BUCKLING OF A COLUMN WITH TEMPERATURE DEPENDENT MATERIAL PROPERTIES

Transcription

1 AMUKKAE ÜNİ VERSİ ESİ MÜHENDİ Sİ K FAKÜESİ AMUKKAE UNIVERSIY ENGINEERING COEGE MÜHENDİ Sİ K Bİ İ MERİ DERGİ S İ JOURNA OF ENGINEERING SCIENCES YI Cİ SAYI SAYFA : : 7 : : BUCKING OF A COUMN WIH EMERAURE DEENDEN MAERIA ROERIES Ömr SOYKASA Afyon Kocatp Univrsity, Faculty of chnical Ecation, İmiryolu 8. Km, Kampüs/Afyon Gliş arihi : 7.4. ABSRAC Buckling of a column with tmpratur dpndnt matrial proprtis is invstigatd. Eulr-Brnoulli thory of thin bams is usd to driv th lmnt matrics by mans of th minimum potntial nrgy principl. mpratur dpndncy of matrial proprtis is takn into account in th formulation. h column is dividd into finit lmnts with th aial dgrs of frdom dfind at th outr fibr of th column. Column lmnts hav simplr drivations and compact lmnt matrics than thos of classical bam-bnding lmnt. Som illustrativ ampls ar prsntd to show th convrgnc of numrical rsults obtaind by th us of nw lmnts. h rsults ar compard with thos of th classical bam-bnding lmnt and analytical solution. h nw lmnt convrgs to th analytical rsults as powrful as th classical bam-bnding lmnt. h tmpratur ffcts on th buckling loads of th column with tmpratur dpndnt matrial proprtis ar also amind. Ky Words : Buckling, Column, Finit lmnt mthod, Stability MAZEME ÖZEİKERİ SICAKIĞA BAĞI BİR KOONUN BURKUMASI ÖZE Malm ölliklri sıcaklığa bağlı bir kolonun burkulması araştırılmaktadır. İnc kirişlr için Eulr-Brnoulli torisi, minimum potansiyl nrji prnsibi vasıtasıyla lman matrislrini çıkarmak için kullanılmaktadır. Formülasyonda malm ölliklrinin sıcaklığa bağımlılığı hsaba katılmaktadır. Kolon, kolonun n dış lifind tarif diln ksnl yönd srbstlik drcsin sahip sonlu lmanlara bölünmktdir. Kolon lmanları, klasik kiriş ğilm lmanından daha basit olarak çıkarılmaktadır v daha küçük lman matrislrin sahiptir. Yni lmanlar kullanarak ld diln sayısal sonuçların yakınsamasını göstrmk için baı örnklr sunulmaktadır. Sonuçlar hm klasik sonlu lman hm d ksin sonuçlarla karşılaştırılmaktadır. Yni lman, analitik sonuçlara klasik kiriş ğilm lmanı kadar güçlü bir şkild yakınsamaktadır. Yin malm ölliklri sıcaklığa bağlı olan bir kolonun burkulma yüklrin sıcaklık tkilri araştırılmaktadır. Anahtar Klimlr : Burkulma, Kolon, Sonlu lman yöntmi, Kararlılık. INRODUCION A column is on of th basic structural lmnts. Eulr gav analytical solutions for th column buckling first, and sinc thn many rsarchrs hav focusd thir attntion on th concpts of buckling and stability of th columns. Finit lmnt mthod is on of th most common mthods in th numrical buckling analysis of structurs (Wavr and Johnston, 984). Coultr and Millr (986) analyd th fr vibrations and buckling of lastic Eulr- Brnoulli bams subjctd to non-uniform aial forcs by th us of various typs of bam finit lmnts. Ali and Sridharan (988) dvlopd a nw formulation to study th intractiv buckling of thin-walld columns having arbitrary cross-sctions. Sakiyama (986) studid th lastic buckling of taprd columns numrically. Rcntly, Goda t all. 39

2 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap (99) dvlopd a finit lmnt cod to invstigat th dynamic latral buckling of thin walld -shapd bam subjctd to impulsiv load. Vairi and Xi (99) proposd a nw numrical modl for analying th buckling of columns with variably distributd aial loads. Mor rcntly, Hlwig and Yura (999) invstigatd torsional buckling of column. Wu (998), Smithpardo and Aristiabalochoa (999) studid postbuckling bhavior of column. In th prsnt papr nw finit lmnts for th column buckling ar introcd with th aial dgrs of frdom (DOF). Eulr-Brnoulli thory of thin bams is usd to driv th lmnt matrics by mans of th minimum potntial nrgy principl. mpratur dpndncy of th column matrial is takn into account in th drivation. h lmnts ar tstd by th buckling analysis of columns with diffrnt boundary conditions, and rsults ar compard with thos of th classical bam-bnding lmnt and analytical solution. h tmpratur ffcts on th stability charactristics of th column ar also amind considring th tmpratur dpndncy of th matrial proprtis of th column.. GOVERNING EQUAIONS AND CONCE FOR HE NEW EEMENS Considr a prfctly straight lastic column with tmpratur dpndnt matrial proprtis (S Figur ). h column is subjctd to an nd aial forc that is applid along its cntroidal ais and a tmpratur load that has a variation along th cntroidal ais. h strain-displacmnt quation of th column can b writtn as y () Figur. Elastic column subjctd to buckling load ε d w dw + α = () whr u and w ar displacmnts along th and w ais; α is th thrmal pansion cofficint dpnding on tmpratur; and is th tmpratur. Strain nrgy of a linar lastic column can b writtn as U Eε dv V = () hn, substituting Eq. () into () w obtain, U 4 d w dw α la d w dw d w dw + α α EdA = d w dw + α (3) whr E is th Young molus, which is also a function of tmpratur. Sinc th tmpratur has a variation along th column ais (no variation ovr th cross-sction), th Young molus and th thrmal pansion cofficint of th column matrial will dpnd on only th coordinat. rforming th intgration ovr th cross-sction of th column and disrgarding th fourth ordr trm, w obtain U d w dw dw EA EI EA F l = (4) whr A and I ar th cross-sctional ara and th momnt of inrtia of th column about cntroidal ais, rspctivly. h thrmal load trms in Eq. (4) can b obtaind prforming th intgration ovr th cross-sctional ara of th column as, Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45

3 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap (5) F = Eα da = EAα = EαdA = EAα A A h potntial nrgy of th aial load and total potntial nrgy of th column bcoms = Ω (6) Π = U + Ω (7) Now, w can invstigat th buckling of th column from th undflctd configuration as u u + u w w + w (8) whr u and w dnot th undflctd configuration; u and w ar infinitsimally small incrmnts. Minimiation of th total potntial nrgy nds th first variation of th total potntial nrgy b ro. aking into account th column is initially straight (i.. w = ), th first variations bcoms δπ = EA + ( ) = (9) Aial displacmnt of th undflctd column is obtaind from Eq. (9) as u ( ) EA = () Using Eq. () th scond variation of th total potntial nrgy of th column is obtaind as follows δ Π d w dw EA EI + = () For intnsional buckling, th scond variation bcoms d w dw EI Π = δ () h matrial proprtis in Eq. () ar dpndnt on tmpratur, and thir dpndncy largly changs with incrasing tmpratur. h influnc of tmpratur-dpndnt matrial proprtis on th thrmomchanical bhavior at lvatd tmpratur and/or high gradint tmpratur is quit significant. In this study, it is assumd that th matrial proprtis of th column hav linar variation with th tmpratur as follows, E() = E + E and α () = α + α (3) By th us of th rotation dfind as φ ( ) = dw in Eq. (), w obtain, d EI φ Π = φ δ (4) From Eq. (4) w s that th scond variation of th total potntial nrgy dpnds on only th rotation. h rotation causs th displacmnts at any distanc from th nutral ais and thy vary linarly with th distanc from th nutral ais. By th us of this approimation of th thin bam thory, w may introc a column lmnt with th aial dgrs of frdom (DOF) dfind at th outr fibr of th column. h nw lmnt has th simplr formulation and mor compact lmnt matrics than thos of th classical bam-bnding lmnt. It has a littl difficulty to apply th boundary conditions. 3. COUMN-BUCKING EEMENS h gomtry of th lmnts usd is shown in Figur. h column is assumd to sustain only aial and flural dformation; shar dformation is disrgardd. h bnding strains and displacmnt fild to th flural bhavior is givn as follows: h y q ( ) q (a) (b) Figur. Gomtry of th lmnts with (a) -DOF and (b) 3-DOF u = h φ (5) d φ = h y q q q3 (6) whr ( ) u is th aial displacmnt of th fibr at = h. Substituting Eqs. (5) and (6) into Eq. (4), th scond variation of th total potntial nrgy bcoms, Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45

4 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap EI δ Π = u (7) h h Now, som column lmnts may b dfind by considring th aial dgrs of frdom on th outr fibr. h displacmnt fild in a column lmnt can b approimatd by th us of intrpolation functions and unknown nodal displacmnts, so that u () = N.q (8) whr N and q dnots th intrpolation function matri and nodal displacmnt vctor of th lmnts. Now, two kinds of column lmnts ar introcd: th lmnts with two nods and thr nods. h intrpolation function matrics and th nodal displacmnts vctors ar obtaind for th lmnt with two nods and for th lmnt with thr nods as follows: N = - q = [ q q ] (9) 3 = N q = [ q q q ] () 3 h column is discrtid by finit lmnts and th displacmnt fild givn by Eq. (8) is usd for th lmnts. hn, th total potntial nrgy of th column can b obtaind by simply summing th potntial nrgy of th lmnts as follows: δ Π = I dn dn q E ( ) q q N N q () h h h first trm givn in th cornr brackt is th lmnt stiffnss matri, k, and th scond on is th lmnt gomtric matri, g. h tmpratur fild within an lmnt can b writtn in trms of nodal tmpraturs, and, as follows () + = () hn, th variation of th Young molus and thrmal pansion cofficint within th lmnt can b writtn for an lmnt as E ( ) = E + E ( ) (3) Substituting Eqs. () and (3) into Eq. (), th stiffnss and gomtric matrics of th -DOF lmnt can b prssd as, k I dn dn E E + + h = (4) rforming th intgration, th lmnt stiffnss matri can b obtaind as k = I + E + E h (5) h gomtrical matri of th column lmnt with -DOF can b writtn as g = 6h (6) For th lmnt with 3-DOF, th stiffnss and gomtric matrics ar obtaind by th us of th intrpolation functions givn by Eq. (), as follows k = 7 E + E h g = 4 3h (7) Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45

5 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap [ K G] Q = (8) whr K is global stiffnss matri; and G is global gomtric matri; Q is th global displacmnts of th column. Various boundary conditions for th column ar considrd: (i) a column clampd at on nd and fr at th othr, (ii) a column with simply supportd at both nds, and (iii) a column clampd at both nds. Boundary conditions for th lmnt with -DOF can b approimatd by th us of Eq. (8). h rotation is ro for a clampd dg. If th column is clampd at =, th boundary condition can b prssd in th local dgrs of frdom considring Eq. (5), so that q = (9) It is obvious that q bcoms ro if th column is clampd at =. Sinc bnding momnt is ro for a simply supportd dg, th drivativ of rotation with rspct to bcoms ro at that dg. For ampl, th simply supportd nd conditions can b approimatd by dn q = = (3) Substituting th intrpolation functions Eq. (9) into Eq. (3), th conditions that will b imposd on th matri quation can b writtn in th local dgrs frdom as q = q (3) h boundary conditions for th lmnt with 3-DOF ar drivd as plaind in dtail abov. For a clampd nd Eq. (9) is valid again (q 3 = if th column is clampd at = ). For a simply supportd nd, th condition will b 4 q or = q q 3 3 ithr = q q3 q (3) th classical bam bnding lmnt (i.. displacmnt and rotation dgrs of frdom ar considrd at ach nod) and th act solutions. Eact valu of th buckling load of a prismatic bam (Brush and Almroth, 975) ar givn by EI cr = β (33) whr β is th instability cofficint for prismatic bams with various boundary conditions. h tmpratur ffcts ar not considrd in th comparison studis. It is obsrvd that convrgnc is slow for th columns with th simply supportd nds whn th column is discrtid by th nw lmnts with qual lngth. h undsird rsult is to waknss in th approimation to th boundary conditions by th us of lmnts with qual lngth. If th lmnts usd at th boundary ar chosn smallr than thos of th intrior, a bttr convrgnc can b obtaind. h variation of th rlativ rror with lngth ratio of lmnts (intrior lmnt lngth to boundary lmnt lngth) is shown in Figur 3 and 4 for fiv and tn lmnts modls of th column, rspctivly. h variations show that th rlativ rror dcrass as th lngth ratio incrass. hrfor, th lmnt lngth ratio 5 is slctd for a bttr convrgnc in th calculations of a column with simply supportd boundary conditions. For a clampd-clampd boundary condition, th lowst ignvalu givs th trivial solution and is ignord in th calculations. Rlativ Error (%) I / B Figur 3. Improvmnt of convrgnc for -DOF modl n= n=5 according to th location of th support bing at lft or at right. 4. NUMERICA SUDIES Rlativ Error (%) 5 5 n= n=5 h columns with diffrnt nd conditions ar discrtid by th us of nw lmnts. And thn th stability charactristics of th columns ar obtaind numrically. h rsults ar compard with thos of I / B Figur 4. Improvmnt of convrgnc for 3-DOF modl Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45

6 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap h rsults of th convrgnc studis ar shown in abls -3 for th nw lmnts and classical lmnt. h rsults obtaind by th us of th nw lmnts, spcially for th clampd cass, ar in good agrmnt with th classical lmnt. abl. Convrgnc and Comparison of th Instability Cofficints β (For a Clampd-Fr Column, Eact Valu of β is.467) Numbr of Elmnt rsnt ( DOF) rsnt (3 DOF) Classical Elmnt 3. () *.486 ().486 ().597 ().469 (4).469 (4) 3.54 (3).468 (6).468 (6) (4).468 (8).468 (8) (5).467 ().467 ().47 ().467 ().467 () * : Numbrs in th parnthsis show th total numbr of DOF abl. Convrgnc and Comparison of th Instability Cofficints β (For a Simply Supportd - Simply Supportd Column, Eact Valu of β is 9.87) Numbr of lmnt rsnt ( DOF) rsnt (3 DOF) Classical Elmnt ()*.7 (5) (6) 4.45 (3).58 (7) (8) 5.3 (4) (9) 9.87 () 6.57 (5) () 9.87 ().94 (9) 9.9 (9) 9.87 () * : Umbrs in th parnthsis show th total numbr of DOF abl 3. Convrgnc and Comparison of th Instability Cofficints β (For a Clampd-Clampd Column, Eact Valu of β is ) Numbr of lmnt rsnt ( DOF) rsnt (3 DOF) Classical Elmnt ()* (5) (4) (3) (7) (6) (4) (9) (8) (5) () () (6) (3) () (9) (9) (8) * : Numbrs in th parnthsis show th total numbr of DOF In th classical lmnt shap functions corrsponding to rotation dgrs of frdom ar quadratic. h nw lmnt with 3-DOF uss quadratic shap functions for aial dgrs of frdom dfind at th outr fibr of th bam, yilding quadratic variation of th rotation along th bam. Sinc boundary conditions for clampd-fr and clampd-clampd cass ar satisfid actly by th us of th nw lmnts, th rsults of classical lmnt and th nw lmnt with 3-DOF would b similar for thos cass as givn in abls and 3. Howvr, thr is som diffrnc for simply supportd cas to th approimat satisfaction of boundary condition. As pctd, th us of highr ordr lmnts in discrtiation givs bttr accuracy. Influnc of th tmpratur dpndncy of th matrial proprtis is studid on th buckling of th bam with th following paramtrs: E = 86 N/cm, E = -598 N/cm C, E()I cr () = β or cr () = β E I (34) hr diffrnt tmpratur loadings hav bn considrd: (i) () = º C (constant tmpratur along th column) (ii) () = (iii) () = variation) 3 4 º C, (linar variation) (35) 4 º C, (quadratic Buckling load dpnding on tmpratur loading will b (36) h calculations ar carrid out for a bam with clampd-fr nds. Rsults obtaind by using of bam lmnts with -DOF ar shown in abl 4. h buckling loads ar lowrd considrably by th thrmal ffct. abl 4. h Instability Cofficints β Without/With mpratur oading Numbr of β (=) lmnt β (i) β (ii) β (iii) CONCUSIONS Nw finit lmnts with aial dgrs of frdom ar proposd for th buckling analysis of columns with tmpratur dpndnt matrial proprtis. Eulr-Brnoulli thory of thin bams is usd to driv th lmnt matrics by mans of th minimum potntial nrgy principl. h tmpratur ffcts ar also amind on th column buckling, and tmpratur dpndncy of matrial proprtis is takn into account in th formulation. h nw column lmnts hav simpl drivations and compact lmnt matrics than th classical bam-bnding lmnt. h boundary condition for Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45

7 Buckling of A Column With mpratur Dpndnt Matrial roprtis, Ö. Soykasap a simply supportd dg is satisfid approimatly. Various nd conditions of columns ar considrd in th ampls. h rsults ar compard with both th classical and act ons. h nw lmnt is as powrful as th classical on, yt it has mor compact matrics. h tmpratur ffcts on th buckling loads of th column with tmpratur dpndnt matrial proprtis ar studid by th us of nw lmnt. h tmpratur loads affcts th lastic molus of column. h buckling loads ar lowrd by th thrmal ffct. 6. REFERENCES Ali, M. A. and Sridharan, S Vrsatil Modl for Intractiv Buckling of Columns and Bam- Columns. Int. J. Solids and Structurs 4, Brush, D. O. and Almroth, B. O Buckling of Bars, lats and Shlls, McGraw-Hill, USA. Coultr, B. A. and Millr, R. E Vibration and Buckling of Bam-Columns Subjctd to Non- Uniform Aial oads. Int. J. Num. Mth. Eng. 3, Goda, K., Watanab O., Motoyama, K. 99. atral Buckling of a hin-walld Bam Subjctd to Impulsiv oad rocdings of th Scond Intrnational Offshor and olar Enginring Confrnc, ublishd by Intrnational Socity of Offshor and olar Enginrs (ISOE), USA, Hlwig,.A. and Yura, J.A orsional Bracing of Columns. J. Struct. Eng. ASCE 5(5), Sakiyama, A Mthod of Analying th Elastic Buckling of aprd Columns. Comp. and Struct. 3, 9-. Smithpardo, J.. and Aristiabalochoa, J. D Buckling Rvrsals of Aially Rstraind Imprfct Bam-column. J. Eng. Mch. ASCE, 5 (4), Vairi, H. H. and Xi, J. 99. Buckling of Columns undr Variably Distributd Aial oads. Comp. and Struct. 45, Wavr, W. and Johnston,. R Finit Elmnts For Structural Analysis, rntic-hall, Nw Jrsy. Wu, B Scondary Buckling of an Elastic Column With Spring-Supports at Clampd Ends. Archiv of Applid Mchanics 68 (5), Mühndislik Bilimlri Drgisi 7 () Journal of Enginring Scincs 7 () 39-45