Machinery's Handbook, 31st Edition
282 FORMULAS FOR COLUMNS Euler Formula.— This formula is for columns that are so slender that bending or buckling action predominates and compressive stresses are not taken into account. P l C IE = total ultimate load, in pounds or newtons 2 2 = π The notation, in English and metric SI units of measurement, is given in the table Ran- kine and Euler Formulas for Columns on page 283 . Factors C for different end condi tions are included in the Euler formulas at the bottom of the table. According to a series of experiments, Euler formulas should be used if the values of l / r exceed the following ratios: Structural steel and flat ends, 195; hinged ends, 155; round ends, 120; cast iron with flat ends, 120; hinged ends, 100; round ends, 75; oak with flat ends, 130. The critical slenderness ratio , which marks the dividing line between the shorter columns and those slender enough to warrant using the Euler formula, depends upon the column material and its end conditions. If the Euler formula is applied when the slenderness ratio is too small, the calculated ultimate strength will exceed the yield point of the material and, obviously, will be incorrect. Eccentrically Loaded Columns.— In the application of the column formulas previously referred to, it is assumed that the action of the load coincides with the axis of the column. If the load is offset relative to the column axis, the column is said to be eccentrically loaded, and its strength is then calculated by using a modification of the Rankine formula, the quantity cz / r 2 being added to the denominator, as shown in the table on the next page. This modified formula is applicable to columns having a slenderness ratio varying from 20 or 30 to about 100. Machine Elements Subjected to Compressive Loads.— As in structural compression members, an unbraced machine member that is relatively slender (i.e., its length is more than, say, six times the least dimension perpendicular to its longitudinal axis) is usually designed as a column because failure due to overloading (assuming a compressive load centrally applied in an axial direction) may occur by buckling or a combination of buck- ling and compression rather than by direct compression alone. In the design of unbraced steel machine “columns” which are to carry compressive loads applied along their longi- tudinal axes, two formulas are in general use: (Euler) (1) (J. B. Johnson) (2) where (3) In these formulas, P cr = critical load in pounds that would result in failure of the column; A = cross sectional area, inches 2 ; S y = yield point of material, psi; r = least radius of gyra- tion of cross section, inches; E = modulus of elasticity, psi; l = column length, inches; and n = coefficient for end conditions. For both ends fixed, n = 4; for one end fixed, one end free, n = 0.25; for one end fixed and the other end free but guided, n = 2; for round or pinned ends, free but guided, n = 1; and for flat ends, n = 1 to 4. It should be noted that these values of n represent ideal conditions that are seldom attained in practice; for example, for both ends fixed, a value of n = 3 to 3.5 may be more realistic than n = 4. If metric SI units are used in these formulas, P cr = critical load in newtons that would result in failure of the column; A = cross-sectional area, mm 2 ; S y = yield point of the material, N/mm 2 ; r = least radius of gyration of cross section, mm; E = modulus of elasticity, N/mm 2 ; l = column length, mm; and n = a coefficient for end conditions. The coefficients given are valid for calculations in metric units. P Q S Ar cr y 2 = P AS cr y = r Q 1 4 2 − a k Q n E S l y 2 2 π =
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