BEA COLUNS Beam columns are structural members that are subjected to a combination of bending and axial stresses. The structural behaviour resembles simultaneousl to that of a beam and a column. ajorit of the steel building frames have columns that carr sizable bending moments in addition to the usual compressive loads.
The sources of this bending moment are shown in Figure 5.1 and explained below: e e e = e a) Out-Of-lumb b) Initial Crookedness c) Eccentric Load Figure 5.1. Sources of Eccentricit in Columns.
It is almost impossible to erect the columns perfectl vertical and centre loads exactl on columns. Columns ma be initiall crooked or have other flaws with the result that lateral bending is produced. In some cases, crane beams parallel to columnsline and other perpendicular beams rest on brackets projecting out from columns. This produces high values of bending moments.
Wind and other lateral loads act within the column height and produce bending. The bending moments from the beams are transferred to columns if the connections are rigid. CONTROLLING DESIGN FACTOR: SECOND ORDER EFFECTS The elastic analsis carried out to calculate deflections and member forces for the given loads is called 1 st order and analsis.
The high axial load present in the column combined with this elastic deflection produces extra bending moment in the column, as is clear from Figure 5.2. The analsis of structure including this extra moment is called 2 nd order analsis. Similarl, other higher order analsis ma also be performed. In practice, usuall 2 nd order analsis is sufficientl accurate with the high order results of much lesser numerical value.
aximum lateral deflection due to bending moment () δ Deflected shape or elastic curve due to applied bending moment () Extra moment = δ, which produces more deflections Figure 5.2. Eccentricit Due to First Order Deflections.
The phenomenon in which the moments are automaticall increased in a column beond the usual analsis for loads is called moment magnification or 2 nd order effects. The moment magnification depends on man factors but, in some cases, it ma be higher enough to double the 1 st order moments or even more. In majorit of practical cases, this magnification is appreciable and must alwas be considered for a safe design.
1 st order deflection produced within a member (δ) usuall has a smaller 2 nd order effect called - δ effect, whereas magnification due to sides-wa ( ) is much larger denoted b - effect (refer to Figure 5.3). -Delta effect is defined as the secondar effect of column axial loads and lateral deflections on the moments in members. The calculations for actual 2 nd order analsis are usuall length and can onl be performed on computers.
For manual calculations, empirical methods are used to approximatel cater for these effects in design. 2 nd order effects are more pronounced when loads closer to buckling loads are applied and hence the empirical moment magnification formula contains a ratio of applied load to elastic buckling load. The factored applied load should, in all cases, be lesser than 75% of the elastic critical buckling load but is usuall kept much lesser than this limiting value.
INTERACTION EQUATION AND INTERACTION DIAGRA The combined stress at an point in a member subjected to bending and direct stress, as in Figure 5.3, is obtained b the formula: Extra oment = A x f = ± ± I x I x Figure 5.3. A Deflected Beam-Column.
For a safe design, the maximum compressive stress (f) must not exceed the allowable material stress (F all ) as follows: A x f = ± ± F all AF all max I x I x + + 1 S x F all S x + + 1 x, max F x all, max This equation is called interaction equation showing interaction of axial force and bending moment in an eas wa.
If this equation is plotted against the various terms selected on different axis, we get an interaction curve or an interaction surface depending on whether there are two or three terms in the equation, respectivel. 1.0 0,0 1.0 Figure 5.4. A Tpical Interaction Curve.
r = required axial compressive strength ( u in LRFD) c = available axial compressive strength = φ c n, φ c = 0.90 (LRFD) = n / Ω c, Ω c = 1.67 (ASD) r = required flexural strength ( u in LRFD) c = available flexural strength = φ b n, φ b = 0.90 (LRFD) = n / Ω b, Ω b = 1.67 (ASD)
AISC INTERACTION EQUATIONS rof. Dr. Zahid Ahmad Siddiqi The following interaction equations are applicable for doubl and singl smmetric members: If r 0.2, axial load is considerable, and c following equation is to be satisfied: r c + 8 9 rx cx + r c 1.0
If r c rof. Dr. Zahid Ahmad Siddiqi < 0.2, axial load is lesser, beam action is dominant, and the applicable equation is: r rx r + + 1.0 2c cx c OENT ADJUSTENT FACTOR (C mx or C m ) oment adjustment factor (C m ) is based on the rotational restraint at the member ends and on the moment gradient in the members. It is onl defined for no-swa cases.
1. For restrained compression members in frames braced against joint translation (no sideswa) and not subjected to transverse loading between their supports in the plane of bending: C m = 0.6 0.4 1 2 where 1 is the smaller end moment and 2 is the larger end moment. 1 / 2 is positive when member is bent in reverse curvature and it it is negative when member is bent in single curvature (Figure 5.5b).
2 2 1 1 a) Reverse Curvature b) Single Curvature Figure 5.5. Columns Bent in Reverse and Single Curvatures. When transverse load is applied between the supports but or swa is prevented, for members with restrained ends C m = 0.85 for members with unrestrained ends C m = 1.0
K-VALUES FOR FRAE BEA-COLUNS K-values for frame columns with partiall fixed ends should be evaluated using alignment charts given in Reference-1. However, if details of adjoining members are not given, following approximate estimate ma be used: K = 1.2 1.5 K = 1 if sideswa is permitted with partiall fixed ends if sideswa is prevented but end conditions are not mentioned
OENT AGNIFICATION FACTORS oment magnification factors (B 1 and B 2 ) are used to empiricall estimate the magnification produced in the column moments due to 2 nd order effects. These are separatel calculated for swa or lateral translation case (lt-case) and for no-swa or no translation case (nt-case). Accordingl, the frame is to be separatel analsed for loads producing swa and not producing swa.
lt = moment due to lateral loads producing appreciable lateral translation. B 2 = moment magnification factor to take care of u effects for swa and deflections due to lateral loads. nt = the moment resulting from gravit loads, not producing appreciable lateral translation. B 1 = moment magnification factor to take care of u δ effects for no translation loads.
r = required magnified flexural strength for second order effects = B 1 nt + B 2 lt r = required magnified axial strength = nt + B 2 lt No-Swa agnification Cm B 1 = 1.0 1 α r e1
where α = 1.0 (LRFD) and 1.60 (ASD) e1 = Euler buckling strength for braced frame = π 2 EI / (K 1 L) 2 K 1 = effective length factor in the plane of bending for no lateral translation, equal to 1.0 or a smaller value b detailed analsis
Swa agnification The swa magnification factor, B 2, can be determined from one of the following formulas: where, B 2 = 1 1 α α = 1.0 (LRFD) and 1.60 (ASD) e2 nt Σ nt = total vertical load supported b the stor, kn, including gravit loads
Σ e2 = elastic critical buckling resistance for the stor determined b sideswa buckling analsis = Σπ 2 EI / (K 2 L) 2 where I and K 2 is calculated in the plane of bending for the unbraced conditions
SELECTION OF TRIAL BEA- COLUN SECTION rof. Dr. Zahid Ahmad Siddiqi The onl wa b which interaction of axial compression and bending moment can be considered, is to satisf the interaction equation. However, in order to satisf these equations, a trial section is needed. For this trial section, maximum axial compressive strength and bending strengths ma be determined.
The difficult in selection of a trial section for a beam column is that whether it is selected based on area of cross-section or the section modulus. No direct method is available to calculate the required values of the area and the section modulus in such cases. For selection of trial section, the beam-column is temporaril changed into a pure column b approximatel converting the effect of bending moments into an equivalent axial load.
eq = equivalent or effective axial load = r + rx m x + r m m x (for first trial) = m (for first trial) = 8.5 0.7K 1x L x 17 1.4K 1 L m x = 10 14(d / 1000)2 0.7K 1x L x m = 20 28(d / 1000)2 1.4K 1 L
The above equation is evaluated for eq and a column section is selected from the concentricall loaded column tables for that load. The equation for eq is solved again using a revised value of m. Another section is selected and checks are then applied for this trial section.
WEB LOCAL STABILITY rof. Dr. Zahid Ahmad Siddiqi For stiffened webs in combined flexural and axial compression: u E If 0.125 λ p = φb 2.75 u 3.76 1 F φb For A36 steel, λ p = 2.75 u 106.7 1 φb u E u If > 0.125 λ p = 1.12 2.33 1.49 φ b For A36 steel, λ p = F φ b 31.8 u 2.33 φb 42.3 E F where λ = h / t w and = F A g