When a column reinforced with longitudinal bars is subjected to pressure, the concrete and steel must shorten together. The relative stresses in the two materials will then be proportional to their moduli of elasticity. From this follows the formula,
P=fc(C +RS) where P = total pressure on column, fc = stress in concrete, C and S = areas of concrete and steel respectively, and R = Es/Ec, or ratio of the modulus of elasticity of the steel to that of the concrete.
In a series of tests of twenty-one columns made by Prof. Gaetano Lanza,1 but three failed under a lower stress than that computed by the above formula. The columns were eight to ten inches square, six to seventeen feet long and reinforced with either one or four bars, the latter being from 3/4 inch to 1 1/4 inches square.
The lowest breaking load was fifty tons on an 8 by 8 inch column with one bar one inch square, and the strongest column, 10 by 10 inches, with four 3/4 inch longitudinal bars, was not crushed with a load of one hundred fifty tons, the limit of the testing machine. The lowest result was twenty per cent, less than that given by the formula, and the greatest excess strength over the theoretical was fifty per cent.
620. While longitudinal reinforcement undoubtedly strengthens a long column against flexure, as well as adds to the resistance to crushing, yet the added strength is gained at the expense of considerable additional Cost. Suppose we have a ten inch square column, twelve feet long, made of concrete with a breaking load of 1,800 lbs. per square inch, or 180,000 lbs. total breaking load. Suppose eight 3/4 inch square bars to be built into this column as longitudinal reinforcement, and that the modulus of elasticity of the steel is ten times that of the concrete. Then the strength of the reinforced column would be, by the formula above,
P = 1,800 (95.5 + (10 X 4.5)) = 252,900.
The longitudinal reinforcement has thus resulted in an increase of strength of 40 per cent., while by the addition of 180 pounds of metal, the Cost of the column has risen from about $3.00 to say $8.50, an increase of about 180 per cent, without counting the Cost of lateral ties, and the additional trouble in building a reinforced column.