COLUMNS The strength of a compression member depends on the ratio of its length to its least lateral dimension, or, what is the same thing, on the ratio of slenderness; that is, the ratio of its length to its radius of gyration. For compression members whose ratio of slenderness does P not exceed 30, the formula s=, for simple stress, may be used. When this ratio exceeds 30, but is not more than 150, s should be reduced by Rankine's formula, in which su is the ultimate strength in compression, which should be divided by a suitable factor of safety; 1, the length; and r, the radius of gyration. Both land r are expressed in the same unit. The values of k which depend on the material of the column and the condition of its ends-that is whether fixed or round—are given in the following table: When the value of - exceeds 150, Euler's formula, which is r given later, should be used. The straight-line formula is more convenient for determining the value of s, and is now in extensive use. It is only approximate, giving values of s that differ somewhat from those obtained by Rankine's formula; but the difference is on the side of safety. For the same notation as before, the straightline formula is s=su-k r The values of Su and k are given in the accompanying: table, in which will also be found the limit of r within which the formula may be used. When exceeds this limit, Euler's EXAMPLE.-What is the ultimate strength per square inch of a medium-steel column 25 ft. long both ends of which are fixed and the radius of gyration of which is 2.5? SOLUTION.-By the straight-line formula, Euler's Formula.-Structural members in compression whose ratio of slenderness exceeds 150 should preferably not be used. Sometimes, however, long columns cannot be avoided, and when exceeds the limits for which the preceding formulas may r be applied, Euler's formula should be used. follows: This formula is as in which E is the modulus of elasticity of the material and n is a constant depending on the end condition, having the value of 1 for columns with both ends pivoted and 4 for columns with both ends fixed. The preceding table gives the values of n2E, expressed in millions of pounds. Formula for Wooden Columns.-The formula for determining the strength of wooden columns having flat or square ends was deduced from exhaustive tests of full-size specimens, made at the Watertown Arsenal, Mass., and may be expressed as follows: UI 100d S=U- in which S is the ultimate strength of column, per square inch of section; U, the ultimate compressive strength of the material, per square inch; 1, the length of the column, in inches; d, the dimension of the least side of the column, in inches. This formula may be applied to all wooden columns, the length or height of which is not under 10 times nor over 45 times the dimension of the least side. In other words, d If the length should not be less than 10 nor more than 45. is less than 10 times the least side, the direct compressive strength of the material per square inch, multiplied by the sectional area of the column, in square inches, will give the strength of the column. If the length is over 45 times the least side, Rankine's formula should be used. COMBINED STRESSES Bending Combined with Compression or Tension.-Assume that P is the axial force acting on the beam; M, the maximum bending moment to which the beam is subjected; A, the crosssectional area of the beam; I, its moment of inertia; and c, the distance from the neutral axis of the most distant fiber, having the same kind of stress (tension or compression) as that caused by P. Then, the working stress should not exceed P Mc In case of compression, s should, in addition, be reduced by one of the compression formulas previously given. The preceding formula for s is the one commonly used in practice, but it is only approximate. When more accurate results are required, the following formula should be used, Here, I is the span; E, the modulus of elasticity, and k, a constant having the following values: For a beam supported at both ends and loaded at For a beam supported at both ends and loaded uni For a beam fixed at both ends and loaded at center..e .32 The minus sign before k is for the case when the direct stress is compressive, and the plus sign, when it is tensile. STRENGTH OF ROPES AND CHAINS Ropes. If C is the circumference of a rope in inches and P the working load in pounds, then, for hemp and manila rope, P=10C2 This formula gives a factor of safety of from 7 for manila or tarred hemp rope to about 11 for the best three-strand hemp rope. For iron-wire rope of seven strands, nineteen wires to a strand, P=600C2 and for the best steel-wire rope of seven strands, nineteen wires to the strand, P=1,000 C2 The last two formulas are based on a factor of safety of 6. Chains. If P is the safe load in pounds and d the diameter of link in inches, then, for open-link chains made from a good quality of wrought iron, and for stud-link chains, P=12,000 d2 P=18,000 d2 Chain Cables.-The strength of a chain link is less than twice that of a straight bar of a sectional area equal to that of one side of the link. A weld exists at one end and a bend at the other, each requiring at least one heat, which produces a decrease in the strength. The report of the committee of the U. S. Testing Board, on tests of wrought-iron and chain cables, contains the following conclusions: "That beyond doubt, when made of American bar iron, with cast-iron studs, the studded link is inferior in strength to the unstudded one. "That, when proper care is exercised in the selection of material, the strength of chain cables will vary by about 5% to 17% of the resistance of the strongest. Without this care the variation may rise to 25%. "That with proper material and construction the ultimate resistance of the chain may be expected to vary from 155% to 170% of that of the bar used in making the links, and show an average of about 163%. "That the proof test of a chain cable should be about 50% of the ultimate resistance of the weakest link." |