AISC lutz1985Q4.pdf
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1、A Unified Approach for Stability Bracing Requirements LEROY A. LUTZ and JAMES M. FISHER Columns and beam flanges in compression often require intermediate lateral bracing to satisfactorily carry the required load. Usually this is a finite number of braces at a spacing S. Many engineers are familiar
2、with the ideal stiffness 4 Pcr/S needed to fully brace the compression member over the length S, and that the required stiffness is generally taken as twice the ideal stiffness, or 8 Pcr/S. If the compression member does not need to be fully braced at each support point, due to either the magnitude
3、of loading required or the size of member used, considerably less bracing stiffness may be required than would be calculated by using 8 Pcr/S. A means to evaluate an accurate value of required stiffness is important for examining structural members whose integrity is questionable because of limited
4、bracing stiffness and strength, and for obtaining economical bracing design for members repeated many times in a structure. No simple technique other than continuous bracing expressions, exists for obtaining the less-than-full bracing integrity. In this paper, the bracing stiffness required for the
5、range from continuous point bracing to a single point brace is presented. The required brace strength is also summarized. The bracing expressions are given in the form most familiar for point bracing. The expressions provide an understanding of the relationship between bracing stiffness and buckling
6、 behavior of the compression member. Furthermore, an expression for tangent modulus of elasticity is proposed as a means to apply the bracing expressions for all levels of critical load less than the yield load. BRACING STIFFNESS FOR CONTINUOUS BRACING SYSTEMS From Timoshenko and Gere, Theory of Ela
7、stic Stability,1 Pcr = (2EI/L2)(m2 + kcL4/m24EI) (1) LeRoy A. Lutz, Ph.D., P.E., and James M. Fisher, Ph.D., P.E., are Vice Presidents of Computerized Structural Design, Milwaukee, Wisconsin. where: L = the overall member length kc= the continuous bracing stiffness m= the number of 1/2 sine waves in
8、 the buckled shape Timoshenko shows kcL4/4EI = m2(m + 1)2 at buckling. Pcr = (2EI/L2)m2 + (m + 1)2 Since m2 + (m + 1)2 = 2m2 + 2m + 1 = 2m(m + 1) + 1, Pcr = (2EI/L2)2m(m + 1) + 1 = +( / ) / / 2 2 2 2 2 1EIL L k EI c = +2 2 2 EIk EI EIL c/ / Thus, P k EI P cr c E = +2 the ideal continuous bracing sti
9、ffness kc = (Pcr PE)2/4EI (2) Neglecting PE for Pcr PE, Pcr = 2 k EI c or kc = Pcr2/4EI. For the case of inelastic action replace E with Et. Using Pcr = 2EtI/L2e L E IP e t cr = / (3) where Le = the effective length of the buckled column. kc = (2Pcr/4)(Pcr/2EtI) = 2Pcr/4Le2. Thus, the ideal continuo
10、us bracing stiffness is: K P L c cr e . / 25 2 (4) BRACING STIFFNESS WITH A FINITE NUMBER OF SUPPORTS For finite braces at a spacing equal to S, where S is small relative to Le, Ki = kcS = (2.5Pcr/Le2)S (5) See Fig. 1. Based on Winters work2 when S equals Le, Ki = 4Pcr/Le(6) Equation 6 is often used
11、 when S is less than Le, with S being substituted for Le. This is not correct. The results FOURTH QUARTER / 1985 163 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of t
12、he publisher. Fig. 1. Finite bracing at spacing S from such calculations can grossly overestimate the bracing stiffness required. One must remember that the length Le in Eq. 6 defines the required buckling length of the member. Using a S which is less than Le in Eq. 6 presumes a shorter mode shape,
13、which requires more stiffness than is necessary. For a small S, relative to Le, Eq. 5 provides an accurate solution for bracing stiffness. However, as S increases relative to Le, using Eq. 5 results in a substantial error. Solutions are presented in Ref. 3 for critical buckling loads (Pcr) for cases
14、 when brace stiffness is less than the stiffness required to force the column to buckle in its highest mode, i.e., the number of one-half waves equal to the number of braces plus one. Solutions are presented for one, two, three and four intermediate brace points, as well as for columns continuously
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