ME 354, MECHANICS OF MATERIALS LABORATORY COMPRESSION AND BUCKLING

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1 ME 354, MECHANICS OF MATERIALS LABATY COMPRESSION AND BUCKLING 01 January 000 / mgj PURPOSE Th purps f this xrcis is t study th ffcts f nd cnditins, clumn lngth, and matrial prprtis n cmprssiv bhaviur and bucling in clumns. EQUIPMENT Slid rds f varius lngths f aluum and stl Univrsal tst machin with grips, cntrllr, and data acquisitin systm PROCEDURE Rpat th fllwing stps fr ach spcimn. Masur th diamtr and lngths f ach spcimn t 0.0 mm. Zr th frc utput (balanc). Activat frc prtct (~50 N) n th tst machin t prvnt vrlading th spcimn during installatin. Install th tp nd f th tst spcimn in th tp grip f th tst machin whil th tst machin is in displacmnt cntrl. Install th bttm nd f th tst spcimn in th lwr grip f th tst machin. In displacmnt cntrl adjust th actuatr psitin f th tst machin t achiv narly zr frc n th spcimn. Dactivat frc prtct. Initiat th data acquisitin and cntrl prgram. Entr th crrct fil nam and spcimn infrmatin as rquird. Initiat th tst squnc via th cmputr prgram. Cntinu th tst until bucling r cmprssiv failur f th tst spcimn ccurs Exa th frc vrsus displacmnt trac fr ach tst. Nt th frc at th nst f bucling r cmprssiv failur (i.., significant dviatin frm linarity)

2 RESULTS Structurs and machins may fail in many ways dpnding n th matrials, inds f lads, and cnditins f supprt. Many machin lmnts can b mdld as unifrm mmbrs undr uniaxial tnsin r cmprssin. Fr tnsil lading, ths mmbrs tnd t slf-align and fail ithr by ductil dfrmatin r brittl fractur dpnding n th matrial. In cmprssin, th failur md is cmplicatd by th pssibility f a gmtric instability, calld bucling, in additin t ductil dfrmatin. Clumns ar structural mmbrs which supprt cmprssiv frcs. Bucling ccurs whn th clumn has a tndncy t dflct latrally, ut f th lin f actin f th frc. Onc bucling initiats, th instability can lad t failur f th clumn bcaus th ccntric frc acts as a mmnt causing gratr strsss and dflctins du t th cmbinatin f th bnding and axial frcs. Th pssibility f bucling incrass fr th fllwing clumn cnditins: 1) lngr, "thinnr" clumns, ) pinnd, fr, r nn-fixd nd cnditins, 3) initial ccntricity f th frc (.g., bnt clumns) and/r 4) lwr lastic mdulus f th clumn matrial. In this xrcis, tw matrials and tw clumn lngths will b studid. Anticipatd bucling r cmprssiv failur frcs will first b calculatd fr varius lngth spcimns and matrials. Fr cmprssiv failur, P and Fr bucling, P cr L I A (1). whr P is th cmprssiv failur frc (yild), is prprtinal limit strss (r yild strngth), A is th initial ara f th gag sctin, P cr is th Eulr critical bucling frc, I is th last mmnt f inrtia f th crss sctin, and L is th ffctiv, unsupprtd lngth f th clumn. Th anticipatd bucling r cmprssiv failur frcs will thn b cmpard t th actual masurd frcs at th nst f instability. Obsrvatins will b mad n th ffcts f nd cnditins, matrial typ, and clumn lngth. Shw all wr and answrs n th Wrsht, turning this in as th In-class Labratry rprt. Rfrncs: "Mchanics f Matrials," J.M. Gr and S.P. Timshn "Mchanics f Matrials," R.C. Hibblr

3 ME 354, MECHANICS OF MATERIALS LABATY COMPRESSION AND BUCKLING WK SHEET 01 January 000 /mgj NAME DATE EQUIPMENT IDENTIFICATION 1) Dtr (l up) th fllwing mchanical prprtis. Tabl 1 Slctd Prprtis fr Tst Matrials 6061-T6 Aluum 1018 Stl (CD) Slctd Mchanical Prprtis (R.T.) Slctd Mchanical Prprtis (R.T.) E (GPa) E (GPa) ( MPa) ( MPa) S UTS ( MPa) S UTS ( MPa) % lngatin % lngatin ) Masur and rcrd th fllwing dimnsins. Clumn Dimnsins fr Aluum Diamtr, d (mm) Lngth 1, L1 (mm) Lngth, L (mm) Tabl Prtinnt clumn dimnsins Clumn Dimnsins fr Stl Diamtr, d (mm) Lngth 1, L1 (mm) Lngth, L (mm) 3) Fr ach clumn, dtr th fllwing gmtric quantitis. Aluum Stl Mmnt f Inrtia: I πd 4 mm 4 Mmnt f Inrtia: I πd 4 mm d Crss sctinal ara: A π d mm Crss sctinal ara: A π mm 4 4 Radius f gyratin squard: I mm Radius f gyratin squard: I mm A A I Radius f gyratin: A mm Radius f gyratin: I A mm 4) Bucling f clumns with pinnd nds is ftn calld th fundamntal cas f bucling. Hwvr, many thr cnditins such as fixd nds, lastic supprts, and fr nds ar ncuntrd in practic. Th critical frcs fr bucling fr ach f ths nd cnditins can b dtrd by applying th apprpriat bundary cnditins and slving th diffrntial quatins. Ths slutins lad t th cncpt f an "ffctiv lngth," L, apprpriat fr ach nd cnditin which is a multipl f th actual lngth, L, f th clumn as shwn in Tabl 3 and Figur 1.

4 Tabl 3 Effctiv clumn lngth fr varius nd cnditins Pinnd/Pinnd Fixd/Fr Fixd/Fixd Pinnd/Fixd L L L L L/ L 0.7 L Figur 1 Illustratin f nd cnditins fr clumns 5) In gnral, axially-ladd cmprssin mmbrs may fail by n f thr mds: crushing; a cmbinatin f crushing r bucling; r bucling aln. Clumns can b placd int thr grups: 1) Shrt clumns - th failur md is by crushing (simpl cmprssiv failur) ) Intrmdiat clumns - th failur md dpnds n simpl cmprssiv and/r bnding strss 3) Lng clumns - th failur md is primarily a functin f th bnding strss (bucling). A paramtr which is mplyd t grup ths clumns is th slndrnss rati, L /. Th imum slndrnss rati L mars th nal transitin frm crushing t bucling. If th axial strss,, is plttd as a functin f slndrnss rati, thn th imum slndrnss rati is th nal transitin frm th cnstant strss fr crushing,, t th strss as functin f L / fr bucling, cr ( L / ). Aluum Stl Elastic mdulus: E MPa Elastic mdulus: E MPa Prprtinal limit strss: MPa Prprtinal limit strss: MPa Minimum slndrnss rati: L Minimum slndrnss rati: L

5 On th fllwing graphs, plt fr L L and cr ( L / ) fr L L. Allwabl Cmprssiv Strss (MPa) Slndrnss rati (L /) Allwabl Cmprssiv Strss (MPa) Slndrnss rati (L /) a) Allwabl cmprssiv strss fr aluum b) Allwabl cmprssiv strss fr stl Figur 1 Allwabl cmprssiv strss fr aluum and stl 6) Dtr th fllwing critical cmprssiv frcs fr th xprimntal clumns Aluum Stl Fr clumn lngth L1, th unsupprtd lngth if ach grip nd is l mm lng such that LL1-(* l) mm Fr clumn lngth L1, th unsupprtd lngth if ach grip nd is l mm lng such that LL1-(* l) mm Effctiv lngth, L using Tabl 3 Effctiv lngth, L using Tabl 3 fr th Fixd/Fixd nd cnditin mm fr th Fixd/Fixd nd cnditin mm Fr L1, slndrnss rati, L / Fr L1, slndrnss rati, L / Minimum slndrnss rati: L Minimum slndrnss rati: L if L L. MPa if L L MPa cr ( L / ) if L L. MPa cr ( L / ) if L L. MPa Crss sctinal ara, A mm Crss sctinal ara, A mm Us th smallr f th strsss calculatd abv. Fr L1, critical frc, P L1 cr A N Us th smallr f th strsss calculatd abv. Fr L1, critical frc, P cr L1 A N

6 Fr clumn lngth L, th unsupprtd lngth if ach grip nd is l mm lng such that LL-(* l) mm Fr clumn lngth L, th unsupprtd lngth if ach grip nd is l mm lng such that LL-(* l) mm Effctiv lngth, L using Tabl 3 Effctiv lngth, L using Tabl 3 fr th Fixd/Fixd nd cnditin mm fr th Fixd/Fixd nd cnditin mm Fr L slndrnss rati, L / Fr L, slndrnss rati, L / Minimum slndrnss rati: L Minimum slndrnss rati: L if L L cr ( L / ). MPa if L L if L L. MPa cr ( L / ) if L L MPa. MPa Crss sctinal ara, A mm Crss sctinal ara, A mm Us th smallr f th strsss calculatd abv. Fr L, critical frc, P cr L A N Us th smallr f th strsss calculatd abv. Fr L, critical frc, P cr L A N 7) Masur th actual critical cmprssiv frcs fr th xprimntal clumns. Fr L1, Aluum Fr L1, Stl Masurd critical cmprssiv frc,p L1 N Masurd critical cmprssiv frc,p L1 N Fr L1, critical frc, P cr A N Fr L1, critical frc, P cr A N % diff % diff Fr L, Aluum Fr L, Stl Masurd critical cmprssiv frc,p L N Masurd critical cmprssiv frc,p L N Fr L, critical frc, P cr A N Fr L, critical frc, P cr A N % diff % diff 8) Cmmnt n hw wll th quatins prdictd th actual critical cmprssin frc. Wr discrpancis rasnabl? If nt, what culd pssibl surcs f rrr b attributd t? (Rcall that th assumptins fr th bucling frcs assum n initial ccntricity, prfctly straight clumns, and n ff-axis lading). 9) As a dsignr, what stps can b tan t rduc th tndncy t bucl, gmtrically? matrial-wis?