. Canadian engineer. given as .0251of the compression load, can only be correct for columns having a ratio of — = , or thereabouts. By referringagain to Table i, Cols. 11 and 12, it will be seen that forapproxmiate ratios of— = 37-8, that the ratio .0251 com-pares very closely indeed with the result of the formulagiven hereafter. Derivation of Formula,—Referring to Fig. i, It hasbeen assumed that cDlumn is hinged top and bottom, andthat a load P is applied top and bottom. -74 THE CANADIAN ENGINEER Voliimi; 30. If tlic coliiinn is very short llie malirial will be crushedand there will he an
. Canadian engineer. given as .0251of the compression load, can only be correct for columns having a ratio of — = , or thereabouts. By referringagain to Table i, Cols. 11 and 12, it will be seen that forapproxmiate ratios of— = 37-8, that the ratio .0251 com-pares very closely indeed with the result of the formulagiven hereafter. Derivation of Formula,—Referring to Fig. i, It hasbeen assumed that cDlumn is hinged top and bottom, andthat a load P is applied top and bottom. -74 THE CANADIAN ENGINEER Voliimi; 30. If tlic coliiinn is very short llie malirial will be crushedand there will he an even pressure over both channels, asshown by Kiy. ^, and equal to Sc per square inch. Fig. 2. Fig. 3. T __1. If the column is long, itwill fail by bending andthere will be tension on oneside and compression on theother, or, by referring toFig. 3, it will be seen thatthere is a difference of stressbetween the two channelsequal to 2k. Or the total stress in chan-nel marked would be A 2k~ , (Uie to Fig. 4. i^et .9c = 16,000 lbs. = allowable pressure per squareinch when column is very short, and S, = allowableaverage pressure per square inch for long columns. Then, (So — S,) = stress per square inch due tobending of column at centre = k, and {So — .S,) A = totalstress in column at centre due to bending of column.[Equation i.] A = area of column. Referring to Merrimans Mechanics of Materials, 1894 edition, page 132, he gives what he considers the most accurate column formula for P -7-= Si, namely, A ~ = ^^ [Equation 2.] m - E r Substituting usual values in Equation 2, 16,0005, = ? ; [Equation 3.] I + 12,000 r Referring to Fig. 3 and Fig. 4 it will be noticed thatthe right-hand channel (marked ) has a stress persquare inch = 5c, while channel marked has a stressper square inch = Sc — 2k, making a difference in stressper square inch between the two channels of afe persquare inch. But fe = (Sc — S,). [Equation 4.]Therefore, 2k = 2(So — S,). [Equ
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