St. Venants Torsion Constant of Hot Rolled Steel Profiles and Position of the Shear Centre

Transcription

1 NSCC2009 St. Vnants Torsion Constant of Hot Rolld Stl Profils and Position of th Shar Cntr M. Kraus 1 & R. Kindmann 1 1 Institut for Stl and Composit Structurs, Univrsity of Bochum, Grmany BSTRCT: Th knowldg of th cross sction proprtis is rquird for th static analysis using bam thory. For arbitrary cross sctions th xact torsional valus can only b dtrmind analytically if th sction has a basic gomtry. For that rason in practic diffrnt formula ar oftn usd to approximat th valus whn rolld sctions ar applid. In contrast th us of numrical mthods, as for instanc th finit lmnt mthod (FEM), allows th dtrmination of th accurat torsional valus. Th approximations partially show comparativly big discrpancis to th accurat valus. For that rason th accurat torsional proprtis of diffrnt hot rolld cross sctions as wll as nw formula basd on th numrical solutions ar prsntd in this ssay. Th nw formula allow a mor prcis approximation of th St. Vnants torsion constant than th xisting ons. 1 INTRODUCTION Bar mmbrs ar oftn subjctd by torsional loadings. This spcially applis for analyss according to 2 nd ordr thory, sinc torsion not schduld usually ariss as shown in th xampl of Figur 1. For th static analysis using bam thory th knowldg of th torsional cross sction proprtis is thrfor ssntial. Figur 1. Dformation of a bar mmbr according to 2 nd ordr thory, Kindmann (2008) 454

2 Figur 2. Warping ordinat of an HEM 200, Kraus (2005) For rolld sctions analytical solutions to dtrmin ths proprtis do not xist. For that rason th cross sctions ar assumd as thin walld in gnral and analyzd with corrsponding thoris and constitutiv modls. This lads to torsional proprtis which show discrpancis in comparison to accurat solutions, which can b dtrmind using th finit lmnt mthod (FEM). Figur 2 xmplifis th diffrncs with th warping ordinat ω for a rolld I profil. Th warping ordinat is a valu, which conncts th torqu arising du to torsional loadings with th dformations u in longitudinal dirction x of a bar: u = ω( y, z) ϑ ( x ) (1) Sinc th cross sction dos not kp a plan constitution whn dforming, th distortions ar rfrrd to as warping. Using formula (1) ω can also b intrprtd as unit warping for ϑ= 1. Th dformation u rsp. th warping ordinat dpnds on th position of th rotation axis, about which th cross sction twists whn subjctd to torsion. In gnral this is th axis of th shar cntr M to which th warping ordinat thrfor rfrs to. Sinc th xact position is usually not known in advanc, an arbitrary rotation axis D is chosn for which ordinats ω can b dtrmind ithr using th simplifid modls (middl lin modl/thin walld thory) or accurat approachs on basis of numrical modls. Th formula 1 ym yd = z ω d, I y 1 zm zd = y ω d I dscrib th position of th shar cntr dpnding on ω. Now th warping ordinat can b dtrmind by th following transformation rlationship: ( ) ( ) ω=ω ω z y y + y z z with k M D M D z 1 ω k = ω d Figur 2 clarifis, that th accurat solution for th warping ordinat shows linarly changing valus ovr th plat thicknss. In contrast th solution using th middl lin modl only provids a singl valu which is assumd as constant rgarding th plat thicknss. This diffrnc not only ffcts th position of th shar cntr whn rgarding cross sctions with lss than two axs of symmtry, but also th St. Vnants torsion constant as wll as th warping constant. ccording to Kindmann & Kraus (2007) ths valus can b dtrmind using th following formula: = ω 2 ω I d (4) ω ω IT = + ( y ym ) ( y ym ) + + ( z zm ) ( z zm ) d (5) z y In th following chaptrs th dtrmination of th torsional cross sction proprtis, spcially th St. Vnants torsion constant will b focusd on. ccurat rsults ar compild for diffrnt cross sctions which wr gaind using th FEM. Th knowldg of ths valus allows a dvlopmnt of formula for approximation which provid bttr rsults than th xisting ons so far. 455 (2) (3)

3 2 FINITE ELEMENT METHOD FOR CROSS SECTIONS Th cross sction proprtis prsntd in this ssay ar calculatd with th program QSW-FE, s Kraus (2005). Th thortical background for th calculation of th proprtis as wll as shar strsss du to shar forcs, primary and scondary torsion using th finit lmnt mthod (FEM) is also dscribd by Kindmann & Kraus (2007) in dtail. For that rason only th basic principls will b shown hr. Figur 3. Discrtization of a cross sction using 9-nodd finit lmnts Th cross sction is dividd into finit lmnts as shown in Figur 3 using curvilinar 9-nodd lmnts. Th lmnt formulation is basd on th isoparamtric concpt, whr for th dformations as wll as th lmnt gomtry an qual st of shap functions is usd. In principl, lmnts with diffrnt numbrs of nods n could also b drivd, th 9-nodd lmnt has xposd as vry fficint in th sns of th numrical ffort though. Th dgr of frdom in ach nod is th warping ordinat ω. Th quilibrium in trms of virtual work for th cross sction dformation du to primary torsion, to which th warping ordinat corrsponds, can b statd for a finit lmnt with th lmnt ara as follows: 9 ( δω) ω ( δω) ω δ W = δωi Txi G + d i= 1 z z y y δω ( ) δω ( ) + G ( y ym ) ( z zm ) d = 0 z y Th first componnt of this quation corrsponds to an xtrnal virtual work, whr th nodal shar * flow T x, which corrsponds to th torqu of ϑ = 1, accomplishs work at th warping ω. This shar flow is usd to formulat th nodal quilibrium with rgard to th whol lmnt msh yilding in th quation systm for th cross sction. Th othr componnts of formula (6) ar gaind from an innr virtual work, corrsponding to ϑ = 1 as wll. From ths an lmnt stiffnss matrix and a kind of load vctor can b formulatd. Th virtual work dmands C 0 -continuos shap functions which ar summarizd in th vctor f. Th so calld Lagrangian polynomials ar applid with a quartic dvlopmnt corrsponding to th 9-nodd lmnt. Equation (6) lads to th following lmnt stiffnss rlationship with th lmnt stiffnss matrix K and th load vctor f : ϑ δω : t = K ω f (7) τ ϑ (6) 1 1 T T f f f f K = G + dt ( J) dη dζ 1 1 z z y y 1 1 T T f f f = G ( ) ( ) dt ( ) η ζ ϑ f y ym f z z M J d d 1 1 z y 456 (8) (9)

4 Th indication δω : in quation (8) is supposd to show, that th complt stiffnss rlationship dpnds on th virtual nodal valus of th warping. Th so calld Jacobi dtrminant dt ( J ) transforms th diffrntial of th ara d = dx dy in dη dζ. In addition th partial diffrntiations occurring in th formula (8) and (9) hav to b transformd as wll, s Kraus (2007) for instanc. Th intgrations can usually not b solvd analytically. For that rason numrical intgrations ar prformd using th Gauss quadratur. ftr stting up th quation systm using ths lmnt stiffnss rlationships th finit lmnt calculation provids th warping ordinat rfrring to a rfrnc point D at first, compar sction 1. In ordr to dtrmin th position of th shar cntr, th St. Vnants torsion constant and th warping constant, th intgrations of th formula (2) to (5) hav to b solvd. For that rason it is ncssary to know th cours of th function ω rsp. ω within th total cross sction. Howvr, th finit lmnt analysis only provids th nodal solutions. For th dscription of th dvlopmnt within th finit lmnts th prviously mntiond shap functions ar applid again, as it is common us in numrical mthods. With th knowldg of th warping ordinat th strsss du to primary torsional momnts can also b dtrmind. pparntly th 9-nodd lmnt taking quartic functions as a basis for th dformation will provid a crtain inaccuracy, sinc th ral distortional bhavior will diffr in gnral. In addition furthr factors will influnc th xactnss of th FEM-solution using two-dimnsional curvilinar lmnts. Dtaild information to this issu is givn by Kindmann & Kraus (2007). With a rfinmnt of th lmnt msh th inaccuracis can b minimizd lading to FEM-rsults with high accuracy. 3 DOUBLESYMMETRIC I-SECTIONS 3.1 ccurat cross sction proprtis In Tabl 1 th accurat rsults of th torsional cross sction proprtis ar compild for rolld I sctions. Th valus ar gaind as dscribd in th prvious chaptr. Kindmann t al. (2008) and Wagnr t al. (1999) hav publishd furthr rsults for a grat varity of cross sctions. Tabl 1. ccurat torsional cross sction proprtis for rolld I-sctions 457

5 Figur 4. Warping ordinat (HEM 600) and shar strsss as a rsult of St. Vnants torsion (HEM 300) Concrning Tabl 1 it should b mntiond, that max ω spcifis th maximum warping ordinat. Howvr, hr th maximum valu of th plat middl lin is compild, vn though th valus at th plat dgs ar biggr. Figur 4 (lft) is supposd to clarify this issu. It shows th numrical rsult for th warping ordinat for an HEM 600. Th maximum valu of 466 cm2 is locatd at th plat dg. Th valu of cm2 spcifid in Tabl 1 is th warping at mid-plat. It is an intrsting aspct that if a middl lin modl is applid, not rgarding th rolld aras going along with a smallr cross sction ara, largr valus for Iω ar dtrmind for th cross sctions of Tabl 1. Kraus (2005) analyzs this phnomnon. It is a rsult of th warping bhavior of th cross sction. Th rolld aras lad to a smallr cross sction warping. Whn intgrating to Iω according to formula (5), this ffct has a largr influnc than th gain of th cross sction ara. 3.2 Nw formula for th St. Vnants torsion constant Figur 4 shows finit lmnt solution for a hot rolld I-sction. Th figur on th right displays th shar strss distribution for a HEM 300 du to a primary torsion momnt Mxp = 1 kncm. In th ara Tabl 2 Past and nw formula for th IT of rolld I-sctions 458

6 of th transition from wb to flang (rolld ara), a concntration of strsss can b rcognizd. It is obvious that this is going along with an incrasd torsion stiffnss rsp. I T, which cannot b covrd by modling th cross sction using rctangular partial plats, which is a common approach for th calculation of I T. For this purpos Trayr & March (1930) dvlop a formula on basis of th mmbran analogy. In this modl th flangs and th wb ar covrd by rctangular partial plats. In addition, to covr th strss concntration within th rolld ara, Trayr & March ovrlap a circl with th diamtr D. Th portion of th torsional constant from this additional part is modifid by a factor α. It should b mntiond, that this procding is not in conformanc to th usual approach of dividing into indpndnt partial aras. Howvr, th priory intrst is th accuracy of th calculation formula. For that purpos th numrical solutions ar comparativly rfrrd to. s shown in Tabl 1, th accuracy of th formula is btwn 97.4 and % for th common Europan rolld sris IPE, HE, HEB and HEM. Howvr, furthr calculations for othr profils show much largr discrpancis. Th largst on noticd is for an HP 320 x 88, whr th formula dlivrs a valu of I T = cm 4, whil th FEM calculation provids I T = cm 4, bing a not ngligibl ovrstimation (129 %). For that rason Kindmann (2006) works out a nw formula, with which th St. Vnants torsion constant can b dtrmind with highr accuracy. Tabl 1 displays th nw formula in comparison to th on of Trayr & March. In th nw modl th partial aras of th cross sction, which ar usd to dtrmin I T, do not ovrlap. 4 NGLES 4.1 ccurat cross sction proprtis and position of th shar cntr Usually angls ar tratd fr of warping, bing a rsult of th assumption of a thin walld cross sction (middl lin modl). Calculations on bass of th FEM show, that du to th actual plat thicknss th angls show warping dformations, although bing rlativly small. Howvr, th dformations hav an influnc on th position of th shar cntr M. Whil rgarding th middl lin modl, M is positiond at th intrsction of th middl lins of th angl lgs. Tabl 4, with rfrnc to Figur 6, givs an ovrviw on whr th accurat position of th shar cntr in comparison to th middl lin modl is. In addition, Tabl 4 contains th rsults for th torsional constant. Figur 6. Position of th shar cntr of qual and unqual lggd angls 4.2 Nw formula for th St. Vnants torsion constant ngls also show strss concntrations du to primary torsional momnts in th rolld ara as shown for I-sctions, compar Figur 7. For th dtrmination of th St. Vnants torsion constant two approachs hav bn followd in th past. In th first on only th partial rctangular plats ar rgardd for ach angl lg lading to th following formula: 3 I = 1/3 ( a+ b t) t (10) T Th scond approach corrsponds to th formula of th I-sctions, whr an additional circl is insrtd to covr th torsional rigidity of th rolld aras, s Ptrsn (1990) for xampl: 3 3 t t ( / ) ( ) t IT = a t a + b t α D 3 3 b t with: D = 2 ( 2 t+ 3 r) 8 ( t+ 2 r ) 2, α = 0.07 r/ t (11)

7 Tabl 3. ccurat torsional cross sction proprtis for rolld angls according to DIN EN Figur 7. Modl for th dtrmination of th nw formula for IT 460

8 Figur 8. Discrpancis of th approximations for I T Th dviation of th rsults gaind by quation (10) and (11), in comparison to th accurat solution of th FEM, is displayd in Figur 8. Using formula (10) th largst discrpancy of about 14 % can b noticd, whr th I T is always calculatd to small. On th othr hand, th I T using formula (11) is always approximatd to larg. For that rason Kindmann & Kraus (2008) dvlop a nw formula, for which th basic ida of th I-sctions is bing pickd up and th cross sction is dividd into thr partial componnts as shown in Figur 7. Th drivation lads to th following formula: 3 4 I ( 2 ) T = a+ b c t / c with: c = t+ 0.4 r 1 (12) Th rsults of this quation show a vry good complianc to th FEM solution btwn 99.9 and % for unqual- rsp and % for qual lggd angls as shown in Figur 8. 5 CONCLUSIONS With th finit lmnt mthod accurat rsults for th torsional proprtis can b obtaind contributing to th quality of th static analysis of bar mmbrs. For many profils th accurat rsults ar tabld in this ssay in ordr to dirctly implmnt thm to ths analyss. In addition formula wr drivd to provid aids for a quick dtrmination of th torsional constant for similar profils. LITERTURE Kindmann, R Nu Brchnungsforml für das IT von Walzprofiln und Brchnung dr Schubspannungn. Stahlbau 75 (2006): Kindmann, R., Kraus, M Finit-Elmnt-Mthodn im Stahlbau. Brlin: Vrlag Ernst & Sohn. Kindmann, R Stahlbau, Til 2: Stabilität und Thori II. Ordnung. Brlin: Vrlag Ernst & Sohn. Kindmann, R., Kraus, M Torsionsträghitsmomnt und Schubmittlpunkt von Winklprofiln. Bautchnik 85 (2008): Kindmann, R., Kraus, M., Nibuhr, H. J. (2 nd d.) STHLBU KOMPKT, Bmssungshilfn, Profiltablln. Düssldorf: Vrlag Stahlisn. Kraus, M Computrorintirt Brchnungsmthodn für blibig Stabqurschnitt ds Stahlbaus. achn: Shakr Vrlag. Kraus, M On th Computation of Hot Rolld Cross Sction Proprtis and Strsss using th Finit Elmnt Mthod. Procdings of th 6 th Int. Conf. on Stl & luminium Structurs (ICSS 2007): Ptrsn, C Stahlbau. Wisbadn: Viwg Vrlag. Trayr, G. W., March, H. W Torsion of mmbrs having sctions common in aircraft construction. NC-Rport 334. Wagnr, W., Saur, R. Gruttmann, F Tafln dr Torsionsknngrößn von Walzprofiln untr Vrwndung von FE-Diskrtisirungn. Stahlbau 68 (1999):