Multiplying this ratio by the total area of the column, 324 square inches, we have 6.93 square inches of steel required in the column. This would very nearly be provided by four bars 11 inches square. Four round bars 11 inches in ,diameter would give an excess in area.
Either solution would be amply safe under the circumstances, pro vided the column was properly reinforced with bands.
Example 2. A column 16 inches square is subjected to a load of 115,000 pounds, and is reinforced by four 1-inch square bars besides the bands. What is the actual compressive stress in the concrete per square inch? Answer. Dividing the total stress (115,000) by the area (256), we have the combined unit-stress C = 449 pounds per square inch. By inverting one of the equations above, we can write: In the above case, the four ;-inch bars have an area of 3.06 square inches; and therefore, 3.06 p = = .012; r = 12.
256 Substituting these values in the above equation, we may write: 449 440 c = — 397 pounds per square inch.
1 — .012 + X 12) 1 .1'32 The net area of the concrete in the above problem is 252.94 square inches. Multiplying this by 397, we have the total load carried by the concrete, which is 100,117 pounds. Subtracting this from 115,000 pounds, the total load, we have 14,SS3 pounds as the com pressive stress carried by the steel. Dividing this by 3.06, the area of the steel, we have 4,864 pounds as the compressive stress in the steel. This is practically twelve times the unit-compression in the concrete, which is an illustration of the fact that if the compression is shared by the two materials in the ratio of their moduli of elasticity, the unit-stresses in the materials will be in the same ratio. This unit stress in the steel is about one-third of the working stress which may properly be placed on the steel. It shows that we cannot economically use the steel in order to reduce the area of the concrete, and that the chief object in using steel in the columns is in order to protect the columns against buckling, and also to increase their strength by the use of bands.
• It sometimes happens that in a building designed to be struc turally of reinforced concrete, the column loads in the columns of the lower story may be so very great that concrete columns of sufficient size would take up more space than it is desirable to spare for such a purpose. For example, it might be required to support a load of
320,000 pounds on a Column 1S inches square. If the concrete (1:3:5) is limited to a compressive stress of 400 pounds per square inch, we may solve for the area of steel required, precisely as was clone in example 1. We should find that the required percentage of steel was 13.4 per cent, and that the required area of the steel was therefore 43.3 square inches. But such an area of steel could carry the entire load of 320,000 pounds without the aid of the concrete, and would have a compressive unit-stress of only 7,400 pounds. In such a case, it would be more economical to design a steel column to carry the entire load, and then to surround the column with sufficient concrete to fireproof it thoroughly. Since the stress in the steel and the con crete are divided in proportion to their relative moduli of elasticity, which is usually about 10 or 12, we cannot develop a working stress of, say, 15,000 pounds per square inch in the steel, without at the same time developing a compressive stress of 1,200 to 1,500 pounds in the concrete, which is objectionably high as a working stress.
311. Hooped Columns. It has been found that the strength of a column is very greatly increased and even multiplied by surrounding the column by numerous hoops or bands or by a spiral of steel. The basic principle of this strength can best be appreciated by considering a section of stovepipe filled with sand and acting as a column. The sand alone, considered as a column, would not be able to maintain its form, much less to support a load, especially if it was dry. But when it is confined in the pipe, the columnar strength is very con siderable. Concrete not only has great crushing strength, even when plain, but can also be greatly strengthened against failure by the tensile strength of bands which confine it. The theory of the amount of this added resistance is very complex, and will not here be given. The general conclusions, in which experimental results sup port the theory, are as follows: 1. The deformation of a hooped column is practically the same as that of a plain concrete column of equal size for loads up to the maximum for a plain column.