Columns and Stability
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1 ARCH 331 Note Set 1. Su01n Columns nd Stilit Nottion: A = nme or re A36 = designtion o steel grde = nme or width C = smol or compression C c = column slenderness clssiiction constnt or steel column design d = nme or dimension, or depth e = eccentric distnce o ppliction o orce (P) rom the centroid o cross section E = modulus o elsticit or Young s modulus = il stress = ending stress criticl = criticl uckling stress in column clcultions rom P criticl = totl stress in the is direction F = llowle il stress F = llowle ending stress F = ield stress I = moment o inerti I min = moment o inerti tht is criticl to the clcultion o slenderness rtio K = eective length ctor or columns L = nme or length L e = eective length tht cn uckle or column design, s is e, L eective = internl ending moment, s is N.A. = shorthnd or neutrl is P = nme or il orce vector, s is P P crit = criticl uckling lod in column clcultions, s is P criticl, P cr r = rdius o grtion T = smol or compression W = designtion or wide lnge section = verticl distnce z = distnce perpendiculr to the - plne = clculus smol or smll quntit = displcement due to ending = ngle = dimeter smol = pi ( rdins or 180) = engineering smol or norml stress Design Criteri Including strength (stresses) nd servicilit (including delections), nother requirement is tht the structure or structurl memer e stle. Stilit is the ilit o the structure to support speciied lod without undergoing uncceptle (or sudden) deormtions. Phsics Recll tht things like to e or preer to e in their lowest energ stte (potentil energ). Emples include wter in wter tnk. The energ it took to put the wter up there is stored until it is relesed nd cn low due to grvit. 1
2 ARCH 331 Note Set 1. Su01n Stle Equilirium When energ is dded to n oject in the orm o push or disturnce, the oject will return to it s originl position. Things don t chnge in the end. Unstle Equilirium When energ is dded to n oject, the oject will move nd get more distured. Things chnge rpidl. Neutrl Equilirium When energ is dded to n oject, the oject will move some then stop.. Things chnge. Column with Ail Loding A column loded centricll cn eperience unstle equilirium, clled uckling, ecuse o how tll nd slender the re. This instilit is sudden nd not good. Buckling cn occur in sheets (like m memor metl cookie sheet), pressure vessels or slender (nrrow) ems not rced lterll. Buckling cn e thought o with the lods nd motion o column hving sti spring t mid-height. There eists lod where the spring cn t resist the moment in it n longer. Short (stu) columns will eperience crushing eore uckling.
3 ARCH 331 Note Set 1. Su01n Criticl Buckling Lod The criticl il 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 sts in the elstic rnge): EI π EI π EA min P criticl L or Pcr 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. Yield Stress nd Buckling Stress The two design criteri or columns re tht the do not uckle nd the strength is not eceeded. Depending on slenderness, one will control over the other. But, ecuse in the rel world, things re rrel perect nd columns will not ctull e loded concentricll, ut will see eccentricit Euler s ormul is used onl i the criticl stress is less thn hl o the ield point stress: EI min P criticl ; L criticl P A criticl F to e used or L e r C c E F where C c is the column slenderness clssiiction constnt nd is the slenderness rtio o column or which the criticl stress is equl to hl the ield point stress. 3
4 ARCH 331 Note Set 1. Su01n 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 ver long column is rced intermittentl long its length, the column length tht will uckle cn e determined. The eective length cn e ound multipling the column length n eective length ctor, K. K L L e 4
5 ARCH 331 Note Set 1. Su01n Loding Loction Centric loding: The lod is pplied t the centroid o the cross section. The limiting llowle stress is determined rom strength (P/A) or uckling. Eccentric loding: The lod is oset rom the centroid o the cross section ecuse o how the P em lod comes into the column. This oset introduces ending long with e il stress. (This cn lso hppen with continuous ems cross column or wind loding.) 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 5
6 ARCH 331 Note Set 1. Su01n Eccentric Loding The eccentricit cuses ending stresses moment o vlue P e. Within the elstic rnge (liner stresses) we cn superposition or dd up the norml nd ending stresses: P A I The resulting stress distriution is still liner. And the n.. moves (i there is one). The vlue o e (or loction o P) tht cuses the stress t n edge to ecome zero is t the edge o the kern. As long s P sts within the kern, there will not e n tension stress. I there is ending in two directions (i-il ending), there will e one more ending stress dded to the totl: With P, 1, nd : B 1 A I z I P z h z e 1 e A P + + = z result P e 1 P e1 B h z e 1 e A P + + = z result P e 1 P e1 6
7 ARCH 331 Note Set 1. Su01n Eccentric Loding Design Becuse there re comined stresses, we cn t just compre the il stress to limit il 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 il stress to llowle stress= 1 (no ending) to the rtio o ending stress to llowle stress = 1 (no il lod). F The interction digrm cn e more sophisticted (represented 1 curve insted o stright line). These tpe o digrms tke the eect o the ending moment incresing ecuse the em delects. This is clled the P- (P-delt) eect. 1 Limit Criteri ethods F 1) 1. 0 F F interction ormul (ending in one direction) ) 1. 0 F F F interction ormul (iil ending) 3) F gniict ion F ctor 1.0 interction ormul (P- eect) 7
8 ARCH 331 Note Set 1. Su01n Emple 1 (pg 346) 8
9 ARCH 331 Note Set 1. Su01n Emple (pg 346) 9
10 ARCH 331 Note Set 1. Su01n Emple 3 (pg357) 10
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