Theme 8 Stability and buckling of members

Transcription

1 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 University Ostrv

2 Stility Stility is n ility to keep or renew the origin equilirium stte o the system without spontneous growing o deormtions. Bsic principles / 33

3 Stte o stility, indierence, unstle stte Q l ) ) c) < > Slender memers, xilly compressed cn turn side rom theirs origin stright position. ) Stte o stility memer gets ck to its origin stte ) Indierence stte (just theoreticl stte) memer stys in deormted position ut deormtions don t grow ny more c) Unstle stte spontneous growing o deormtions. Bsic principles 3 / 33

4 Buckling Consider pin-ended column xilly compressed y the orce. When this lod is inesed, the column ecomes unstle, trnsverse delection occurs nd the result is the collpse. This phenomenon is clled uckling. I the column is slender, uckling occurs t stress elow the yield limit (this is clled iticl stress - is not relted to the strength o mteril). l This is very complicted prolem, which is ected y: mteril chrcteristics geometricl chrcteristics lod type o supports origin stte o stress internl strin mnucturing or ssemling imperections Bsic principles / 33

5 Euler s solution o stility o stright memer is or the simplest model. Conditions o solution: idel elstic mteril stright memer xil lod theory o smll deormtions - ε << 1 delection w lrge, stticl eects re set on deormted memer - theory nd order Leonhrd Euler ( ) Loss o the stility ecomes when the lod reches the vlue o iticl orce π. E I. ( L ) l l l l Euler s solution o stility o stright memer 5 / 33

6 Theory o the II-nd order in uckling The term Smll deormtions theory : Chnges o shpe o structure re smll with the spect to its sie (dimensions). Thn we cn use lot o mthemticl simpliictions, which usully led to liner dependency. Theory o smll deormtions Theory o lrge deormtion (inite deormtion) H w << l H w l Theory o the II-nd order - Geometric nonlinerity Theory I-st order l w l Leds to derivtion o iticl orce π. E I. ( L ) M y H.l M y H.l+.δ The equilirium conditions we set on the deormted structure (uckling o columns). 6 / 33

7 Euler s solution o stility o stright memer Criticl orce: π. E I. ( L ) Buckling lenght: L β l L - Buckling lenght is equl to the lenght o the sinus hlwve o n elstic curve ter deormtion the lenght o inlex points. Euler s solution o stility o stright memer I The em in uckling will turn side rom its origin stright position in direction o the smllest strength (I min ). 7 / 33

8 Criticl orce, Criticl stress, Buckling, Slenderness σ E. I π. E. I E. i E σ π. π. π. A L A. L L λ or n idel elsto-plstic mteril (steel) the reltion o iticl stress cn e represented in reerence system y s cuic hyperol, clled Euler s hyperol: simple pressure σ π E. λ λ L i slenderness o memer [-] chrcteristic o the r in wht concern its stility, depending on the length nd the supports o the r, the shpe nd the mgnitude o its oss section. i I A rdius o inerti [m] This hyperol is theoreticl one, ecuse it neglects the imperections o the rel r. Euler s solution o stility o stright memer 8 / 33

9 Criticl orce, Criticl stress, Buckling Slenderness The iticl stress t uckling isn t constnt o the mteril, ecuse it depends on the slendernessλ. The hyperol is limited t the vlue o the yield limit y nd in wht concern the slendernessλ, the hyperol is limited to λ mx. 1) Limittion rom the let: Gr is limited y y. Simple pressure is deciding the r ils y yielding. simple pressure λ 0 Limittion o the Euler s hyperol rom the let nd rom the right. σ E π. λ E λ π π y, ) σ hs smller vlues s y - prolem o stility - Euler s hyperol; the r ils y uckling. 1) ) Euler s solution o stility o stright memer 3) 3) Limittion rom the right: λ Slenderness is so ig, tht the crrying cpcity in uckling is erro nd the memer loses its crrying unction λ mx 90 9 / 33

10 Tesco, Ostrv Columns 10 / 33

11 Tesco, Ostrv Detil o column supports 11 / 33

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

13 Exmple Column: IPN00 nd L,0 m. 1) Determine slenderness o the column (decided vlue is the lrger one), iticl orce, iticl Eulers stress ) Compre the solution with the simple pressure. x y IPN00 l m 1. Buckling: β 1 L β l 1 m Slenderness: i i y 80mm λ 18,7 mm y λ L i y L i 50 rom tles: 13, 9 γ i i I I e M yk 360 / S35 1,00 35MP A 330mm y y 1, 10 1, mm 18,7mm 6 6 mm mm λ is igger, the em turns side perpendiculrly to xes. 13 / 33

14 Exmple Euler orce: E I π L 151, 56 kn l m Euler stress: E σ π. λ σ 5,38MP A π E or : 5,38MP λ y x IPN00. Simple pressure: γ yk M 35 1,00 35MP mx mx A ,9 10 N 78,9kN A Conclusion: uckling is deciding, mximl orce is 150kN ccording to uckling. Some higher vlue o orce would cuse the lost o stility. 1 / 33

15 Determine dimeter d o the steel r centrlly loded y the orce ccording to simple pressure nd lso ccording to uckling. 5 0 kn, 35 MP, l,0m, E,1 10 MP, d Exmple 3 yk λ M 1,0 l m d? 1. Simple pressure: yk 35MP γ M Amin A πd A πd d req π 0,010m 15 / 33

16 Exmple 3. Buckling: d? π d I π d L 6 π E req L 6 L d req 3 6 E E π π I πd 6 l m d req 0,030m uckling decides Assessment: 1) ) Design: d 3mm I N N N Rd Rd Rd N Ed A ,11kN N 6 Ed πd d π 6 E I π 6,6kN 0kN L π req 0, Stisying. 0,03 Stisying. 7 m 166, N 16 / 33