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1、ENGINEERING JOURNAL / SECOND QUARTER / 2001 / 65 ABSTRACT T he AISC-LRFD Specification provides a conservative prediction of the strength of singly symmetric I-shaped beam-columns bent about the axis of symmetry if the com- pression flange is larger than the tension flange. This paper presents a mor
2、e economic method that bases the in-plane capacity on the attainment of the full plastic capacity of the cross-section, and the lateral-torsional strength on an inelastic modification of the elastic buckling solution. The proposed derivations are based on essentially the same phi- losophy as the one
3、s for the doubly symmetric wide-flange design rules in the AISC Specification. However, for singly symmetric shapes it is not possible to arrive at simple approximate empirical interaction equations. The proposed methods are of necessity spreadsheet oriented, and an appendix to this paper provides a
4、 sample calculation scheme using the MATHCAD software as the vehicle of calculation. INTRODUCTION The Load and Resistance Factor Design Specification of the American Institute of Steel Construction (AISC, 1999) has an accurate way of treating the determination of the strength of doubly symmetric wid
5、e-flange beam-columns under compressive axial load. By using the interaction equations in Chapter H in conjunction with the provisions in Chapter E (columns), and Chapter F (beams) an efficient design pro- cedure is available to the structural engineer. An extension of these design criteria to the s
6、ingly symmetric (another term used frequently in the literature is mono-symmetric) I- shaped beam-column results in quite conservative designs when the compression flange is larger than the tension flange. This is especially so for the lateral-torsional buck- ling limit state. The predictions of the
7、 axial compression capacity and the in-plane and the lateral-torsional bending capacity in Chapters E and F of the AISC Specification, respectively, are reasonably accurate for these singly sym- metric members. However, the beam-column interaction equations connecting the pivotal points of only axia
8、l force and only bending moment in the axial force-bending moment interaction space are too conservative for these shapes. This paper will present an alternate method that is more appropriate to the actual conditions existing in singly symmetric beam-columns. The method can deal rationally with the
9、case of uniform bending about the axis of symme- try (the x-axis of the cross section). The cases of non-uni- form bending and bi-axial bending are not discussed. The paper will first consider the fully plastic in-plane capacity of a zero-length member subject to an axial force applied through the g
10、eometric centroid of the cross section and a bending moment applied in the plane of symmetry. The cross section is assumed to be fully yielded under the applied axial force and bending moment. The cross section capacity is then modified to account for the fact that under compression the strength of
11、a real member of a given length is reduced from the fully yielded case. This reduction is a proportion of the in-plane column strength formulas in Chapter E of the AISC Specification. The length effect is approximately accounted for by rotating the zero length interaction curve about the pivot P = 0
12、 and M = Mpuntil it ends at the point P = Pcrand M = 0. The next portion of the paper considers the lateral- torsional buckling strength of a singly symmetric beam- column subjected to axial force through the geometric centroid of the cross section, and equal applied end-bending moments. The elastic
13、 lateral buckling behavior is defined by a quadratic equation that expresses the relationship between the applied axial force and the end-bending moment (Galambos, 1968 and 1998). For a given moment this equation can be solved for the elastic axial capacity. This capacity is then reduced to approxim
14、ate the inelastic strength by using the procedure given in Appendix E3 of the AISC Specification (AISC, 1999). The applied bending moments are moments that are amplified to account for second-order bending of the mem- ber and the story (B1and B2, respectively, in Chapter C of the AISC Specification,
15、 or by explicit second-order analy- sis). The proposed methods are of necessity spreadsheet ori- ented, and an appendix provides a sample calculation scheme using the MATHCAD software as the vehicle of calculation. The example can be used to set up computa- tional schemes by other spreadsheet progra
16、ms, such as Excel or Quattro Pro. IN-PLANE BEHAVIOR The cross section of the singly symmetric shape is made up of three plates welded together into a wide-flange shape, as shown in Figure 1. The following derivation will develop the equations relating the axial load and the plastic bending moment wh
17、en the steel is perfectly plastic, that is, the stress Strength of Singly Symmetric I-Shaped Beam-Columns THEODORE V. GALAMBOS Theodore V. Galambos is emeritus professor of structural engineering, University of Minnesota, Minneapolis, MN. 2000-6.qxd 7/12/2001 10:00 AM Page 65 2003 by American Instit
18、ute 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 the publisher. 66 / ENGINEERING JOURNAL / SECOND QUARTER / 2001 is equal to the yield stress Fyeverywhere in the cross sec- tion. All three p
19、lates have the same yield stress. The fully yielded cross section can have its plastic neu- tral axis in the top flange (Figure 2), in the web (Figure 3) or in the bottom flange (Figure 4). When the neutral axis is in the top flange, equilibrium of forces requires that: This equation can now be solv
20、ed for yp, the distance of the neutral axis from the top of the top flange: where The limits of applicability of this equation are: yp= 0 when P = AFy(or p = 1, i.e. the whole cross section is yielded in tension); and yp = tf1when: The plastic moment can be obtained by taking moments about the centr
21、oid of the cross section, where the axial force P is assumed to act: The equations of the location of the plastic neutral axis and the plastic moment can be similarly derived for the other cases. The equations are given below. Plastic neutral axis in the web: bf1 tf1 dh tw tf2 bf2 y X Y Fig.1. Cross
22、 section of the singly symmetric shape. P Mpc centroidal axis yp Fy Fy Fig. 3. Plastic neutral axis in the web. 1 2 =+ yfp PFAby(1a) 1 1 2 =+ p f A yp b (1b) = y P p AF 111 22 11= = fff t bA p AA P Mpc centroidal axis yp Fy Fy Fig. 2. Plastic neutral axis in the top flange. P Mpc centroidal axis Fy
23、Fy yp Fig. 4. Neutral axis in the bottom flange. () 2 111 2 12 22 22 + + = + pfp fpffp pcy f wff yty byybtyy MF t h AtyAdy (2) () () 1 11 1 22 1 -for -1-1 2 + =+ fw ff pf ww AAAA A yptp ttAA (3) () () 1 1 1 1 1 1 2 2 - 22 2 2 2 + + + =+ + fp pf fw fp pcyfp f f ty yty t Ayt thy MFthyy t Ady (4) 2000-
24、6.qxd 7/12/2001 10:00 AM Page 66 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 the publisher. ENGINEERING JOURNAL / SECOND QUARTER / 2001 / 67 Plastic neutral axis
25、in the bottom flange: The interaction curves for the full plastic capacity of a zero-length tee-shape is shown in Figure 5. The plastic limit envelope is the heavy solid line. The axial force P is applied through the geometric centroid of the cross section. The directions of the applied forces are s
26、hown in Figure 6 for the four quadrants of the interaction space. Also shown in Figure 5 are the curves representing the relationship between axial force and bending moment when the stress at the extreme fiber is equal to the yield stress, and the inter- action curve according to the specification o
27、f the American Institute of Steel Construction (AISC, 1999). The axial force axis shows the non-dimensional ratio P/Py, where Py= AFyis the axial force when the whole cross section is yielded in compression or tension. The bending moment axis shows the non-dimensional ratio M/Mp, where the plas- tic
28、 moment Mpis defined by the formulas given in Appen- dix 1. The formulas for the first yield interaction relationship for the first quadrant (P is compressive and M causes com- pression on the top of the top fiber) are as follows: For compressive yielding at the top of the top flange For tensile yie
29、ld at the bottom of the bottom flange Similar expressions hold for the other three quadrants shown in Figure 6. The AISC-type interaction equations for the first quadrant of the zero-length member are: In viewing Figure 5 it is evident that for a zero-length member the AISC interaction equations are
30、 conservative in the first and third quadrants, while in the second and fourth quadrant they are slightly unconservative for this particular cross section. Equations 9 and 10 are not strictly valid for tee-shapes according to the AISC Specification Section F1.2c. There the nominal moments are limite
31、d to 1.5Myand My, respectively, depending on whether the flange or the stem are in compression (Myis the yield moment). Because the Equations 9 and 10 are not exactly in conformance they are named “AISC-type formulas” herein. The previous derivations, equations, and plots are appli- cable for the in
32、-plane strength of the cross section, that is, for a zero-length member. In the AISC Specification (AISC, 1999) the formulas for the cross section are generalized by redefining the terms in the interaction equations: the applied moment, M, is defined as the amplified moment obtained from a second-or
33、der analysis of the frame. Alter- nately, M may be determined by the approximate procedure ;y MOMENT CAPACITY, M/Mp -1.5-1.0-0.50.00.51.01.5 AXIAL CAPACITY, P/Py -1.0 -0.5 0.0 0.5 1.0 PLASTIC MOMENT ELASTIC MOMENT AISC Fig. 5. Interaction diagram for a zero-length tee-shape.Fig. 6 Direction of axial
34、 force and bending moment on the cross section. ()()1 2 2 1 for 11 2 fw p f AA A p ydp bA + =+ () () 1 11 1 1 2 22 2 2 + + = + f fwf pf pcypf f p p t h AyAty yth MFythy b dy dyy (5) (6) Plastic Moment Capacity of Cross Section tf1= 0.75 in.;tf2= 0.75 in.;tw= 0.5 in.;h= 20 in.; bf1= 15 in.;bf2= 0.5 i
35、n.;Fy= 50 ksi M P 12 34 y x PMy F AI += (7) () = y x M dyP F AI (8) 1.0 for 0.2 2 += ypy PMP PMP (9) 8 1.0 for 0.2 9 += ypy PMP PMP (10) 2000-6.qxd 7/12/2001 10:00 AM Page 67 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be
36、reproduced in any form without the written permission of the publisher. 68 / ENGINEERING JOURNAL / SECOND QUARTER / 2001 given in Chapter C, that is, M = B1Mnt+B2Mlt, where B1and B2are amplification factors, and Mntis the moment due to applied forces causing no lateral translation, and Mltis due to
37、forces causing lateral translation of the story under con- sideration. Mntand Mltare obtained in this case by perform- ing a first-order analysis of the frame. The maximum value of the axial force P is taken in the AISC procedure as the yield strength of the cross section, Py= AFy, when the axial fo
38、rce is in tension, and Pmax= Pcr, the in-plane critical load computed by the column formulas in Section E2 (AISC, 1999). In effect, then, the zero-length interaction curve is rotated so that its end at M = 0 is anchored to the point P = Pcr. This is shown in Figure 7 for a singly symmetric mem- ber.
39、 The geometric rotation of the interaction curve is accom- plished by assuming that the compressive stress varies from the yield stress Fyat P = 0 to the critical stress Fcr when P = Pcr. For any value of P between P = 0 and P = Pcr linear reduction is assumed, that is The equations for the plastic
40、capacity (Equations 1 through 6) are consequently modified as follows: Plastic neutral axis in the top flange: Plastic neutral axis in the web: Plastic neutral axis in the bottom flange: The axial capacity Pcrin these equations is the critical load in the plane of symmetry (the y-y axis in Figure 1,
41、 i.e. buckling is about the x-x axis). The value of Pcris deter- () = yycr cr P FFF P (11) Fig. 7. Rotation of interaction curve. Geometrical Approximation of the Effect of Member Length, In-Plane Behavior 1 1 11 for 0 11 + = + f p f yy A App y bA FF (12) () 1 1 11 2 12 2 2 22 = + + + p pcfp fp ffp
42、y f wff y Mbyy ty btyy F t h AtyAdy (13) 111 1 11 for 11 + + =+ + ffwf pf ww yy AAAA App yt ttAA FF () () 1 1 1 1 1 1 2 2 2 22 2 2 f pcf fp p f w fp fpy f fy t MAy ty y ty t thy thyy F t Adyf = + + + + 1 1 22 1 1 1 1 for 1 1 + + =+ + + + + fw pf ff y fw y AA Ap yth bb F AA p A F () () 1 11 1 1 2 22
43、2 2 =+ + + f pcfwf pf pf f p py t h MAyAty yth ythy b dy dyy F (14) (15) (16) (17) 2000-6.qxd 7/12/2001 10:00 AM Page 68 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
44、of the publisher. ENGINEERING JOURNAL / SECOND QUARTER / 2001 / 69 mined by the column formulas in Section E2 of the AISC Specification: Typical interaction diagrams are presented in Figures 8 and 9. The curves in Figure 8 are for built-up shapes: a dou- bly symmetric wide-flange shape, a cross sect
45、ion for which the bottom flange is half as wide as the top flange, and a tee shape. The solid lines represent in-plane strength, while the dashed lines are for lateral-torsional buckling behavior (to be discussed in the next section of this paper). The heavy lines are depicting the curves of the ful
46、ly plastic capacity defined by Equations 12 through 17, and the thin lines rep- resent the AISC-type interaction equations. The information in Figure 9 is for rolled shapes: a 2L843/4double angle, and a WT1867.5 tee section. From these typical curves it can be observed that: The AISC-type interactio
47、n equations are an excellent approximation for a doubly symmetric wide-flange shape; In the second and fourth quadrants (see Figure 6 for definition of the directions of the cross-sectional forces), assuming that the top flange is larger than the bottom flange, the AISC-type curves are close to the fully plastic curves; and In the first and third quadrant the AISCtype approach can be quite conservative, especially for tee-shapes. tf1=0.75in; tf2=0.75in; tw=0.5in; h=20in; bf1=15in; bf2=15in; L=15ft; Fy=50ksi; MOMENT CAPACITY, M/Mp -1.5-1.0-0.50.00.
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