FINITE ELEMENT STRESS ANALYSIS OF ELASTIC BEAMS UNDER NON-UNIFORM TORSION

Transcription

1 Damjan anić Goan ukalj osip nić DO: 0.78/OF FE ELEME RE LY OF ELC EM UDER O-UFORM ORO umma his pap psnts a two-dimnsional finit lmnt fomulation fo th stss analsis of lastic bams subjctd to non-unifom tosion. h lmnt stiffnss mati and load vctos a divd using th pima and sconda waping functions. h pima function cosponds to that occuing with unifom tosion poblms. h sconda function is intoducd to tak into account ffcts causd b th staind waping. hus sha stsss a dividd into th pima and sconda ons kping th sam maning as th waping functions. h poposd finit lmnt modl nabls th stss analsis to b caid out gadlss of cosssctional shaps. h matial is assumd to ob Hook s law. h ffctivnss of th psntd finit lmnt algoithm is validatd though two bnchmak ampls. K wods: non-unifom tosion waping function sha stss finit lmnts. ntoduction asic tosion tho fd to as t. Vnant s o unifom tosion tho is wll dscibd in litatu [-4] and has bn fo a long tim a fundamntal tho applid b man sachs in th fild of tosion. f th waping ffcts a to b takn into account th non-unifom tosion tho has to b applid [5 6]. h dvlopmnt of th computational scinc and of numical analsis sultd in numous mthods.g. finit lmnt mthod finit diffnc mthod finit volum mthod tc. Mging tosion thois with numical analsis opns up nw possibilitis in pdicting and modlling vaious tps of stuctus and machin lmnts submittd to a tosion tp of load. om numical solutions to th unifom tosion poblm a psntd in Rfs. 7 and 8 whil thos usd in th nonunifom tosion poblm solving a givn in Rfs n modn ngining pactic ngins sking pactical solutions oftn tnd to idaliz thi input paamts. uch a pocdu oftn givs good appoimat sults although it dos not giv a al insight into th bhaviou of th stuctual mmb discussd. Whn talking about tosion ths idalizations usuall pf to nglct th ffcts of staind waping which alwas occu whn bams hav paticula staints. lthough th fact is that in th long bams o in th bams having low tosional and high sctoal igiditis such ffcts a usuall ngligibl. hfo som authos to simplif thi analss f to th tps of coss-sctions: solid closd thin-walld and opn thin-walld sctions and th point out th opn thin-walld sctions whn daling with staind waping []. RCO OF FME XL- (06) 7

2 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams his pap psnts a finit lmnt fomulation fo th lina stss analsis of bams subjctd to non-unifom tosion. h applid tosion tho is a modifid tho oiginall psntd b apountzakis [9]. h dscibd numical modl uss an isopaamtic quadilatal nin-nod lmnt psntd b Pilk []. h lmnt stiffnss mati and load vctos a divd using th pima and sconda waping functions. h pima waping function cosponds to that occuing with unifom tosion poblms. h sconda waping function is intoducd to account fo th sha stss du to staind waping ffcts. hus th sha stss is dividd into th pima and th sconda on wh th fom appas du to unifom tosion and th latt du to staind waping. h poposd finit lmnt modl nabls th stss analsis to b caid out gadlss of coss-sctional shaps. h matial is assumd to ob Hook s law. h comput pogam is wittn using th Pthon pogamming languag. h ffctivnss of th algoithm discussd is validatd though two bnchmak ampls.. asic considations ight-handd Catsian coodinat sstm ( z) is chosn in such a wa that th z- ais coincids with th bam ais passing though th cntoid of ach coss-sction. h sha cnt of th coss-sction is dfind b th position coodinats and. h displacmnt fild is takn as uzz vz z () w z ' z ν ''' z wh u v and w a th igid-bod tanslations of th coss-sction in th - - and z- dictions spctivl; φ is th angl of twist about th sha cnt; ν is Poisson's atio; and a th pima and sconda waping functions spctivl. h sconda waping function is intoducd to obtain th sconda sha stss du to staind waping [9]. h non-zo stain componnts a found fom th fist od stain-displacmnt lations as and ε '' ν '''' z w z u w z v w z γ ' ν ''' γ ' ν ''' f th fouth od divativ of th angl of twist is lft out fom Eq. () stss componnts of a lina lastic matial a calculatd as σ E ε E'' z z G γ G ' E ''' G γ G ' E ''' h sha stss is dcomposd into a pima and a sconda pat du to th unifom and th non-unifom tosion spctivl i.. () (3) (4) 7 RCO OF FME XL- (06)

3 Finit Elmnt tss nalsis of Elastic ams D. anić G. ukalj. nić G ' G' (5) E''' E''' Fom Eq. (4) quilibium quations fom th following pssions: and z z G'' G'' E'''' E'''' 0 0 z 0. G' E''' E''' (7) z Futhmo Eq. (7) is dcomposd into two patial diffntial quations: 0 (8). t can b obsvd that both waping functions and a modlld b Eqs. (8) and (9). ounda conditions fo patial diffntial quations (8) and (9) a spctivl n n 0 (0) n n 0. () h waping functions dfind in Eqs. (8) and (9) a calculatd with spct to th sha cnt. tss sultants acting on ach coss-sction of th bam can b dfind as follows: M z z d E '' d E'' d σ d () (3) wh is th bimomnt whil psnts th waping momnt of intia. h tosional momnt M z consists of two pats: and psnting t. Vnant s o th unifom tosion momnt and th waping o th non-unifom tosion momnt spctivl. Fom Eqs. (5) and (3) th componnts of tosional momnt can b pssd as and t d G ' d t d E ''' d (6) (9) (4) (5) RCO OF FME XL- (06) 73

4 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams n Eq. (4) t dnots th t. Vnant tosional constant whil fom Eq. (5) psnts th waping momnt of intia du to th sconda waping function analogous to. h diffntial quation fo th non-unifom tosion follows fom th inclusion of Eqs. (4) and (5) into Eq. (3) M z E ''' G ' (6) t and if pssd in tms of th continuous tosional momnt m z m z t m z dm z E '''' G ''. (7) dz solving th diffntial quation fom Eq. (6) o (7) th pssion fo th angl of twist is divd. h analtical solution to Eq. (7) is takn fom Chn and tsuta [3] as mz z Gt 3cosh 4sinh. (8) Gt E z C C z C z C ( z) 3. Finit lmnt fomulation n th stss analsis pocdu th fist stp is th calculation of th pima and sconda waping functions followd b th dtmination of th tosional constant and th waping momnt of intia. Onc obtaind ths valus a usd fo th calculation of th angl of twist fom Eq. (8) and lastl of th stsss fom Eqs. (4) and (5). o calculat th waping functions th pocdu continus with th disctization of th coss-sction with finit lmnts. sopaamtic quadilatal nin-nod lmnts a usd wh th Lagangian shap functions a: ηζ ηζ ηζ η ζ ηζ η η ζ 7 4 ηζ η η ζ ηζ ζ η ζ ηζ 4 η ζ 6 ηζ η ζ ηζ ηζ η ζ ηζ η η ζ ηζ η η ζ h waping functions a calculatd b solving th patial diffntial quations fom Eqs. (8) and (9) with th bounda conditions fom Eqs. (0) and (). ft appling th wightd sidual mthod to Eq. (8) it follows that d i j s n g n n n i n j ds 0 g 7 i j wh th scond intgal in Eq. (0) implis th bounda condition in vcto notation. h intgal fom in Eq. (0) is fd to as a wak fomulation. f th tst functions and a chosn to b idntical and if Gn s fist idntit and th divgnc thom a usd th wak fomulation taks th final fom d g d 0. () ow if th finit lmnt fom is applid b th inclusion of () 8 9 (9) (0) 74 RCO OF FME XL- (06)

5 Finit Elmnt tss nalsis of Elastic ams D. anić G. ukalj. nić th following pssion is obtaind:. 0 d g (3) nd fo ach lmnt th is a fom. 0 p k p k (4) h solution to th pima waping function is obtaind b solving Eq. (3) fo ach lmnt. h wak fomulation fo th scond patial diffntial quation Eq. (9) is takn as j i n j i n s n n ds d 0 (5) Onc again if th tst functions and a chosn to b idntical and Gn s fist idntit and th divgnc thom a usd th wak fomulation taks th final fom. 0 d (6) h finit lmnt fom is applid b th inclusion of (7) and it is obtaind that. 0 d (8) h solution to th scond waping function is obtaind b solving Eq. (8) fo ach lmnt. h stiffnss mati is of th sam fom fo both patial diffntial quations and it is calculatd as. dd dd k (9) o mak th tansfomation fom th al domain ( ) to th fnc domain (η ζ) fist th acobian mati has to b calculatd. (30) f tansfoming th intgal on has to add th dtminant of th acobian mati o th acobian und th intgal. Fo ampl if tansfoming th pssion fo th sufac of th lmnt fom th al domain Ω to th fnc domain Ω it is calculatd as. dd dd (3) h gadint of th shap function mati is calculatd as (3) nd if tansposd thn RCO OF FME XL- (06) 75

6 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams. (33) f Eqs. (3) (3) and (33) a includd into Eq. (9) th final pssion fo th stiffnss mati is divd as. d d d d k (34) h load vcto which cosponds to Eq. (8) is. s s s s dd d d g p (35) h load vcto which cosponds to Eq. (9) is. dd d p (36) On th basis of Eq. (36) to solv Eq. (9) fist on has to solv Eq. (8) to obtain th pima waping function. Whn th stiffnss matics and th load vctos of ach lmnt a combind into a global stiffnss mati and a global load vcto gnal solutions a obtaind b solving lina sstms of quations. P K P K (37) ft calculating th waping functions th tosional constant and th waping momnt of intia a calculatd fom Eqs. (4) and () o (5) spctivl. h angl of twist is calculatd analticall fom Eq. (8) in which b stting th bounda conditions dpnding on th tp of th bam suppots unknown constants (C C 4 ) a dtmind. Fom th it is as to calculat th stsss b using Eqs. (4) and (5). 4. umical Eampls On th basis of th poposd finit lmnt pocdu a comput pogam fo stss analsis is dvlopd. h accuac of th psntd modl is illustatd b two ampls. tsss calculatd b th comput pogam a th aial stss σ z and th sultant sha stss which is givn as. (38) Fom Eq. (38) on should not that th sultant sha stss will alwas b positiv sinc it onl psnts th intnsit of th stss and not th diction. Eampl 4. Figu shows a smmtic -bam fid at both nds. h otation about th z- ais is full staind whil th waping is compltl f. h bam is loadd b unifoml distibutd toqu m z = km/m. h lastic moduli a E = 0 GPa and G = GPa. inc th coss-sction is smmtic th sha cnt coincids with th cntoid of th cosssction. 76 RCO OF FME XL- (06)

7 Finit Elmnt tss nalsis of Elastic ams D. anić G. ukalj. nić Fig. mmtic -bam fid at both nds (a) (b) (c) (d) () Fig. Distibution of (a) th angl of twist φ (b) of th bimomnt (c) of th tosional momnt M z (d) t. Vnant s momnt () th staind waping momnt ov th lngth of th -bam Distibutions of th angl of twist φ bimomnt tosional momnt M z and its componnts ov th lngth of th bam a shown in Figu. inc th intnsit of th sultant sha stss is onl calculatd and sinc th is smmt it is nough to obsv th bam fom z = 0 to z = l/. h highst lvl of th aial stss in th coss-sction occus at z = l/ sinc th bimomnt valu is highst th. ha stss has its minimum/maimum at z = 0 and z = l. Distibutions of th highst aial and sha stsss b its componnts a shown in Figu 3. h coss-sction is disctizd b 756 lmnts. h gaphs abov th flangs in Figu 3 (a) and (b) show th distibution of th dscibd stss ov th cnt lin of th flang. n Figu 3 (c) th gaph abov th flang and th gaph on th ight show th distibution of pima sha stss ov th upp contou of th flang and ov th ight contou of th wb spctivl. inc th cons of th coss-sction a shap on can notic th appaanc of v high stss concntations on th gaph on th ight in Figu 3 (c). fillting o chamfing th cons such stss concntations can b significantl ducd. RCO OF FME XL- (06) 77

8 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams (a) (b) (c) Fig. 3 Distibution of (a) th aial stss σ z at z = l/ (b) th sconda sha stss at z = 0 (c) th pima sha stss at z = 0 ov th sufac of th -bam coss-sction his ampl was solvd analticall b Šimić [7]. h numical solution is compad with th analtical solution at th diffnt coss-sctional cuts fo th diffnt coss-sction positions (makd in Figu ). h compad valus a shown in abls and 3 wh th valus of th aial stss and of th sconda sha stss (σ z ) a shown fo th cnt lin of th coss-sctional cut whil th valus of th pima sha stss ( ) a shown fo on of th contous of th coss-sctional cut. 78 RCO OF FME XL- (06)

9 Finit Elmnt tss nalsis of Elastic ams abl Compaison btwn th numical and analtical valus of th aial stss σ z on vaious cuts of th -bam umical naltical z = z = l/ z = l/ abl Compaison btwn th numical and analtical valus of th sconda sha stss on vaious cuts of th -bam umical naltical z = z = l/ z = l/ D. anić G. ukalj. nić abl 3 Compaison btwn th numical and analtical valus of th pima sha stss on vaious cuts of th -bam umical naltical z = 0 C C z = l/4 C C z = l/ C C Eampl 4. Figu 4 shows a cantilvd U-bam loadd with th tosional momnt M z = 0 m at th f nd. h lastic moduli a E = 0 GPa and G = GPa. Fig. 4 Cantilvd U-bam Distibutions of th angl of twist φ bimomnt tosional momnt M z and its componnts ov th lngth of th bam a shown in Figu 5. h highst lvl of aial stss occus in th coss-sction with th highst valu of th applid bimomnt and that is in th coss-sction at z = 0. h sconda sha stss has its maimum in th sam coss-sction i.. at z = 0. h highst lvl of pima sha stss occus in th coss-sction at z = l. Distibutions of th highst aial and th highst sha stss b its componnts a shown in Figu 6. h coss-sction is disctizd b 884 lmnts. h gaphs in Figu 6 (a) and (b) show th distibutions of th dscibd stss ov th cnt lin of th U pofil whil thos in Figu 6 (c) show th distibutions of th pima sha stss ov th out contou of th U pofil. simila ampl is solvd analticall b Chn and tsuta []. h numical solution is compad to th analtical on at si diffnt coss-sctional cuts fo th diffnt coss-sction positions (makd in Figu 4). h compad valus a shown in abls 4 5 and 6 wh th valus of th aial stss and of th sconda sha stss (σ z ) a shown fo th cnt lin of th coss-sctional cut whil th valus of th pima sha stss ( ) a shown fo on of th contous of th coss-sctional cut. RCO OF FME XL- (06) 79

10 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams (a) (b) (c) (d) () Fig. 5 Distibution of (a) th angl of twist φ (b) th bimomnt (c) th tosional momnt M z (d) t. Vnant s momnt () th staind waping momnt ov th lngth of th U-bam abl 4 Compaison btwn th numical and analtical valus of th aial stss σ z on vaious cuts of th U-bam umical naltical C C.7.8 z = 0 E E F F C C z = l/0 E E F F C C E E F F z = l abl 5 Compaison btwn th numical and analtical valus of th pima sha stss on vaious cuts of th U-bam umical naltical z = D D z = l/0 D D z = l D D abl 6 Compaison btwn th numical and analtical valus of th sconda sha stss on vaious cuts of th U-bam umical naltical.36.7 z = 0 D D z = l/0 D D z = l D D RCO OF FME XL- (06)

11 Finit Elmnt tss nalsis of Elastic ams D. anić G. ukalj. nić (a) (b) (c) Fig. 6 Distibution of (a) th aial stss σ z at z = 0 (b) th sconda sha stss at z = 0 (c) Distibution of th pima sha stss at z = l ov th sufac of th U-bam coss-sction 5. Conclusions his pap has psntd a modl fo th lastic stss analsis of bams und nonunifom tosion. gnuin solution has bn achd using th finit lmnt mthod. h liabilit of th poposd numical modl has bn vifid b studing two ampls which showd a ctain dg of accuac. Du to stss concntations th solution poposd in this pap slightl diffs fom th analtical solution but th diffnc can b ducd b RCO OF FME XL- (06) 8

12 D. anić G. ukalj. nić Finit Elmnt tss nalsis of Elastic ams appling fillts o chamfs on th cons of th coss-sction. his solution is abl to displa stss distibutions ov th sufac of th coss-sction. Futu sach will tnd th poposd algoithm to a modl with th matial and gomtic nonlinait. Fo that pupos th angl of twist will b solvd numicall. inc th modl is abl to show stss distibution th ida is to displa gadual plastification ov th sufac of th coss-sction. cknowldgmnt h authos gatfull acknowldg th financial suppot of th Coatian cinc Foundation (pojct o. 6876) and th Univsit of Rijka ( ). REFERECE [] W. D. Pilk: nalsis and Dsign of Elastic ams Computational Mthods. ohn Wil & ons w Yok 00. DO: 0.00/ [] W. F. Chn. tsuta: ho of am-columns Volum : pac bhavio and dsign.. Ross Publishing Fot Lauddal FL 008. [3].. Conno: nalsis of tuctual Mmb stms. Ronald Pss 976. [4]. imoshnko. Goodi: ho of Elasticit. McGaw-Hill Co. Y 970. [5] V. Z. Vlasov: hin-walld Elastic as (in Russian). nd d. Fizmatgiz Moscow 959. [6] Gjlik: h ho of hin Walld as. ohn Wil & ons w Yok 98. [7]. nić G. ukalj M. Čanađija: ha stss analsis in ngining bams using dplanation fild of spcial -D finit lmnts. Mccanica vol. 45 (00) pp DO: 0.007/s z [8] tfan M. Lupoa D. Constantin C. aciu: umical Dtminations with Finit Diffncs Mthod of Pismatic ams ubjctd to osion. Pocdings of th Wold Congss on Engining 0 Vol WCE 0 London U.K. [9] E.. apountzakis: as und osional Loading: Gnalizd am ho ppoach. R Civil Engining vol. 03 ticl D pags 03. DO: 0.55/03/9658 [0] R. E. Fatmi: on-unifom waping including th ffcts of tosion and sha focs. Pat : gnal bam tho. ntnational ounal of olids and tuctus vol. 44 (007) pp DO: 0.06/j.ijsolst [] R. E. Fatmi: on-unifom waping including th ffcts of tosion and sha focs. Pat : naltical and numical applications. ntnational ounal of olids and tuctus vol. 44 (007) pp DO: 0.06/j.ijsolst [] E.. apountzakis V. G. Mokos: Waping sha stsss in nonunifom tosion b EM. Computaional Mchanics vol. 30 (003) pp 3-4. DO: 0.007/s [3] E.. apountzakis V. G. Mokos: Dnamic analsis of 3-D bam lmnts including waping and sha dfomation ffcts. ntnational ounal of olids and tuctus vol. 43 (006) pp DO: 0.06/j.ijsolst [4] Chang-w Chn: h waping tosion ba modl of th diffntial quadatu lmnt mthod. Computs and tuctus vol. 66 ssus -3 (998) pp [5] Z. Wang. Zhao D. Zhang Gong: Rstaind tosion of opn thin-walld bams including sha dfomation ffcts. ounal Zhjiang Univ-ci (ppl Phs & Eng) vol. 3 ssu 4 (0) pp [6] G. ukalj. nić: onlina tabilit nalsis of hin-walld Fams Using UL-E Fomulation. ntnational ounal of tuctual tabilit and Dnamics vol. 4 o. (004) pp DO: 0.4/ [7] V. Šimić: Otponost Matijala. Školska knjiga Zagb 995. ubmittd: ccptd: ssociat pt Damjan anić Pof. d. sc. Goan ukalj Pof. d. sc. osip nić Univsit of Rijka Facult of Engining Dpatmnt of Engining Mchanics Vukovaska Rijka Coatia 8 RCO OF FME XL- (06)