Let d=effective diameter of column in inches; a=area of the steel bar to be used, in square inches; s spacing of the bands or spirals, in inches; p=ratio of steel to concrete in column.
Then From this we may obtain the size of bars necessary for steel of re quired spacing, or the spacing required for bars of given size.
Example 24.—A concrete column is to carry a load of 225,000 pounds and be reinforced with 1 per cent of spiral steel and 2 per cent of longitudinal steel. Using the stresses recommended by the Joint Committee for concrete of 2000 pounds compressive strength, find dimensions for concrete and steel.
Solution.—Without hooped reinforcement, the value of would be limited to 22.5 per cent of the compressive strength, or 450 lb./ in. This may be increased 55 per cent when spiral reinforcement is used or 700 lb., Using Table XVI, for p=.02 and n=15, from which L7. 100X1.28 251 and diameter of column is 18 inches. Longitudinal steel, .02X251=5.02 in2 From Table X we see that five 1-inch square bars, spaced about 11 inches apart about the circumference of the column, or nine 1-inch square bars spaced about 6 inches apart may be used. For the spiral steel, we find from (51) that if the spacing be made 2z inches, a=.01X18X2z,d =.112 and ;-inch round bars may be used.
122. Eccentrically Loaded Columnso When the center of gravity of the load upon a column does not coincide with the gravity axis of the column, bending stresses are produced which must be taken into account in designing the column. In some cases, lateral forces may be acting upon a column, producing bending moments, as in wall columns carrying the ends of beams which are firmly attached to the columns. When these conditions exist, the maximum unit com pression due to both direct thrust and bending moment at any sec tion must not exceed the safe values for the concrete, and any tensions which may occur must be taken by proper reinforcement.
Let Fig. 59 represent the section of a column under eccentric load.
A= area of section of column; area of longitudinal steel in section; P = longitudinal load on column; e= eccentricity of load; moment of inertia of section about its gravity axis; I, = moment of inertia of steel area about same axis; v = distance gravity axis to most remote edge of section; moment on section, Pe; L= maximum unit compression on concrete; f',= minimum unit compression on concrete.
made up of two parts—that clue to direct thrust and that due to bending moment, and is When the stress due to moment is greater than that clue to direct thrust, f', becomes negative, showing the stress to be tension. Tensions in columns, if occurring at all, should be very small and need not be specially provided for. The stresses in steel are always less than and therefore within safe limits.
If the section is symmetrical about its gravity axis, u= d/2, and for rectangular sect ions, I c= 12 and in which is distance between centers of steel on the two sides of column. For circular sections, and where d is the diameter of the column and is diameter of the circle containing the centers of the steel bass.
Example 2i.—A wall column, 12X16 inches in section, carries the end of a beam which brings longitudinal load of 00,000 pounds and a bending moment of 1S0,000 in.-lb. upon the column. The column is reinforced with four 1-inch square steel bars at the corners, the centers of steel being 2 inches from surfaces of concrete. n=15. Find the unit stresses on the concrete.
12=4096. I,=4X12X12,'4=144.
60000180000 XS {-235=177 lb. lb, Complete discussions of the principles of reinforced concrete design with applications to structures is given in " Concrete, Plain and Reinforced," by Taylor and Thompson, and in " Principles of Reinforced Concrete Construction," by Turneaure and Maurer.