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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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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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INSTABILITIES IN BEAMS AND COLUMNS 30.12
LOAD CAPABILITY CONSIDERATIONS
TABLE 30.1 Influence of Eccentricity and Slenderness Ratio on Allowable Load
30.6 BEAM-COLUMN ANALYSIS
A member that is subjected to both a transverse load and an axial load is frequently called a beam-column. To apply the immediately preceding stress-limiting criterion to a beam-column, we first determine the moment distribution, say, Mtr, and the corresponding deflection, say, Ytr, resulting from the transverse load acting alone. Suppose that the transverse load is symmetrical about the column midpoint, and let Ytr, mid and Mtr, mid be the values of Ytr and Mtr at the column midpoint. Then the only modifications necessary in the preceding development are to replace Y by Y + Ytr in Eqs. (30.17) and (30.20), and to replace Ymid by Ymid + Ytr, mid and add Mtr, mid on the right side of Eq. (30.23). If the transverse load is not symmetrical, then it is necessary to determine the maximum moment by using an approach which will now be developed. Note that, at any point, x, the moment about the z axis is M(z) = P(e y Y) + M(z)tr (30.24)
where Y includes the deflection due to M(z)tr. M(y) has the same form as Eq. (30.24), but with the roles of y and z interchanged.The maximum stress for any given value of x is given by Eq. (30.23). We seek to apply this equation at that value of x which yields the maximum value of . A method that is reasonably efficient in locating a minimum or maximum to any desired accuracy is the golden-section search. However, this method is limited to finding the minimum (maximum) of a unimodal function, that is, a function which has only one minimum (maximum) in the interval in which the search is conducted. We therefore have to conduct some exploratory calculation to find the stress at, say, a dozen points on the beam-column in order to locate the unimodal interval of interest within which to apply the golden-section search. The actual number of exploratory calculations will depend on the individual case. For example, in a simply supported case with a unimodal transverse moment, there is clearly only one maximum. But, in general, we must check enough points to be sure that a potential maximum is not overlooked. The golden-section search procedure is as follows: Suppose that we seek the minimum value of F(x) in Fig. 30.5 within the interval D (note that if we sought a maximum in Fig. 30.5, we would have to conduct two searches). We locate two points x(1) and x(2). The first is 0.382D from the left end of the interval; the second is 0.382D
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