Method for including restrained warping in traditional frame analyses

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1 thod for incuding rstraind arping in traditiona fram anayss PCJ Hoognboom and A Borgart Facuty of Civi Enginring and Goscincs Facuty of Architctur, Dft Univrsity of Tchnoogy, Dft, Th Nthrands Rstraind arping is important for th torsion dformation and aia strsss of thin-a opn sctions Hovr, this phnomnon is not incudd in commony usd fram anaysis programs This papr prsnts a simp mthod to incud th ffct of rstraind arping of mmbr nds in a fram anaysis In this mthod, prior to th fram anaysis th structura dsignr incrass th sction torsion stiffnss by a factor Aftr th anaysis th structura dsignr obtains th trm bi-momnts and aia strsss from th computd torsion momnt Th mthod is dmonstratd in thr amps Th importanc of rstraind arping is shon Ky ords: Torsion, fram anaysis, rstraind arping, dsign mthod Introduction Whn a bam is oadd in torsion th cross-sctions dform in th aia dirction In othr ords pan cross-sctions do not rmain pan (Fig ) This phnomnon is cad arping Thr dimnsiona fram anaysis programs usuay appy th torsion thory of D Saint Vnant This thory assums fr arping of th mmbr sctions In raity many joints ar such that thy prvnt th mmbr nds from arping fry This incrass th mmbr torsiona stiffnss and introducs aia strsss spciay in th mmbr nds For soid and thin-a cosd sctions ths ffcts can b oftn ngctd Hovr, for thin-a opn sctions rstraind arping is oftn important Th torsion thory of Vasov incuds th ffct of rstraind arping [Vasov 959, 96, Zbirohoski-Koscia 967] Compard to th traditiona bam thory, th Vasov thory introducs to tra quantitis; th arping constant C and th bi-momnt B Th arping constant is a cross-sction proprty and a masur for th ffort ndd to rduc arping Th unit of th arping constant is [ngth 6 ] Th bi-momnt is a masur for th strsss ndd to rduc arping Th unit of th bi-momnt is [forc ngth ] For an I-sction th bi-momnt is th momnt in th fangs tims thir distanc (Fig ) For othr cross-sction shaps th intrprtation is much mor difficut Th Vasov torsion thory can b impmntd in fram anaysis programs [ijrs 998] Th adaptations incud on tra dgr of frdom (th arping) and an tra forc (th bimomnt) for vry mmbr nd In th Appndi th stiffnss matri is incudd for impmntation of th Vasov thory in a fram anaysis program If impmntd th HERON, Vo 50, No (005) 55

2 structura dsignr oud b ab to spcify for vry mmbr nd hthr arping is, fr, fid or connctd If fr arping oud b sctd for a mmbr nds th fram anaysis oud b acty th sam as a traditiona D Saint Vnant anaysis If fid arping or connctd arping oud b sctd th mmbr oud b stiffr and a bi-momnt distribution nds to b dispayd Such a fram program oud aso ao distributd torsion momnt oading, hich oftn occurs in practic Hovr, non of th commony usd fram anaysis programs incud th Vasov torsion thory Th rason might b that Vasov torsion thory is not knon as yt and that nginrs ar not ducatd in th ffcts of rstraind arping To th authors opinion th Vasov torsion thory shoud b standard functionay in thr dimnsiona fram anaysis programs a a b B = a Figur : Fr arping (a) and rstraind arping (b) of an I-sction oadd in torsion Figur : Intrprtation of th bi-momnt in an I-sction Nonthss, th ffcts of rstraind arping can arady b takn into account in a traditiona fram anaysis To this nd, a simp mthod is prsntd in Chaptr 5 Th mthod incuds prvntd arping at on or both mmbr nds Th mthod is not vaid for connctd arping, for amp in continuous bams or at a chang of cross-sction (non-prismatic mmbrs) Th mthod aso dos not ao distributd torsion momnt oading For connctd arping and distributd torsion oading th structura dsignr nds to us a fram program that incuds Vasov torsion thory In Chaptr th Vasov torsion thory is summarisd In Chaptr 3 th anaysis mthod is drivd In Chaptr 6 th mthod is dmonstratd for thr structurs It is notd that th torsion thory of D Saint Vnant is aso rfrrd to as uniform torsion or circuatory torsion Th torsion thory of Vasov is aso rfrrd to as non-uniform torsion or arping torsion 56

3 Vasov Torsion Thory In 940, VZ Vasov dvopd a torsion thory in hich rstraind arping is incudd [Vasov 959] Th rotation j of th bam cross-sction foos from th diffrntia quation d 4 d = m 4 d d, () hr, is th torsion stiffnss, is th arping stiffnss and m is a distributd torsion momnt aong th bam Th torsion stiffnss and arping stiffnss ar cross-sction proprtis that can b computd ith spciaisd softar [ShapDsignr, ShapBuidr] Tabs ist for oftn occurring cross-sctions [Timoshnko 96, Young 989] Th torsion momnt is d db t = d + d Th bi-momnt is () B d = d (3) Th bam-nds can hav ithr an imposd rotation = 0 or an imposd torsion momnt d t = t0 At th sam tim thy hav ithr an imposd arping = o or an imposd bimomnt B=B 0 Th Vasov thory rducs to th thory of D Saint Vnant if th arping d stiffnss is zro, th distributd momnt is zro and arping is fr 3 Drivation of th Anaysis thod Suppos that arping is prvntd at both sids of a bam Than th boundary conditions ar = = 0, d = 0 d = 0 = =, d = 0 d = (4) Th soution of diffrntia quation () is ( ) ( ) + ( ) = ( ) + m + + ( ) ( ) hr (5) 57

4 = (6) Using, = - t B B and B = B th mnt stiffnss matri 0, = 0, = = = t = = is drivd In hich, th indics, rfr to th ft and right of th bam, rspctivy = ( + ) m + + ( ) + (7) Th bi-momnts at both bam-nds can b prssd in trms of th torsion momnt B m + = t tanh( ) + B = m t tanh( + ) (8) W assum that m =0 In th prvious stiffnss matrics obsrv that th torsion stiffnss is incrasd by th factor ( + ) + + ( ) (9) For > 5 this can b approimatd ith % accuracy by (0) For > 6 th prssion for th bi-momnt can b approimatd ith % accuracy by B = B = t () Suppos that arping is prvntd at on sid of a bam ( = 0) Than th boundary conditions ar = = 0, d = 0, d = 0 = =, d = 0 d = () Th soution of th diffrntia quation is too arg to incud in this papr Th mnt stiffnss matri is 58

5 = ( + ) + + ( ) Th bi-momnts at both bam-nds ar (3) B m + = t tanh( ) +, + m + ( ( ) ) ( + + ( ) ) ( ( + ) ) B = 0 (4) W assum that m =0 In th prvious stiffnss matrics obsrv that th torsion stiffnss is incrasd by th factor ( + ) + + ( ) (5) For > 3 this can b approimatd ith % accuracy by (6) For > 3 th prssion for th bi-momnt can b approimatd ith % accuracy by B t (7) Cross-sction Strip t = b t b b 6 +ν I-sction t t t h t h h 8 +ν Squar cross-sction Round cross-sction Round tub h b b 39 +ν Tab : numbrs for svra cross-sctions 59

6 4 Numbr Th numbr is dfind as (6) =, hich is a mmbr proprty that dtrmins to hat tnt torsion according to D Saint Vnant can dvop hn arping is rstraind Tab givs an ovrvi of numbrs For most bams is argr than 6 Th approimations in th prvious sction ar accurat for > 6 In this cas th dviations from th act vaus ar smar than % For short thin-a opn sctions can b smar than 6 As an amp, considr an I-sction ith a ngth of 4000 mm Th hight is 300 mm, th idth is 50 mm and th a thicknss is 0 mm Poisson s ratio is ν = 035 Using Tab find = 3 For this cas th act formua nds to b usd instad of th approimations Not that for sma vaus of a arg incras in mmbr torsiona stiffnss is obtaind It can b shon that th mmbr torsiona stiffnss cannot b argr than GI p /, hr GIp is th poar momnt of inrtia In thory th drivd mmbr torsiona stiffnss can bcom argr than this maimum for vry short mmbrs Hovr, this has no practica consquncs bcaus th diffrnc btn a vry stiff mmbr and an trmy stiff mmbr givs no diffrnc in structura forc fo 5 Rstraind Warping thod ) For incuding th ffct of rstraind arping nd to kno th torsion constant J and th arping constant C of th mmbr cross-sction any fram programs incud sction ibraris ith commony usd sctions and thir proprtis If th torsion proprtis ar not avaiab a program for cross-sction anaysis can b usd to comput thir vaus [ShapDsignr, ShapBuidr] ) Subsqunty th numbr is cacuatd (6) =, hr is th mmbr ngth, G is th shar moduus and E is Young's moduus G is dfind as G = E + ( ν), (8) hr ν is Poisson s ratio If is argr than 6 th subsqunt formuas ar vaid If is smar than 6 ths formuas might not b vaid and mor aborat formuas might b ndd, hich 60

7 ar drivd in Sction 3 3) Bfor th fram anaysis th torsion stiffnss is ntrd or changd in th fram program If arping is prvntd at on mmbr-nd th torsion stiffnss J nds to b mutipid by (6) If arping is prvntd at to mmbr-nds th torsion stiffnss J nds to b mutipid by (0) 4) Aftr th fram anaysis th argst bi-momnt is cacuatd ith (), (7) B = t at on bam nd ( = 0) if it is rstraind thr, B =+ t at th othr bam nd ( = ) if it is rstraind thr 5) Finay, th rsuting strss stat in th cross-sction can b computd by a program for crosssction anaysis [ShapDsignr, ShapBuidr] This program can aso tak into account th strsss du to othr sction momnts, shar forcs and th norma forc 6 Appications 6 Cantivr A bam is fid at on nd at hich rotation and arping ar prvntd (Fig 3) At th othr nd th bam is oadd by a torqu T = 6 knm hi arping can occur fry Th matria and cross-sction data of th bam ar E = N/mm, G = N/mm, J = mm 4 and C = mm 6 This amp is aso anaysd in [Chn 977] 5,7 67 mm mm T =,6 knm Figur 3: Thin-a cantivr oadd in torsion For this bam th numbr is = = = 666 6

8 Th torsion stiffnss incudd in th fram anaysis is 666 = = J mm Th computd rotation at th mmbr nd is 03 rad and th computd torsion momnt is t = Nmm Th rotation is 8% smar than th sam bam ith unrstraind arping Subsqunty, th bi-momnt is 540 B = t = = Nmm 666 Using th torsion momnt and bi-momnt a program for cross-sction anaysis [ShapDsignr] cacuats th shar strss distribution and th norma strss distribution Th argst shar strss is 09 N/mm, hich occurs at th oadd nd aong th circumfrnc of th crosssction Th trm norma strss is -458 N/mm, hich occurs at th campd nd in th ft of th top fang Th computd aia strss vau is 0% argr (in absout sns) than thos found in [Chn 977] Th anaysis in [Chn 977] is basd on acty th sam Vasov torsion thory but th cross-sction anaysis is prformd anayticay using th thin-a assumption hi th crosssction program usd in this papr is suitab for both thin- and thick-a anaysis Thrfor, th diffrncs ar causd by diffrncs in th cross-sction anaysis hich is not th subjct of this papr A noticab dtai is that th arping constant computd by th program is 7% smar than found in th thin-a anaysis Nonthss, th strsss diffr much ss 6 Bo-Girdr A bo-girdr bridg has a ngth = 60 m Th cross-sction dimnsions and proprtis ar shon in Figur 4 Th concrt Young s moduus is E = N/m and Poisson s ratio ν = 05 At both nds th bridg is supportd hi arping is fr (Fig 5) In th midd th bridg is oadd by a torqu T = Nm This oading occurs hn th bridg is supportd at mid span by to tmporary coumns of hich on fais du to an accidnt Thrfor, th torqu T is du to th support raction of th rmaining tmporary coumn This amp is adaptd from ctur nots by drir C van dr Vn on rinforcd and prstrssd concrt dsign for th Dutch Concrt Association 6

9 0,50 A = 74 m y z,50,700 Iy = 95 m 4 I z = 86 m 4 J = 06 m 4,5 0,500 7,750 0,500,5 0,00 C = 3944 m 6 3,000 m Figur 4: Bridg cross-sction T Figur 5: Structura mod of th bridg incuding fork supports and torsion oading Du to symmtry th arping in th midd sction is prvntd Th structura mod consists of to bam mnts Th numbr is = = J ( + ν) C = = Both mnts hav arping rstraind at on sid Thrfor, th torsion constant for th fram anaysis is 4 30 = = J m Th fram program computs a rotation of rad in th midd of th bridg This is 4% smar than th unrstraind bhaviour Th program computs th torsion momnt t = Nm for th ft bam and = Nm for th right-hand bam Thrfor, th argst bimomnt in th right-hand nd of th ft bam is B = t = ( ) = Nm 4 30 Th argst bi-momnt in th ft nd of th right-hand bam is 60 B = t = = Nm

10 Using th torsion momnt and bi-momnt a program for cross-sction anaysis cacuats th shar strss distribution and th norma strss distribution Th argst shar strss is -74 N/mm, hich occurs vryhr in th bottom fang ft of th symmtry pan Th argst norma strss is 78 N/mm, hich occurs in th symmtry pan in th ft sid of th bottom fang (Fig 6) It shos that aso for cosd thin-a sctions th strsss du to rstraind arping can b important In th bridg aso strsss du to th fura momnt, shar and prstrssing occur but ths hav not bn considrd in this amp 78 N/mm - 78 N/mm Figur 6: Distribution of th norma strsss du to rstraind arping in th midd cross-sction 63 Traffic sign Th structur of a traffic signpost consists of to incind coumns and a horizonta bam (Fig 7) A mmbrs ar st sctions IPE400 Th top joint and th foundation joints ar such that arping cannot occur At th fr nd of th horizonta bam arping can occur no arping 5 kn 4 kn 3 3 knm IPE400 A = 8450 mm 0 m fr arping I y = mm 4 Iz = mm 4 J = mm 4 z y no arping C = mm 6 Figur 7: Structura mod of a traffic sign Th ngth of th incind coumns is 005 m Young s moduus is E = 0 5 N/mm and Poisson s ratio is ν = 035 A vaus ar assumd to b dsign vaus, thrfor, safty factors ar arady incudd Th numbrs of th coumns and th bam ar, rspctivy c = = J = 0050 ( + ν) C = 60, = 3000 = 8 85 b

11 Th torsion constants for th fram anaysis ar rspctivy c 60 4 J = = mm, c 50 b b( + ) b b + + ( b ) ( + ) 4 J = = mm Th fram program computs a rotation of th fr bam nd of 007 rad in th dirction of th momnt oading This is 5% smar than computd ith fr arping of a mmbr nds Th horizonta dispacmnt of th fr bam nd is 07 mm, hich is % smar than ith fr arping Th torsion momnt in th bam is 300 knm Th torsion momnt in th coumns is 05 knm Consqunty, th bi-momnts ar 0 05 Bc = t = = knm in th fid bam nd, 300 Bb = tanh( ) t = =46kNm in th coumn nds 85 With a cross-sction anaysis program th norma strss is computd du to th bi-momnts Th argst norma strss du to B = -46 knm is σ = 64 N/mm It occurs at th fid nd of th horizonta bam in th top and bottom fangs It is notd that for I-bams a formua can b drivd for rstraind arping strsss [Young 989] B σ,ma = ( h t) tb 6 (9) hr h is th hight of th cross-sction, b is h idth of th cross-sction and t is th thicknss of th fangs Th thicknss of th b is not rvant 7 Rmarks Though arg strsss can occur du to rstraind arping, this dos not ncssariy man that traditiona fram anaysis is unsaf for th utimat imit stat Th rason is that traditiona fram anaysis provids a forc fo that is vryhr in prfct quiibrium Th or bound thorm of pasticity thory stats that any quiibrium systm that dos not vioat th utimat strss conditions provids a saf soution for th imit oad of th structur In othr ords, th matria i yid and th pak rstraind arping strsss i not occur Thrfor, traditiona fram anaysis is saf providd that th matrias usd ar sufficinty ducti to ao rdistributions Th advantags of using Vasov torsion thory is mainy ratd to th srvicabiity imit stat 65

12 It can b usd to sho that th dformations i b smar than found ith th traditiona thory It can aso b usd for crack contro in structura concrt bo-girdrs 8 Concusions A simp mthod is prsntd to incud th ffct of rstraind arping of mmbr-nds in traditiona fram anayss Th mthod consists of just 5 stps and th formuas can b asiy mmorisd Rstraind arping can substantiay rduc th structura dformation It can aso introduc arg aia strsss in th rstraind sctions Rstraind arping is not ony important for thin-a opn sctions Aso in thin-a cosd sctions th norma strsss du to th rstraind somtims can b considrab Dspit th arg strsss that can occur hn arping is rstraind, a traditiona fram anaysis is a saf mthod of computing th utimat oad providd that th matrias appid ar sufficinty ducti Rfrncs Chn, WF, T Atsuta (977) Thory of Bam-Coumns, Vo : Spac Bhaviour and Dsign, N York, cgra-hi ijrs, SJH (998) Th Torsion Thoris of D Saint-Vnant and Vasov, Sc thsis, Dft Univrsity of Tchnoogy ShapDsignr (004) softar, onin, ShapBuidr (004) softar, onin, Timoshnko, SP, J Gr (96) Thory of Eastic Stabiity, N York, scond dition, cgra- Hi Vasov, VZ (959) Thin-Wad Eastic Bams (in Russian), osco, nd d, Fizmatgiz Vasov, VZ (96) Thin Wad Eastic Bams, Engish Transation, Washington DC, Nationa Scinc Foundation Young, WC (989) Roark s Formuas of Strss & Strain, 6th dition, N York, cgra-hi Zbirohoski-Koscia, K (967) Thin Wad Bams, From Thory to Practic, London, Crosby Lockood & Son Ltd 66

13 Appndi Stiffnss matri for th Vasov Torsion Thory This matri can b usd for impmntation of rstraind arping in structura anaysis softar For arg vaus of th stiffnss matri can b approimatd as B B + m = G J d d B B d d t = = = = = = = = 0,,,,,, B B B B t = = + m λ λ E = η η λ λ η η λ λ λ λ C ξ µ λ λ µ ξ = + + η ( ) + = + + = ( ) ( ) λ µ ξ + + = ( ) ( ) ( ) = 67

14 If to zro from abov th stiffnss matri rducs to B B + = m

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