CIVL473 Fundamentals of Steel Design CHAPTER 4 Design of Columns- embers with Aial Loads and oments Prepared B Asst.Prof.Dr. urude Celikag 4.1 Braced ultistore Buildings - Combined tension and oments Interaction curves for strength under combined loading F A e p Eqn 1 Eqn 2 c The furthest point on an of the aes from the origin represent the member s capacit under that form of Loading acting singl. An point falling within one of the surfaces represents a load combination that can safel be carried. BS 5950 uses the linear approimation shown in the above figure b the full lines and given b c F A p e c c 1 Eqn 1 A e /p = aial capacit c, c = moment capacit about major and minor aes Dr.urude Celikag 1
ore sophisticated analsis of this problem using the principles of plastic theor has shown that for compact cross-sections Equation 1 ma be replaced b z1 z 2 1 Eqn r r in which r, r are the reduced moment capacities about the major and minor aes respectivel in the presence of the aial load F z 1 2.0 for I- and H-sections, solid or hollow circular sections 5/3 for solid or hollow rectangular sections 1.0 in all other cases z 2 1.0 for all sections ecept solid or hollow circular sections 2.0 solid or hollow circular sections 5/3 for solid or hollow rectangular sections 2 The values of r, r can be obtained from the SCI s Blue Book. Alternativel the following equations can be used to calculate reduced moments. S r = (1-2.5n 2 )S S r = 1.125(1-n)S S r = (1-0.5n 2 )S S r = 1.125(1-n 2 )S for n<0.2 for n>0.2 for n<0.447 for n>0.447 S r, S r = reduced plastic modulus in the presence of aial load F S, S = plastic modulus for zero aial load n = F/Ap r, r should not eceed 1.2p Z and 1.2p Z respectivel Slenderness ratio,, should not eceed the following: for members resisting loads other than wind loads 180 for members resisting self-weight and wind loads onl 250 for an member normall acting as a tie but subject to reversal of stress resulting from the action of the wind 350 Dr.urude Celikag 2
4.2 Braced ultistore Buildings - Compression and Bending Condition I: Condition II: Columns braced in both directions and subject onl to nominal moments applicable to simple construction. Columns braced in both directions and subject applied moments other than nominal moments. Use iterative process to select and check a trial section. When subject to compressive loading the member s strength ma be limited b either of the two conditions: local capacit at the most heavil loaded cross- section overall buckling In the most general case, the beam-column is subject to compression plus moments about both aes. a) Loading and/or the beam arrangement is different at different levels, so the moments will not be the same at both ends b) If some beams are absent or when similar beams on opposite sides of the column carr identical loads then the moments will balance and the loading ma reduce to a simpler form Three distinct cases ma be identified as: Case 1: interaction between column buckling and simple uniaial beam bending. Case 2: interaction between column buckling and beam buckling. Case 3: interaction between column buckling and biaial beam bending. Case 3 is the most general case 1. The load is applied with an eccentricit about the minor ais. ember will collapse b ecessive deformation in this plane. 2. The load is applied with an eccentricit about the major ais. ember fails b a combination of bending about the weak ais and twisting, i.e. similar to beam lateral torsional buckling. F F F e e 3. The load is applied with an eccentricit about both aes. ember will collapse b combined bending and twisting. Dr.urude Celikag 3
Table 24. Nominal effective length, L E, for a strut (BS 5950:Part 1:1990) Conditions of restraints at ends (in plane under consideration) Effectivel held in Restrained in direction at both ends position at both ends Partiall restraint in direction at both ends Restrained in direction at one end Effective Length, L E 0.7L 0.85L 0.85L One end Effectivel held in position and restrained in direction NOT restrained in direction at either end Other end Effectivel restrained in Not held in position direction Partiall restrainewd in direction NOT restrained in direction 1.0L 1.2L 1.5L 2.0L Effective Lengths of Struts Dr.urude Celikag 4
The BS 5950 Compressive Strength Tables (Table 27 a, b, c and d) The are the combination of the original Perr Robertson curve of BS 449 the inclusion of section shape variation the allowance of residual stresses the allowance of stock column effect Compressive strength curves for different values of α Note. Stock columns take loads in ecess of the theoretical (ield stress area). The curves from which Table 27 are tabulated are shown in the figure with the empiricall modified values 0.001/r coefficient (residual stresses) Condition I- Design Procedure-Compression embers With Nominal oments 1) Calculate the factored beam reactions = 1.6 LL + 1.4 DL from the beam bearing onto the column from each ais at the level considered. It ma also be necessar to calculate the reactions for different load factors for different load combinations. 2) Calculate the factored aial load F on the column at level being considered. 3) Choose a section for the lowest column length from the following: 203 UC for buildings up to 3 stores high 254 UC for buildings up to 5 stores high 305 UC for buildings up to 8 stores high 356 UC for buildings from 8 to 12 stores high Otherwise use UB sections from Table 16 (anual) and use the alternative design procedure given at point g) 4) Calculate the nominal moments applied to the column. Factored beam reactions multiplied b the distance from the center of the beam + 100 mm 5) Obtain and applied to each length of the column above and below the beam connections according to the stiffness I/L of each length. When the ratio does not eceed 1.5 the moment ma be divided equall. Dr.urude Celikag 5
6) Choose a trial section and grade of steel such that the following equation is satisfied F c p c = factored aial load on the column = compressive strength A g = gross cross-sectional area, = nominal moment about major and minor ais bs = buckling resistance moment for simple columns Z p Fc A p g c = elastic modulus about minor ais = design strength bs p Z 1 Alternative design procedure for calculation of P c P c of a column ma be obtained from: P c =A g p c A g = gross cross sectional area of the trial section p c = compressive strength a) Choose a trial section avoiding slender UB sections and obtain p. b) Calculate the slenderness,, LE/r c) Determine pc from Table 27. d) Calculate Pc = Ag pc I-section D>1.2 B H-section D< 1.2 B Alternative design procedure for calculation of bs Buckling resistance moment capacit bs = p b S (Table 11) LT = 0.5 L/r Dr.urude Celikag 6
Condition II- Design Procedure - Compression embers Subject to Applied oments other than Nominal oments. 1) Calculate the factored beam reactions = 1.6 LL + 1.4 DL from the beam bearing onto the column from each ais at the level considered. It ma also be necessar to calculate the aial load using different load factors for different load combinations. 2) Calculate the factored moments and from the most unfavourable combination of dead and imposed loads using the load factors and load combinations from Table 1 (anual). 3) Calculate the ratios of the moments applied about both aes at each end of the column, and then determine the equivalent uniform moment factor m and m from Table 9 (anual). 4) Choose a trial section avoiding slender UB sections. If the m and m are equal to 1.0 then no need to carr out the local capacit check. Local capacit check: Check to be carried out at the location of the greatest bending moment and aial load. 1. Determine the design strength p from Table 2 2. Calculate F/A g p 3. Calculate the b/t ratio for the flange outstand and d/t ratio for the web. (t is the thickness of the element concerned. If the b/t ratio eceeds 15 or the d/t ratio eceeds 39 where = (275/p ) 1/2. Use strength reduction from BS 5950. 4. If a new section has been chosen then recalculate F/A g p 5. Obtain moment capacities c and c from the blue book and then calculate using the above formulae. 6. Finall, check that Fc A 1 g p c c c Overall buckling check: F c p c = factored aial compressive load on the column = compressive strength A g = gross cross-sectional area, = equivalent uniform moment-major and minor ais F A c g p c b p Z 1 b = buckling resistance moment b, as in beam condition III & IV, b < p Z Z, Z = elastic modulus about major and minor ais p = design strength for m is the greater of m LT and m for m = m [Table 9] Dr.urude Celikag 7