Theme 9 Stability and buckling of members

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1 Elsticit n plsticit Theme 9 Stilit n uckling o memers Euler s solution o stilit o stright elstic memer uner pressure Deprtment o Structurl Mechnics cult o Civil Engineering, VSB - Technicl Universit Ostrv

2 Stilit Stilit is n ilit to keep or renew the origin equilirium stte o the sstem without spontneous growing o eormtions. Bsic principles / 33

3 Buckling Slener memers, uner pressure cn turn sie rom theirs origin stright position. The ilit ginst it uckling. l This is ver complicte prolem, which is ecte : mteril chrcteristics geometricl chrcteristics lo tpe o supports origin stte o stress internl strin mnucturing or ssemling imperections Bsic principles 3 / 33

4 Stte o stilit, inierence, unstle stte Q l ) ) c) < > ) Stte o stilit memer gets ck to its origin stte ) nierence stte (just theoreticl stte) memer sts in eormte position ut eormtions on t grow n more c) Unstle stte spontneous growing o eormtions. Bsic principles / 33

Euler s solution o stilit o stright memer is or the simplest moel Conitions o solution: iel elstic mteril stright memer xil lo theor o smll eormtions stticl eects re on eormte memer (theor n orer)

5 Euler s solution o stilit o stright memer is or the simplest moel Conitions o solution: iel elstic mteril stright memer xil lo theor o smll eormtions stticl eects re on eormte memer (theor n orer) eonhr Euler ( ) oss o the stilit ecomes when the lo reches the vlue o iticl orce E. ( ) l l l l The em in uckling will turn sie rom its origin stright position in irection o the smllest strength ( min ). Euler s solution o stilit o stright memer 5 / 33

6 Euler s solution o stilit o stright memer Criticl orce:. E. ( ) Buckling lenght: β.l - Buckling lenght is equl to the lenght o the sinus hlwve o n elstic curve ter eormtion the lenght o inlex points. Euler s solution o stilit o stright memer 6 / 33

7 Criticl orce, Criticl stress, Buckling Slenerness σ E. E. E. i E. σ... A A. λ i A rius o inerti [m] simple pressure λ i slenerness o memer [-] imittion o the Euler s hperol rom the let n rom the right. imittions: λ 0 Gr is limite Simple pressure is eciing λ Slenerness is so ig, tht the crring cpcit in uckling is erro n the memer loses its crring unction λ mx 90 Euler s solution o stilit o stright memer 7 / 33

8 Tesco, Ostrv Columns 8 / 33

9 Tesco, Ostrv Detil o column supports 9 / 33

10 Aul VŠB-TU Ostrv Column supports 10 / 33

11 Exmple 1 Determine the mximum o i k 35MP, γ M 1,0, E, MP. Clculte simple pressure n uckling n compre oth vrints. 0,05 m x h 0,05 m l m Solution: 1/1. h 3 1/1 h. 3 6, m min ( >, sot xes, verticll to xes ) A. h 1, m Simple pressure: Buckling: γ k M N mx A A k A mx mx A λ Buckling length ter the ens o rs: β 0,7 β l 0,7 1, m E E 68, 8kN σ M 93,75kN Mximl orce is 68,8 kn. 11 / 33

12 Exmple Determine the iticl orce or PN00 n,0 m. Determine slenerness o the em (ecie vlue is the lrger one), iticl orce, iticl Eulers stress n compre the solution with the simple pressure. l m e 360 / S35, γ M 1,00, k 35MP Solution: 1. Buckling: β 1 β l 1 m Slenerness: A 330mm i i 1, 10 1, mm 18,7mm 6 6 mm mm x i 80mm λ i 50 PN00 i 18,7 mm λ 13, 9 i λ is igger, the em turns sie verticll to xes. 1 / 33

13 Exmple Euler orce: E 151, 56 kn l m Euler stress: σ 5,38MP A E or : 5,38MP λ x PN00. Simple pressure: γ k M 35 1,00 35MP mx mx A ,9 10 N 78,9kN A Conclusion: uckling is eciing, mximl orce is 150kN ter uckling. Some higher vlue o orce woul cus the lost o stilit. 13 / 33

14 Exmple 3 Determine imeter o the steel r uner the lo o the orce ter uckling n lso ter simple pressure. 5 0 kn, k 35 MP, l,0m, E,1 10 MP, λ M 1,0 l m? 1. Simple pressure: k 35MP γ M Amin A A req 0,010m 1 / 33

15 15 / 33 E E req E E req req E. Buckling: Assessment: 6 l m? Design: Exmple 3 Stising. uckling ecies kn kn E m mm m req 0 6, , ,030 7 > N kn N A N N N E R E R 116, ,11 0, Stising.

Theme 8 Stability and buckling of members

Theme 8 Stability and buckling of members Elsticity nd plsticity Theme 8 Stility nd uckling o memers Euler s solution o stility o n xilly compressed stright elstic memer Deprtment o Structurl Mechnics culty o Civil Engineering, VSB - Technicl