Risto Koivula, MSc Laboratory of Structural Engineering, Tampere University of Technology P.O. Box 600, FIN Tampere, Finland

Transcription

1 A THN-WALLED RETANGULAR BOX BEA UNDER TORSON: A OPARSON OF THE KOLLBRUNNER- HAJDN SOLUTON WTH A SOLUTON BY DVDNG THE BEA NTO TWO GUDED VLASOV BEAS WTH OPEN ROSS- SETON Rito Koivla, Sc Laoratory of Strctral Engineering, Tampere Univerity of Technology P.O. Bo 600, FN Tampere, Finland ABSTRAT De to the dole ymmetry of the o eam, the field of normal tree given rie y the aial force or the ending moment are orthogonal with thoe given rie y the torion (and ditortion). Th the former can e independently perpoed on the latter. Then for the antiymmetric warping like torion and ditortion the o ide mid-ae can e regarded a contino non-compreile aial pport preventing the aial diplacement only. The eam can e ct along thee ae into two (or for) gided Vlaov eam with open cro-ection, which can e analyed ing the formal method, 3, 4 of the theory of gided eam. The idea of the theory of gided eam i the fact that for thin-walled eam, the deformation of which are retricted y eternal contino lateral or longitdinal retraint, the mtally independent effective force and diplacement qantitie (warping in term of the generalized ending theory), which can e determined independently y only eternal load, are defined in a olving ytem of ae and pecial point, which doe not coincide with the fndamental ytem with the origin at the centroid, the pole at the hear centre, and the ae parallel to the (central) principal direction. The choice of the adeqate ytem of ae and pecial point i analogo to eparating correctly a grop of mtally dependent differential eqation to mtally independent eqation. The main aim of thi tdy i to create an eact oltion of the prolem in term of the theory of gided Vlaov eam. Eact mean that no information i lot regard to the premie of the theory. KEYWORDS Thin-Walled Rectanglar Bo Beam, Vlaov Beam, Gided Beam, Bimoment, Warping, Torion, Sectorial Area, Kollrnner-Hajdin Theory, Torional Ai, Ditortion, Ditortional Ai

2 THE KOLLBRUNNER-HAJDN THEORY n the Kollrnner-Hajdin theory of o eam with a rigid-in-plane cro-ection i eparated the hear deformation, given rie y the contant hear tre flow arond the te, and the aial warping deformation from normal tree caed y the deplanation of the cro-ection. Shear deformation from the hear tree given rie y warping are neglected a mall compared with the former. The diplacement and force qantitie are given in a right-handed ytem vw, t the coordinate yz with the cro-ectional integral in a left-handed ytem, ecae that ytem i ed in the theory on gided Vlaov eam. The hear deformation of the wall midline (profile line) i then: υ γ + γ (1) n torion the tranlation field in the profile line direction i υ h (, () where h ( i the perpendiclar ditance of the pole from the profile line tangent drawn at point. o (4a+4) 8a - o r ( a,, z t a - o y Figre 1. The deplanation form of the rectangle o cro-ection according to the Kollrnner-Hajdin theory o

3 Then the aial tranlation field, in other word the deplanation at the profile line i: 0 [ h ( γ ] d (, (3) from which i otained (4a + 4) γ. Then: (4) 4( ) (4a + 4) a ( ( h ( d h ( d ( (5) 4( ) 4( ) 0 a a 0 ( ) a a a (6) 1, eq..31 Uing the method of the variational calcl, the following differential eqation wa otained (4) ρe G m (7) t One more preppoition i that the moment load m () i linear: in mot general form in thi formla i inclded etra term, inclding m () and firt derivative of imoment load, c. The ectorial qadratic moment 4 ( a ) ( da ta (8) 3 A The torional moment of inertia of a cloed cro-ection 4tΩ 4t16a 16ta t S 4( ) The radial qadratic moment h h ( da 4ta( ) A The coefficient in the differential eqation (7) h 4t( ) a ρ (11) h t 16ta a 4t( ) a So, for a rectanglar o profile with contant wall-thickne t eqation (7) take the form 4 (4) 16ta Eta ( ) m (1) 3 1, eq..37 The imoment i now otained from eqation B ρe (13) 1, eq..4 The normal tree according to thi theory are B σ (14) S (9) (10)

4 THE GUDED SYSTE OF AXES AND SPEAL PONTS FOR THE HALF PROFLE The eam i divided into two gided Vlaov eam with a U cro-ection along one of the two plane of ymmetry of the eam. The part eam are U profile which are gided y non-compreile ae along the edge of the flange, in other word along the mid-line of the o ide. Thee edge of flange are gided alo y lateral retraint in the ctting plane to keep the ditance from the original o centre (), arond which the half o o i gided to rotate. Figre. v D D D D w D F wd D a r D ( D a π/4 + - B D () a - B F wb a B N N z y D the gided ectorial pole of the half o, in other word one ditortional centre of the o B the centre of twit for the StVenant torion of the half o the centre of torion of the entire o and of the laterally gided half o, the gided origin N the non-compreile edge a D ( the ectorial area arond the ditortional centre D. ( D ( Figre. The half-o eam a a gided eam.

5 n the eginning two independent degree of freedom mt e regarded for the U profile with the noncompreile ae along the edge, t withot the lateral pport: Firt, caed y the normal tree given rie y the imoment warping, the cro-ection tend to rotate arond a ectorial pole, arond which it i poile to draw a ectorial area with zero vale at the non-compreile ae. Thi gided ectorial pole (D) i placed on the ymmetry ai of the U cro-ection at the flange length ditant from the we ehind the we (otide the o). Deformed y the warping normal tree only the cro-ection rotate arond thi gided torional ai. Loading that eam throgh thi ai in w- direction doe not give rie to normal tree. Second, for cae of gided eam with two non-compreile ae, the hear deformation from the contant hear flow etween thee ae mt e taken into conideration a an independent degree of freedom. aed y the contant hear flow deformation, thi aially gided along the flange edge, t laterally non-gided U cro ection tend to rotate arond a St Venant torional centre (B), which i placed at the interectional point of the iector of the angle (at eqal perpendiclar ditant from all three ide of the cro-ection) if the wall thickne i contant. The cro-ection rotate arond the point B withot any tendency to deplanation and normal tree. The lateral giding, fiing the cro ection to rotate arond the o centre ai, contitte a copling etween thee two for a laterally non-gided profile independent degree of freedom. To match the deformation from the warping torion arond D and the twiting torion arond B, we mt match the tranlation of point (ai from thee two degree freedom of the cro-ection to zero. Like the moment i eparated into warping and twiting moment, alo the hear force mt e eparated into two component, one placed at the ectorial pole D and proportional to the StVenant twiting moment arond B, and the other placed at the StVenant hear centre B and proportional to the warping moment (the derivative of the imoment) arond the ectorial pole D. The gided StVenant torional qadratic moment arond B and arond, and the ectorial qadratic moment arond D are: 8ta t tb ( ) t t D a ( ) (15), (16), (18) 3 The angle of twiting torion and warping torion arond the correponding pecial ae are a a B (19), (0) Uing thee we otain the eparate twiting and warping moment a fnction of the angle of twit () arond the o centre ai (the tar denote a StVenant entity): a B G tb B G tb (1) (3) a (3) D BD ED D ED () The hear force are otained aed on thee moment a fnction of the nknown (): a (3) FwB D /( ) E D (3) ( ) a FwD B /( ) G tb (4) ( )

6 The two moment can e tranformed according to the theory of gided eam to the o centre ai : D + af wb + B + ( a ) F a (3) a ( ) E D + ( ) GtB (3) E + G t (5) The moment eqilirim arond the o ai give the eqation for the angle of rotation: (4) () E G + t m (6) Thi give the aic eqation for (): wd (4) E G t m (7), t, and m () are now divided y compared with eqation (7). ompared with the Kollrnner- Hajdin eqation (7) the coefficient ρ 1, while for the Kollrnner-Hajdin eqation ρ (11) a The eqation of the angle on twit according to oth of the theorie can e epreed with one formla (4) ( α 3 ) ( ) a ± Eta ( a ± ) m, where 6 α 1 + ν (8) n the ign ± the ign elong to the gided eam theory, and the ign + elong to the Kollrnner- Hajdin theory. ν i the Poion koefficient. The imoment, torional moment and normal tree according to the theorie are following: The theory of gided eam a BD E D D E D E (9) a a (3) () B D + E + G t (30) σ B ( D D (31) D The Kollrnner-Hajdin theory B ( ) E E D (3) a + (3) () B ρe + G (33) B ( σ (14) t

7 L Figre 3.: A cantilever eam, loaded with a moment at the free end. For the cantilever eam in figre 3. with the end condition ( 0) 0, ( 0) 0, B( L) 0 ( L) 0 (34), (35), (36) () - B () + (3) () E + G (37) t the oltion fnction for eqation (8), when m () 0, i α α + a± a± Ae + Be + D, (38) which give for the given end condition α( L) inh a ± a ± αl + tanh +, where (39) α αl a ± coh a ± a 3 ( + ) Eta α G t For the gided eam oltion (-), when α a a, then in eqation (9) B D 0 like BD m e (41) α and the normal tree according to (31) σ 0, too. The greater the ide difference i, the cloer the oltion are one to another, t they never coincide. DSUSSON t i een in eqation (39) that the Kollrnner-Hajdin (+) oltion give non-adeqate relt when a : it introdce normal tree for a qare o eam in torion, which i not tre, ecae for a qare cro-ection o the ectorial area ( 0 according to (5) and (6). oefficient ρ in eqation (11) i wrong. The gided eam oltion give the right relt alo for the qare o. The oltion according to the theory of gided eam can e generalized to accont the ditortional degree of freedom y copling the already introdced warping and twiting degree of freedom from (18) - () with a pring connection. The notion of the ditortional centre for the point D in thi article i introdced y profeor Erkki Niemi from Lappeenranta Univerity of Technology, independent of the theory of gided eam, a a point loaded throgh which antiymmetrically in torion, the o eam doe not ditort. Thi notion epree the eence of thee for econdary pecial point of a o eam. He ha alo introdced thi ytem of ae in which all ign are poitive in the formla of normal tree, which i efl featre for generalized warping in the theory on gided eam. (40)

8 REFERENES 1. Kollrnner, rt F., Hajdin, Nikola (197): Dünnwandige Stäe, Band 1, Zürich/Belgrad SBN Koivla, Rito (1998): Tranformation of Force and Diplacement Qantitie etween Different Sytem of Ae and Special Point, Reearch and Development; Proceeding of the nd nternational onference of Thin Walled Strctre, Singapore, SBN Koivla, Rito (000): Analyi of Ditortion of a Thin-Walled rectangle Beam Uing the Theory of Gided Vlaov Beam, Proceeding of the V Finnih echanical Day, Tampere, SBN Koivla, Rito (003): Derivation of the Kollrnner-Hajdin Theory of the Thin-Walled Rectanglar Bo Beam nder Torion y Dividing the Beam into Gided Vlaov eam with Open ro-section. Proceeding of the V Finnih echanical Day, Epoo, SBN