = 490 X 0.015 = 7.35 sq in. Ten 1-in round rods are used. The smallest column that can be employed under this code to carry the required load, with f'. = 2 500 lb per sq in, 1% of spiral, and 4% vertical steel is, by Formula (6), This section-area, by Table II, requires a column with a core-diameter of 23 in (gross diameter 27 in), and with an effective section-area of 415 sq in. The chosen spiral reinforcement, from Table VI, is a 16-in wire with a 21⁄2-in pitch, and the section-area of the vertical steel is Ao = 415 X 0.04 = 16.60 sq in Fourteen 14-in round rods are used. Example 4. Design based upon the recommendations of the American Concrete Institute. If f,= 16 000 lb per sq in, fe vertical steel p = = 625 lb per sq in, n = 12, minimum ratio of 1%, maximum ratio of spiral steel p' 2%, axial load = = 500 000 lb, unstayed height = 12 ft, and fireproofing 2 in; then, with 2% vertical and 2% spiral reinforcement, the effective cross-sectional area of the column is By Table II the next larger effective area is 380 sq in, corresponding to a corediameter of 22 in (gross diameter 26 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 not more than 3 in. For the spiral the choice is a 5%-in wire with a 2%-in pitch (Table IX, percentage 1.94). This percentage is a little low, but is still adequate, because the column-section is a little larger than necessary. This may be proved by applying Formula (8), making A = 380, and solving for p. For the vertical reinforcement Ao = 380 X 0.02 = 7.60 sq in Ten 1-in round rods are used. The smallest column that can be employed under this code to carry the required load, with fe=625 lb per sq in, 2% of spiral, and 4% of vertical steel is, by Formula (8), |