2. Further loading of a hooped column still further increases the shorten ing and swelling of the column, the bands stretching out, but without causing any apparent failure of the column.
3. Ultimate failure occurs when the bands break or, having passed their elastic limit, stretch excessively.
Hooped columns may thus be trusted to carry a far greater unit load than plain columns, or even columns with longitudinal rods and a ew bands. There is one characteristic that is especially useful for a column which is at all liable to be loaded with a greater load than its nominal loading. A hooped column will shorten and swell very perceptibly before it is in danger of sudden failure, and will thus give ample warning of an overload.
Considhre has developed an empirical formula based on actual tests, for the strength of hooped columns, as follows: Ultimate strength = c' A + 2.4s' pA (42) in which, c' = Ultimate strength of the concrete; s' = Elastic limit of the steel; p = Ratio of area of the steel to the whole area; A = Whole area of the column.
This formula is applicable only for reinforcement of mild steel. Applying this formula to a hooped column tested to destruction by Professor Talbot, in which the ultimate strength (c') of similar con crete was 1,380 pounds per square inch, the elastic limit of the steel (s') was 48,000 pounds per square inch; the ratio of reinforcement (p) was .0212; and the area (A) was 104 square inches; and sub stituting these quantities in Equation 42, we have, for the computed ultimate strength, 409,900 pounds. The actual ultimate by Talbot's test was 351,000 pounds, or about SO per cent.
Talbot has suggested the following formulae for the ultimate strength of hooped columns per square inch: Ultimate strength = 1,600 + 65,000 p (for mild steel). . (43) = 1,600 + 100,000 p (for high steel). .(44) In these formulae, p applies only to the area of concrete within the hooping; and this is unquestionably the correct principle, as the concrete outside of the hooping should be considered merely as fire protection and ignored in the numerical calculations, just as the con crete below the reinforcing steel .of a beam is ignored in calculating the strength of the beam. The ratio of the area of the steel is com puted by computing the area of an equivalent thin cylinder of steel which would contain as much steel as that actually used in the bands or spirals. For example, suppose that the spiral reinforcement con sisted of a 1-inch round rod, the spiral having a pitch of 3 inches. A -1-inch round rod has an area of .196 square inch. That area for 3 inches in height would be the equivalent of a solid band .0653 inch
thick. If the spiral had a diameter of, say, 11 inches, its circum ference would be 34.56 inches, and the area of metal in a horizontal section would be 34.56 X .0653 = 2.257 square inches. The area of the concrete within the spiral is 95.0 square inches. The value of p is therefore 2.257 ÷, 95.0 = .0237. If the ?-inch bar were made of high-carbon steel, the ultimate strength per square inch of the column would be 1,600 + (100,000 X .0237) = 1,600 4,- 2,370 = 3,970. The unit-strength is considerably more than doubled. The ultimate strength of the whole column is therefore 95 X 3,970 = 377,150 pounds. Such a column could be safely loaded with about 94,300 pounds, provided its length was not so great that there was danger of buckling. In such a case, the unit-stress should be reduced according to the usual ratios for long columns, or the column should be liberally reinforced with longitudinal rods, which would increase its transverse strength.
312. Effect of Eccentric Loading of Columns. It is well known that if a load on a column is eccentric, its strength is consider ably less than when the resultant line of pressure passes through the axis of the column. The theoretical demonstration of the amount of this eccentricity depends on assumptions which may or may not be found in practice. The following formula is given without proof or demonstration, in Taylor and Thompson's treatise on Concrete: Let e = Eccentricity of load; L = Breadth of column; f = Average unit-pressure; — Total unit-pressure of outer fibre nearest to line of vertical pressure Then, (45) = + 6e ) As an illustration of this formula, if the eccentricity on a 12-inch column were 2 inches, we should have b = 12, and e = 2. stituting these values in Equation 45, we should have = 2 f, which means that the maximum pressure would equal twice the average pressure. In the extreme case, where the line of pressure came to the outside of the column, or when e = 1 b, we should have that the maximum pressure on the edge of the column would equal four times the average pressure.
Any refinements in such a calculation, however, are frequently overshadowed by the uncertainty of the actual location of the center of pressure. A column which supports two equally loaded beams on each side, is probably loaded more symmetrically than a column which supports merely the end of a beam on one side of it. The best that can he done is arbitrarily to lower the unit-stress on a column which is probably loaded somewhat eccentrically.