DESIGN OF BEAM-COLUMNS - II

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1 DESIGN OF BEA-COLUNS-II 14 DESIGN OF BEA-COLUNS - II 1.0 INTRODUCTION Beam-columns are members subjected to combined bending and axial compression. Their behaviour under uniaxial bending, biaxial bending and torsional flexural buckling were discussed in art I on this topic in the previous chapter. It was shown that a range of behaviour varing from flexural ielding to torsional flexural or flexural buckling is possible. In this chapter evaluation of strength of beam-columns is discussed. The steps in the analsis of strength of beam-column are presented along with an example. 2.0 STRENGTH OF BEA-COLUNS The discussions in the part I of this topic, clearl indicated that the behaviour of beamcolumns is fairl complex, particularl at the ultimate stage and hence exact evaluation of the strength would require fairl complex analsis. However, for design purposes, simplified equations are available, using which it is possible to obtain the strength of members, conservativel. These are discussed below. 2.1 odes of Failure The following are the possible modes of failure of beam-columns Local section failure This is usuall encountered in the case of short, stock beam columns (λ/r << 50) with relativel smaller axial compression ratio (/ d < 0.33) and beam-columns bent in reverse curvature. The strength of the end section reached under combined axial force and bending moment, governs the failure. The strength of the section ma be governed b plastic buckling of plate elements in the case of plastic, compact and semi-compact sections or the elastic buckling of plate elements in the case of slender sections (see the chapter on plate buckling) Overall instabilit failure under flexural ielding This tpe of failure is encountered in the case of all members subjected to larger compression (/ d > 0.5) and single curvature bending about the minor axis as well as not ver slender members subjected to axial compression and single curvature bending about the major axis. Version II 14-1

2 DESIGN OF BEA-COLUNS-II The member fails b reaching the strength of the member at a section over the length of the member, under the combined axial compression and magnified bending moment. In the case of weak axis bending of slender members (λ/r > 80), the failure ma be b weak axis buckling, or failure of the maximum moment section under the combined effect of axial force and magnified moment. The section failure ma be due to elastic or plastic plate buckling depending on the slenderness ratio (b/t) of the plate (See the chapter on plate buckling) Overall instabilit b torsional flexural buckling This is common in slender members (λ/r>80) subjected to large compression (/ d >0.5) and uniaxial bending about the major axis or biaxial bending. At the ultimate stage the member undergoes biaxial bending and torsional instabilit mode of failure Design Equations as per IS: 800 The design code specifies, as given below, the linear interaction equations to check the section strength to prevent local section failure as well as member failure b flexural ielding and torsional flexural buckling. These are conservative simplifications of the non-linear failure envelopes discussed in the previous chapter Local section failure The Indian Standard Specification IS: 800 clearl deals with all tpes of members and mainl classifies the members in two groups, namel lastic and Compact Sections and Semi-Compact sections. The strength of Slender embers ma be analsed b following the procedure discussed in the chapters on cold-formed steel members. IS: 800 states that under combined axial force and bending moment section strength as governed b material failure and member strength as governed b buckling failure have to be checked as given below; 1. Section Strength lastic and Compact Sections In the design of members subjected to combined axial force (tension or compression) and bending moment, the following should be satisfied nd α 1 + nd α 2 Conservativel, the following equation ma be used under combined axial force and bending moment d + d + d ( 1 ) ( 2 ) Version II 14-2

3 DESIGN OF BEA-COLUNS-II, = factored applied moments about the minor and major axis of the cross section, respectivel nd, nd = design reduced flexural strength under combined axial force and the respective uniaxial moment acting alone = factored applied axial force d = design strength in compression due to ielding given b d = A g f / γ m0 d, d = design strength under corresponding moment acting alone A g = gross area of the cross section α 1, α 2 = constants as given in Table 1 n = N / N d Table: 1 Constants α 1 and α 2 Section α 1 α 2 I and Channel 5n 1 2 Circular tubes 2 2 Rectangular tubes 1.66/(1-1.13n 2 )< /(1-1.13n 2 ) < 6 Solid rectangles n n 3 For plastic and compact sections without bolts holes, the following approximations ma be used for evaluating nd and nd : a) lates nd = d (1-n 2 ) b) Welded I or H sections 2 n a nd = d 1 d 1 a nd = d (1-n) / (1-0.5a) d n = / d and a = (A - 2 b t f ) / A 0.5 c) For standard I or H sections for n < 0.2 nd = d\ for n>0.2 nd = 1.56 d (1-n) (n+0.6) nd = 1.11 d (1-n) d d) For Rectangular Hollow sections and Welded Box sections When the section is smmetric about both axes and without bolt holes nd = d (1-n) / (1-0.5a f ) d Version II 14-3

4 DESIGN OF BEA-COLUNS-II nd = d (1-n) / (1-0.5a w ) d a w = (A - 2 b t f ) / A 0.5 a f = (A-2 h t w ) / A 0.5 e) Circular Hollow Tubes without Bolt Holes nd = 1.04 d (1-n 1.7 ) d Semi-compact section In the absence of high shear force semi-compact section design is satisfactor under combined axial force and bending, if the maximum longitudinal stress under combined axial force and bending, fbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb x, satisfies the following criteria. f x f /γ m0 For cross section without holes, the above criteria reduces to + + d d d d, d, d are as defined earlier ( 3 ) 2. Overall ember Strength embers subjected to combined axial compression and moment shall be checked for overall buckling failure as given below: d d + k C + 0.6k m d C m + k d LT + k d Cm d ( 4 ) C m, C m = Equivalent uniform moment factor as per Table 2 = applied axial tension or compression under factored load, = maximum factored applied bending moments about and -axis of the member, respectivel. d, d = design strength under axial tension or compression as governed b buckling about minor () and major () axis respectivel. d, d = design bending strength about (minor) or (major) axis of the cross section K = 1+(λ -0.2)n n K = 1+(λ -0.2)n n Version II 14-4

5 0.1λ LT n 0.1n K LT = 1 1 C 0.25 C 0.25, n, n = C mlt = ( ) ( ) mlt mlt DESIGN OF BEA-COLUNS-II ratio of actual applied axial force to the design axial strength for buckling about the and axis, respectivel and Equivalent uniform moment factor for lateral torsional buckling as per Table 2 corresponding to the actual moment gradient between lateral supports against torsional deformation in the critical region under consideration. ore accurate evaluation of beam-column strength is possible b resorting to non-linear - analsis. In this case, the actual axial compression and bending moments as obtained from such an analsis ma be used in the interaction equation and the swa effects ma be disregarded in evaluation of u, ex and e. These methods of analsis and design are beond the scope of this chapter and are not discussed herein. 3.0 STES IN ANALYSING A BEA-COLUN (i) Calculate the cross section properties. Area, principal axes moments of inertia, section moduli, radii of gration, effective lengths and slenderness ratios. (ii) (iii) (iv) Evaluate the tpe of section based on the (b/t) ratio of the plate elements, as plastic, compact, semi-compact, or slender. Check for resistance of the cross-section under the combined effects as governed b ielding (Eq. 1,2 or 3). Check for resistance of member under the combined effects as governed b buckling (Eq. 4). Version II 14-5

6 DESIGN OF BEA-COLUNS-II Table: 2 Equivalent Uniform oment Factor Bending moment diagram Range Uniform loading C m, C m, C mlt Concentrated load 1 ψ ψ 0.4 h ψ 0 α s 1 1 ψ α s α s 0.4 s ψ h α s = s / h 1 α s 0 0 ψ α s α s ψ 0 0.1(1 ψ) 0.8 α s (1 ψ) 0.8 α s α s 1 1 ψ α h α h h s ψ h 0 ψ α h α h 1 α s 0 1 ψ α h (1+2 ψ) α h (1+2 ψ) For members with α h = swa h / s buckling mode the equivalent uniform moment factor C m = C m = 0.9. C m, C m, C mlt shall be obtained according to the bending moment diagram between the relevant braced points oment factor Bending axis oints braced in direction C m - - C m - - C mlt - - for C m for C mlt for C m Version II 14-6

7 DESIGN OF BEA-COLUNS-II 4.0 SUARY This chapter presented equations for the design of beam-columns and an example design. The behaviour and design of beam-columns are contained in the two parts on this topic. The following are the important points discussed in these chapters. The beam-column ma fail b reaching either the ultimate strength of the section (in the case of smaller axial load and shorter members) or b the buckling strength as governed b weak axis buckling or lateral torsional buckling. At lower loads, the failure is likel to be after the formation of the plastic hinges, especiall in the case of shorter members. In slender beam-columns with larger axial compression, either weak axis or lateral torsional buckling would control failure. The interaction formulae given for the design are conservative and simple, considering the complicated nature of beam-column failure. In the design of beam-columns in frames, the magnification of moment due to -δ and - effects are to be considered. 5.0 REFERENCES 1. IS: 800(2007), General Construction in Steel Code of ractice, Bureau of Indian Standards, New Delhi, Dowling.J, Knowles and Owens, G.W., Structural Steel Design, Butterworth, London, Eurocode 3: 1992, Design of Steel structures, British Standards Institution. Version II 14-7

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