Strength of Columns OR STRUTS Machinery's Handbook, 31st Edition
281
COLUMNS Strength of Columns or Struts
Structural members which are subject to compression may be so long in proportion to their diameter or lateral dimensions that failure may be the result 1) of both compression and bending; and 2) of bending or buckling to such a degree that compression stress may be ignored. In such cases, the slenderness ratio is important. This ratio equals the length l of the column in inches or millimeters, according to the unit system in use, divided by the least radius of gyration r of the cross section. Various formulas have been used for designing columns which are too slender to be designed for compression only. Rankine or Gordon Formula.— This formula is generally applied when slenderness ra- tios range between 20 and 100, and sometimes for ratios up to 120. The notation, in Eng- lish and metric SI units of measurement, is given on page 283 . , lb/in 2 or N/mm 2 p K r l S 1 ultimate load 2 = + = ` j Factor K may be established by tests with a given material and end condition, and for the probable range of l / r . If determined by calculation, K = S / C π 2 E . Factor C equals 1 for either rounded or pivoted column ends, 4 for fixed ends, and 1 to 4 for square flat ends. The fac tors 25,000, 12,500, etc., in the Rankine formulas, arranged as on page 283 , equal 1/ K , and have been used extensively. Straight-Line Formula.— This general type of formula is often used in designing com pression members for buildings, bridges, or similar structural work. It is convenient espe cially in designing a number of columns that are made of the same material but vary in size, assuming that factor B is known. This factor is determined by tests. p S B r l ultimateload, lb/in 2 y = − = ` j S y equals yield point, lb/in 2 , and factor B ranges from 50 to 100. Safe unit stress = p ÷ factor of safety. Formulas of American Railway Engineering Association.— The formulas that follow apply to structural steel having an ultimate strength of 60,000 to 72,000 lb/in 2 . For building columns having l / r ratios not greater than 120: allowable unit stress = 17,000 − 0.485 l 2 / r 2 For columns having l / r ratios greater than 120:
, r l 1 18 000 18 000 , 2
allowable unit stress
=
2
+ ⁄ For bridge compression members centrally loaded and values of l / r not greater than 140:
, , 15 000 4 1 15 000 3 1 − −
r l r l
2
Allowable unit stress, riveted ends Allowable unit stress, pin ends
=
2
2
=
2
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