Rigid Frames - Compression & Buckling

Transcription

1 ARCH 614 Note Set 11.1 S014n Rigid Frmes - Compression & Buckling Nottion: A = nme or re d = nme or depth E = modulus o elsticity or Young s modulus = xil stress = ending stress z = stress in the x direction F = llowle xil stress F = llowle ending stress F x = orce component in the x direction F y = orce component in the y direction FBD = ree ody digrm G = reltive stiness o columns to ems in rigid connection, s is I = moment o inerti with respect to neutrl xis ending k = eective length ctor or columns = length o em in rigid joint L c L e M = length o column in rigid joint = nme or length = eective length tht cn uckle or column design, s is e = internl ending moment = nme or moment vector P = nme or xil orce vector, s is P P crit = criticl uckling lod in column clcultions, s is P criticl, P cr r = rdius o gyrtion V = internl sher orce y = verticl distnce = displcement due to ending = pi (180) = summtion symol = reltive stiness o columns to ems in rigid connection, s is G Rigid Frmes Rigid rmes re identiied y the lck o pinned joints within the rme. The joints re rigid nd resist rottion. They my e supported y pins or ixed supports. They re typiclly stticlly indeterminte. Frmes re useul to resist lterl lods. Frme memers will see sher ending xil orces nd ehve like em-columns. 1

2 ARCH 614 Note Set 11.1 S014n Behvior The reltion etween the joints hs to e mintined, ut the whole joint cn rotte. The mount o rottion nd distriution o moment depends on the stiness (EI/L) o the memers in the joint. End restrints on columns reduce the eective length, llowing columns to e more slender. Becuse o the rigid joints, delections nd moments in ems re reduced s well. Frmes re sensitive to settlement ecuse it induces strins nd chnges the stress distriution. Types Gled hs pek Portl resemles door. Multi-story, multiple y portl rmes re commonly used or commercil nd industril construction. The loor ehvior is similr to tht o continuous ems. Stggered Truss Full story trusses re stggered through the rme ys, llowing lrger cler stories. Connections Stggered Truss Steel Flnges o memers re ully ttched to the lnges o the other memer. This cn e done with welding, or olted pltes. Reinorced Concrete Joints re monolithic with continuous reinorcement or ending. Sher is resisted with stirrups nd ties. Brced Frmes Brced rmes hve ems nd columns tht re pin connected with rcing to resist lterl lods.

3 ARCH 614 Note Set 11.1 S014n Types o Brcing knee-rcing digonl (including eccentric) X K or chevron digonl X sher wlls which resist lterl orces in the plne o the wll sher wlls K (chevron) Compression Memers - Columns Including strength (stresses) nd servicility (including delections), nother requirement is tht the structure or structurl memer e stle. Stility is the ility o the structure to support speciied lod without undergoing uncceptle (or sudden) deormtions. A column loded centriclly cn experience unstle equilirium, clled uckling, ecuse o how tll nd slender they re. This instility is sudden nd not good. Buckling cn occur in sheets (like my memory metl cookie sheet), pressure vessels or slender (nrrow) ems not rced lterlly. Buckling cn e thought o with the lods nd motion o column hving sti spring t mid-height. There exists lod where the spring cn t resist the moment in it ny longer. Short (stuy) columns will experience crushing eore uckling. 3

4 ARCH 614 Note Set 11.1 S014n Criticl Buckling Lod The criticl xil lod to cuse uckling is relted to the delected shpe we could get (or determine rom ending moment o P ). The uckled shpe will e in the orm o sine wve. Euler Formul Swiss mthemticin Euler determined the reltionship etween the criticl uckling lod, the mteril, section nd eective length (s long s the mteril stys in the elstic rnge): EI π EI π EA P criticl or Pcr L Le Le r nd the criticl stress (i less thn the norml stress) is: Pcriticl EAr E criticl A A Le Le r where I=Ar nd L e r is clled the slenderness rtio. The smllest I o the section will govern, i the eective length is the sme or ox xes. Yield Stress nd Buckling Stress The two design criteri or columns re tht they do not uckle nd the strength is not exceeded. Depending on slenderness, one will control over the other. But, ecuse in the rel world, things re rrely perect nd columns will not ctully e loded concentriclly, ut will see eccentricity Euler s ormul is used only i the criticl stress is less thn hl o the yield point stress, in the elstic uckling region. A trnsition ormul is used or inelstic uckling. 4

5 ARCH 614 Note Set 11.1 S014n Eective Length nd Brcing Depending on the end support conditions or column, the eective length cn e ound rom the delected shpe (elstic equtions). I very long column is rced intermittently long its length, the column length tht will uckle cn e determined. The eective length cn e ound y multiplying the column length y n eective length ctor, K. K L L e 5

6 ARCH 614 Note Set 11.1 S014n Bending in Columns Bending cn occur in column like memers when there re trnsverse lods such s wind nd seismic lods, when the column is in rme, or when the column lod does not go through the xes. This sitution is reerred to s eccentric loding nd the moment is o size P x e. e P T T T C C C C T Wind Lod Let Tension Let Seismic Lod Let Tension Let Eccentric Lod Let Tension Let Wind Lod Let Brced t Top 6

7 ARCH 614 Note Set 11.1 S014n P- (delt) Eect The ending moment on column will produce lterl delection. Becuse there is n xil lod P on the column, there will e n ddition moment produced o the size P x, which in turn will cuse more delection, incresing the moment, etc.. This non-liner increse in moment is clled the P- eect. Design methods usully tke this into ccount with mgniiction ctors. Comined Stresses Within the elstic rnge (liner stresses) we cn superposition or dd up the norml nd ending stresses (where M cn e rom Pe or clculted): x P A My I x The resulting stress distriution is still liner. And the n.. cn move (i there is one)! x Interction Design Becuse there re comined stresses, we cn t just compre the xil stress to limit xil stress or ending stress to limit ending stress. We use limit clled the interction digrm. The digrm cn e simpliied s stright line rom the rtio o xil stress to llowle stress= 1 (no ending) to the rtio o ending stress to llowle stress = 1 (no xil lod). The interction digrm cn e more sophisticted (represented y curve insted o stright line). These types o digrms tke the eect o the ending moment incresing ecuse the em delects. This is clled the P- (P-delt) eect. F F

8 M+ ARCH 614 Note Set 11.1 S014n Limit Criteri Methods 1) 1. 0 F F interction ormul (ending in one direction) x y ) 1. 0 F F F x y interction ormul (ixil ending) ( Mgniict ion ctor ) 3) 1. 0 F F x interction ormul (P- eect) Rigid Frme Anlysis Structurl nlysis methods such s the portl method (pproximte), the method o virtul work, Cstiglino s theorem, the orce method, the slope-displcement method, the stiness method, nd mtrix nlysis, cn e used to solve or internl orces nd moments nd support rections. Sher nd ending moment digrms cn e drwn or rme memers y isolting the memer rom joint nd drwing ree ody digrm. The internl orces t the end will e equl nd opposite, just like or connections in pinned rmes. Direction o the em-like memer is usully drwn y looking rom the inside o the rme. P M+ M+ Frme Columns Becuse joints cn rotte in rmes, the eective length o the column in rme is hrder to determine. The stiness (EI/L) o ech memer in joint determines how rigid or lexile it is. To ind k, the reltive stiness, G or, must e ound or oth ends, plotted on the lignment chrts, nd connected y line or rced nd unrced mes. G EI l EI l c 8

9 ARCH 614 Note Set 11.1 S014n where E = modulus o elsticity or memer I = moment o inerti o or memer l c = length o the column rom center to center l = length o the em rom center to center For pinned connections we typiclly use vlue o 10 or. For ixed connections we typiclly use vlue o 1 or. Brced non-swy rme Unrced swy rme 9

10 + M + V ARCH 614 Note Set 11.1 S014n Lterl Buckling in Bems With compression stresses in the top o em, sudden popping or uckling cn hppen even t low stresses. In order to prevent it, we need to rce it long the top, or lterlly rce it, or provide igger I y. Torsionl uckling cn result with simultneous twisting nd ending, which cn e prolem with thin wlled, non-symmetric sections. Exmple 1 (pg 15) 3. + V + M + V + M 10

11 + M + V ARCH 614 Note Set 11.1 S014n Exmple (pg 154) 4. + V + M + V + M 11

12 ARCH 614 Note Set 11.1 S014n Exmple 3 Find the column eective lengths or steel rme with 1 t columns, 15 t em when the support connections re pins or ) when it is rced nd ) when it is llowed to swy. The reltive stiness o the em is twice tht o the columns (I). 1 t I I I 15 t 1