Failure of slender and stocky columns (2nd year)

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Lecture slides on the mathematical derivation and application of the Euler's buckling load for slender column, as well as the Rankine's failure load.

Text of Failure of slender and stocky columns (2nd year)

  • 1. FailureofslenderandstockycolumnsDrAlessandroPalmeri

2. TeachingscheduleWeek Lecture 1 Staff Lecture 2 Staff Tutorial Staff1 Beam Shear Stresses 1 A P Beam Shear Stresses 2 A P --- ---2 Shear centres A P Basic Concepts J E-R Shear Centre A P3 Principle of VirtualforcesJ E-R Indeterminate Structures J E-R Virtual Forces J E-R4 The CompatibilityMethodJ E-R Examples J E-R Virtual Forces J E-R5 Examples J E-R Moment Distribution -BasicsJ E-R Comp. Method J E-R6 The Hardy CrossMethodJ E-R Fixed End Moments J E-R Comp. Method J E-R7 Examples J E-R Non Sway Frames J E-R Mom. Dist J E-R8 Column Stability 1 A P Sway Frames J E-R Mom. Dist J E-R9 Column Stability 2 A P Unsymmetric Bending 1 A P Colum Stability A P10 Unsymmetric Bending 2 A P Complex Stress/Strain A P UnsymmetricBendingA P11 Complex Stress/Strain A P Complex Stress/Strain A P ComplexStress/StrainA PChristmasHoliday12 Revision1314 Exams152 3. Mo@va@ons(1/5) Load-carryingstructuresmayfailinavarietyofways,dependingupon: Typeofstructure(truss,frame,) Condi@onsofsupport(pinned,fixed,) Loadsapplied(sta@c,dynamic,) Materialsused(briQle,duc@le,) Failuresarepreventedbydesigningstructuressothatmaximumstresses(strengthcriterion)andmaximumdisplacements(s,ffnesscriterion)remainwithinadmissiblelimits3 4. Mo@va@ons(2/5)4 ForthefansofTheBigBangTheory: SheldonandHowardhavegotthisseriouslywrong! YoucantusetheYoungsmodulustoquan@fythestrengthofmaterial,butitss,ffness! 5. Mo@va@ons(3/5) S@ffnessandstrengthofmaterials Inthestress-straincurveforaduc@lematerial(e.g.steel),theYoungsmodulusEdefinesthes@ffness,whiletheyieldstressyrepresentsthestrength5 6. Mo@va@ons(4/5) S@ffnesscriterion:SlenderColumn6 Strengthcriterion:ShortColumn 7. Mo@va@ons(5/5)7CoventryCathedralSlendercolumnDetailofthesupport 8. LearningOutcomes Whenwehavecompletedthisunit(2lectures+1tutorial),youshouldbeableto: DerivetheEulerscri@calloadforslenderpinned-pinnedcolumnsincompression Predictthemodeoffailureforbothshortandslendercolumnsincompression8 9. Furtherreading RCHibbeler,MechanicsofMaterials,8thEd,Pren@ceHallChapter13onBucklingofColumn THGMegson,StructuralandStressAnalysis,2ndEd,ElsevierChapter21onStructuralInstability(eBook)9 10. ShortandSlenderStruts10 Increasingthelengthofastrutreducesitsbucklingload Forinstance,amatchs,ckisreasonablystrongincompression(lek),butalongers,ck,withthesamecrosssec@onandthesamematerial,wouldbeweakerandbucklesincompression(right) 11. Buckling,i.e.LateralInstability(1/2)11 Thatis,ifacolumnisrela@velyslender,itmaydeflectlaterallywhensubjectedtoacompressiveforceP(Fig(a))andfailbybending(Fig(b)),ratherthanfailingbydirectcompressionofthematerial 12. Buckling,i.e.LateralInstability(2/2)12 Pcritistheso-calledcri,calbucklingload IftheaxialloadPislessthanPcrit,bendingiscausedbylateralloadsonly IfPisgreaterthanPcrit,therulerbendsevenwithoutlateralloads 13. EulersCri@calLoadforPinned-PinnedSlenderColumns OneoftheLearningOutcomesofthisUnitisforyoutobecomeabletomathema@callyderive(andrememberaswell)theexpressionofthecri@calloadPcritforpinned-pinnedslendercolumn Pcrit=PEisokencalledEulersbucklingload AkertheSwissmathema@cianLeonhardEuler(1707-1783)Pcrit =! 2 EIminL213 14. Mathema@calDeriva@on:BendingEqua@on Whatstheequa@onrulingthebeamsdownwarddeflec@on,uz(x),foragivenbendingmomentdiagram,My(x)? Weusedthissecond-orderdifferen@alequa@oninpartAtocalculatethebeamsdeflec@onundertransverseloads where,asusual: E=Youngsmodulus Iyy=Secondmomentofareaaboutthehorizontalneutralaxis14EIyyd2uz (x)dx2 = My (x) 15. Mathema@calDeriva@on:SignConven@on Doyourememberfromlastyear?15Sign convention Mostly vertical loads act vertically Downward deflection () is +ve Already chosen bending moment convention Sagging moment (l ) is +ve We MUST reconcile these two choices:xyloadxslopeyxy d 0d>xcurvaturey 2yxd >0d2But this is theshape of hoggingbending moment,i.e. M f = Materials yield stress 32. StrengthandS@ffnessCriteria(1/2) Strengthcriterion Stockycolumnstendtofailbecausetheelas@climitofthematerialisreached Thesafetychecksis: S,ffnesscriterion Slendercolumnstendtofailbecausetheelas@cconfigura@onisunstable Thesafetycheckis:322P P EIminP < P = f A< = y y E L2eBothmustbesa0sfied 33. StrengthandS@ffnessCriteria(2/2) ForbriQlematerialssuchasconcrete,theyieldingstressfyisreplacedwiththecrushingstressfc Thesafetycheckthenreads:332P P EIminP < P= fA< = c c E L2eBothmustbesa0sfiedfc 34. StrengthandS@ffnessCriteria34 TheRankinesfailureloadPRcombinesthesetwodifferentcriteria,thereforetakingintoaccountbothmaterialandgeometricalnonlineari@es PR=Pyfor=0 PRapproachesPEasgoestoinfinity0 100 200 300 4002.01.51.00.50.0lPPyPyP PERPR =Py PEPy + PEP/PyRankine(1820-1872)wasaScoushcivilengineer,physicistandmathema@cian 35. Ul@mateNormalStress35 experimentallyderived(dots)forwide-flangesteelcolumns asafunc@onoftheslenderness=kL/rmin 36. KeyLearningPoints1. Columnsincompressionmayfailbecause Insufficientbendings@ffness:Lateralbuckling Insufficientaxialcapacity:Yielding/Crushing2. EulersbucklingloadPEdependson: Minimumsecondmomentofarea,Imin Lengthofthecolumn,L Boundarycondi@ons3. Interac@onbetweenlateralbucklingandaxialcapacitycanbetakenintoaccountthroughtheRankinesformula36Effec@velength,LePR =Py PEPy + PEPE = 2 EIminL2e