# barcode generator vb.net source code INSTABILITIES IN BEAMS AND COLUMNS 30.11 in Software Generator EAN13 in Software INSTABILITIES IN BEAMS AND COLUMNS 30.11

INSTABILITIES IN BEAMS AND COLUMNS 30.11
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INSTABILITIES IN BEAMS AND COLUMNS
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Then we have, from Eqs. (1) through (4), allow = P P 4P 4 + 1+ A [(EA2)/(4 )]( /L)2 P A (5)
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Usually P and L are given, and E are the properties of chosen material, and is determined from the clearances, tolerances, and kinematics involved, so that Eq. (5) is reduced to a cubic in A. At the moment, however, we are interested in comparing the allowable nominal column stress P/A with the allowable stress of the material allow for columns of different lengths. Keeping in mind that the radius of gyration r of a circular cross section of geometric radius R is R/2, we will define R =r 2 allow =p P/A E =q allow Then Eq. (5) may be written as p = 1 + 4 + 16 [ 2pq(r/L)2 1] (7) (6)
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The first term on the right side of Eq. (7) is due to direct compressive stress; the second term is due to the bending moment produced by the load eccentricity; the third term is due to the bending moment arising from the column deflection. When is small, p will be close to unity unless the denominator in the third term on the right side of Eq. (7) becomes small that is, the moment due to the column deflection becomes large. The ratio L/r, whose reciprocal appears in the denominator of the third term, is called the slenderness ratio. Equation (7) may be rewritten as a quadratic in p. Thus, 2q r 2 2 r p (1 + 4 ) 2q L L
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+ 1 p + (1 + 4 )
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16 =0
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We will take for q the representative value of 1000 and tabulate 1/p for a number of values of L/r and . To compare the value of 1/p obtained from Eq. (8) with the corresponding result from Euler s formula, we will designate the corresponding result obtained by Euler s formula as 1/pcr and recast Eq. (30.9) as 1 r = 2q pcr L
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To interpret the results in Table 30.1, note that the quantities in the second and third columns of the table are proportional to the allowable loads calculated from the respective equations. As expected, the Euler formula is completely inapplicable when L/r is 50. Also, as expected, the allowable load decreases as the eccentricity increases. However, the effect of eccentricity on the allowable load decreases as the slenderness ratio L/r increases. Hence when L/r is 250, the Euler buckling load, which is the limiting case for which the eccentricity is zero, is only about 2 percent higher than when the eccentricity is 2 percent.
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