Consideration of Slenderness Effect in Columns

Transcription

1 Conideration of Slenderne Effect in Column Read Ainment Text: Section 9.1; Code and Commentary: 10.10, General Short Column - Slender Column - Strenth can be computed by coniderin only the column ection propertie. One whoe trenth i le than that computed baed on ection propertie; axial load and moment capacitie are inificantly affected by lenth, loadin condition of column Concentrically Loaded Column Euler potulated the phenomenon of elatic bucklin a: π EI t Pcr = ( KL) π EI t fcr = ( KL / r) where P cr = aximum poible axial load E t = Tanent modulu of column material at bucklin I = oment of inertia of the ection K = A calar to adjut for column end condition L = Column unupported lenth r = Radiu of yration of ection r = I / A Bucklin Load Veru Slenderne Ratio: Fiure 1. Effect of Slenderne on Strenth of Axially Load Column. 1

2 If the tre-train curve of hort piece of the iven member i of the hape of (a) below, a it would be for reinforced concrete column, E t i equal to Youn modulu, provided the bucklin tre P c /A i below the proportional limit f p.. If it i larer than f p, bucklin occur in the inelatic rane. In thi cae E t i the tanent modulu (the lope of the tre-train curve). A the tre increae E t decreae. A plot of the bucklin load v. the lenderne ratio, a o-called column curve (Fiure 1.b above), which how the reduction in bucklin trenth with increain lenderne. If the lenderne ratio i maller than (kl/r) min failure occur by cruhin. If the lenderne ratio i larer than (kl/r) min failure occur by bucklin, bucklin load or tre decreain for reater lenderne.

3 Evaluation of the "k" Coefficient Fiure. Bucklin and Effective Lenth of Axially Loaded Column. Comment on Axially Loaded Column A column miht be conidered hort under ome load condition and end condition, lender under other. Column braced aaint ide way have effective lenth between 0.5L and L. Column not braced aaint ide way alway have effective lenth reater than L. Hih trenth teel and concrete make lender column more common. Conideration of lenth effect become more important. Evaluation of k will be conidered in more detail in the next ection. 3

4 In reinforced concrete tructure we are not uually concerned with inle member but rather with riid frame of variou confiuration. See Fiure 3, if ided ay i prevented a indicated by a brace, the bucklin confiuration will be a hown in Fiure 3.a. The buckled hape of the column correpond to Fiure.c., except the lower end i hined. The unbraced lenth kl will be maller than l. On the other hand if no ide way bracin i provided to to an identical frame, the bucklin will look like Fiure 3.b. The column i in a ituation imilar to that of.d upide down, except that the upper end i not fixed but only partially retrained by the irder. Even thouh both frame in Fiure 3 are identical, the unbraced frame will buckle at a radically maller load than the braced frame. Fiure 3. Riid Frame Bucklin: (a) Laterally Braced; (b) unbraced. 4

5 Conideration of Second-Order Effect - Axial Load and Bendin A column under the influence of axial load and bendin will have a deformation at midpan (and in addition a maximum moment) which will be affected by the lenth and tiffne of the column (or "beam-column" a it may approximately be called). Conider a column bent in inle curvature by either end moment or lateral load: Fiure 4. oment in Slender ember with Compreion plu Bendin, Bent in Sinle Curvature. I there a method by which the influence of axial load may be related to oriinal deflection? It ha been hown by Timohenko and Gere that Where y = y y = y o = P = P cr = p / pcr Elatic deflection of beam-column, inle curvature Deflection of correpondin beam without axial load Applied axial load The critical axial load for the column without exterior moment 5

6 Johnon howed that with implified aumption, the maximum moment for the beam column could be written a 1 max = 0 1 p / pcr where max = aximum moment in the inly curved beam-column o = aximum moment in beam, axial load where 1/(1 - P/P cr ) i known a a moment manification factor, which reflect the amount by which the beam moment o i manified by the preence of a imultaneou axial force P. Implication A lenderne ratio increae, P cr decreae and max increae Fiure 5. Effect of lenderne and Effect of Axial Load on Column oment. Keep in mind that our interaction diaram, derived earlier for a ection i valid reardle of column lenth. We mut reconider it ue in liht of thee modification to load condition. Thu, we ee hih moment manification in column with inle curvature. What would occur in the cae of column with end moment of oppoite ene? 6

7 Reultin in Double Curvature. oment Diaram may take one of the followin eneral hape with maximum moment at or near end: Fiure 6. oment in Slender ember with Compreion Plu Bendin, Bent in Double Curvature. A a reult, our moment modification i mall. The eneral moment manification cae may then be written a: where cm max = 0 1 p / p cr C m = a factor of moment diaram relation 1 = member braced aaint ide way no tranvere loadin = 1.0 ide way, other cae. 7

8 i the larer moment: 1 i poitive if have inle curvature 1 i neative if have double curvature Fiure 7. Fixed Portal Frame, Laterally Unbraced. Fiure 8. Fixed Portal Frame, Laterally Braced. 8

9 ACI Read. (a) odulu of Elaticity ACI (b) oment of Inertia Beam I Column I Wall Uncracked I Cracked I Flat Plate and Flat Slab I (c) Area 1.0 A ACI Radiu of yration r = 0.30h for rectanular member, where h i in the direction tability i bein conidered, or r = 0.4Dfor circular member, where D i the diameter of the compreion member. How do we find the column riidity EI? Due to the fact that a reinforced column i a non-homoeneou member conitin of teel and concrete and concrete i ubjected to creep and hrinkae while teel i not, it i not eay to find EI exactly. If we try to do an exact analyi to find the EI, the value we find will be a ood a our aumption. ACI Section ay: EI 0.EI + EI = 1+ β c S e d ACI pae 18 Or conervatively EI 0.4EI c = 1+ β d ACI 10-1 pae 18 Where E c = odulu of elaticity of concrete, pi E = odulu of elaticity of teel, (9,000,000 pi) I = oment of inertia of ro ection (in 4 ) I = oment of inertia of reinforcement about the centroidal axi of member cro ection (in 4 ) β d = Ratio of maximum factored dead load moment to maximum factored total load moment, alway poitive. 9

10 factor β d account for the effect of creep in the concrete. Therefore, it i more appropriate to apply the term 1+β d to the term E c I /5 only becaue concrete i the one which creep. Eq i not unreaonable for lihtly reinforced concrete member, but reatly underetimate the effect of reinforcement of heavily reinforced member. ACI CODE CONSIDERATION OF LENGTH EFFECTS IN COLUNS A. Braced Frame. For moment reitin frame that i effectively braced aaint ide way by hear wall or diaonally braced frame: c = δ n ACI 10-8 where the moment-manification factor i iven a: cm δ n = 1.0 Pu 1 ACI P c P cr π EI = ACI ( kl ) t u where l u i the unupported lenth of compreion member For the frame braced aaint ide way and without load between upport (ACI 318 Sect ): C m 1 = ACI i the larer of ( 1 and ) 1 / > 0 1 / < 0 Sinle curvature Double curvature Other cae C m = 1.0 For column with no or very mall applied moment (i.e., axially or nearly axially loaded column), increain lenderne alo, reduce trenth. ACI ,min = Pu ( h) where 0.6 and h are in inche. 10

11 B. Unbraced Frame Becaue ide way can occur only for all column of a tory imultaneouly, rather than for any individual column, the ACI Code pecifie that in framed not braced aaint ide way, the value of amplification factor that pertain to the load cauin way hould be computed for the entire tory actin on unbraced frame. 1 = 1n + δ 1 ACI pae 130 = n + δ ACI pae 130 The moment manification factor are: (a) ACI The manified way moment δ hall be taken a the column end moment calculated uin a econd order analyi baed on the member tiffnee detailed above (ACI ). (b) ACI δ = 1 Q If δ calculated in thi way exceed 1.5, δ hall be calculated uin ACI or ACI (c) ACI Read ACI δ = Pu P ACI c Criteria for Nelect of Slenderne (ACI 10.1.) For compreion member braced aaint ide way, the effect of lenderne may be nelected when 1 Braced Frame: klu 34 1 ACI 10-7 pae 18 Unbraced Frame: kl u < ACI pae 130 and for all compreion member with 100 u kl < an analyi a defined by Section hall be made. 11