Nonuniform Torsion Without Shear Deformation Effect Using FEM

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 8 NONUNIFORM TORSION WITHOUT SHEAR DEFORMATION EFFECT USING FEM Dang-Bao TRAN1,2, Ja osla NAVRÁTIL1 1 Depa men o S uc u es, Facul y o Ci il Enginee ing, VSB–Technical Uni e si y o Os a a, Lud íka Podéš ě 1875/17, Os a a, Czech Republic. 2 Depa men o Ci il Enginee ing, Facul y o A chi ec u e, Thu Dau Mo Uni e si y, T an Van On 06, Binh Duong P o ince, Vie nam. dang.b[email p o ec ed], ja osla [email p o ec ed]. DOI: 10.35181/ ces-2021-0006 Abs ac . This pape p esen s he use o a ini e elemen me hod (FEM) o analyze nonuni o m o sion wi h an a bi a y c oss-sec ion wi h homogeneous elas ic ma e ial wi hou shea de o ma ion e ec . Beams a e cons ained by he mos common ypes o suppo s, such as ixed, pinned, and olle , and a e subjec ed o any applied o sional load, concen a ed, and dis ibu ed. The p esen ed FEM ans o ms he 3D analysis o nonuni o m o sion beams in o sepa a ed 2D c oss-sec ional and 1D modeling. The geome ic cons an s o he c oss-sec ion a e i s ly de i ed om 2D FEM, which uses a 9-node isopa ame ic elemen . Then, a 1D FEM, which uses he He mi ian shape unc ion, is de eloped o compu ing he wis angle, he de i a i e o wis angle, and s ess esul an s. Finally, he s ess ield is ob ained om he local analysis on he 2D-c oss sec ion. A MATLAB p og am is execu ed o alida e he nume ical me hod h ough examples. The alida ion examples ha e p o en he eliabili y o he au ho 's nume ical me hod o analyzing p oblems de ined abo e. Keywo ds Nonuni o m o sion, hin-walled s uc u e, shea de o ma ion e ec , wa ping es ain , ini e elemen me hod. 1. In oduc ion In enginee ing p ac ice, we o en encoun e beam s uc u es loaded in o sion. When he wa ping o a membe 's c oss-sec ion is no es ained, he s ess ield o a p isma ic beam wi h homogeneous iso opic ma e ial can be de i ed om he Sain -Venan heo y s ic ly [1-4]. In p ac ice, because i) bounda y condi ions a e imposed, ii) geome ical cha ac e is ics o he sec ion, he wa ping o beam's c oss-sec ion is es ained, which leads o addi ional no mal and shea s esses, which he Sain Venan heo y does no ake in o accoun . Vlaso [5] was he i s o o mula e he p oblem o nonuni o m o sion. Bensco e [6] imp o ed Vlaso 's heo y, which neglec s he shea de o ma ion e ec , leading o e o s wi h closed c oss-sec ions [7]. Many au ho s es ablished FEM o conside nonuni o m o sion aking in o accoun he shea de o ma ion e ec [7- 14]. Howe e , he abo e esea ch [7-14] used he app oxima ions o Thin Tube Theo y [5] o de e mine he ba 's geome ic cons an s, es ic ing he accu acy and applicabili y o he o mula ions. El Fa mi [15,16] p oposed a beam heo y wi h a nonuni o m wa ping, including he e ec s o o sion and shea o ces o a bi a y c oss-sec ions made o homogeneous iso opic elas ic ma e ial. Mokos and Sapoun zakis [17] p esen ed a nonuni o m o sion heo y conside ing he shea de o ma ion e ec o gene al c oss-sec ions implemen ed by Bounda y Elemen Me hod. The pu pose o his pape is o es ablish a nume ical me hod using FEM o sol e he nonuni o m o sion p oblem wi hou he shea de o ma ion e ec o he p isma ic beam wi h a bi a y c oss-sec ion unchanged h oughou he leng h wi h homogeneous iso opic elas ic ma e ial, based on displacemen and s ain ields de i ed om Sapoun zakis, EJ e al. [17]. This pape is a p elimina y esea ch s ep o u u e esea ch, which conside s he shea de o ma ion e ec in nonuni o m o sion. 2. A b ie in oduc ion o he heo e ical o mulas o nonuni o m o sion Le us conside a p isma ic beam wi h a bi a y c oss- sec ion, cons an along he leng h L wi h he modulus o elas ici y E, and shea modulus G. The longi udinal axis is he x-axis, and he c oss-sec ions lie in he y–z plane. S is he shea cen e o he c oss-sec ion. The pa allel sys em , S zzz=+ S yyy=+ in e sec s a O, he a bi a y poin . CXYZ is he pa allel sys em wi h Sxyz h ough he cen e SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 9 o g a i y C. The mul iply connec ed domain Ω is bounded by n cu es, 1,Γ 2,Γ..., 1, n− Γ , n Γ as Fig. 1. Tangen ec o wi h associa e coo dina e s and no mal ec o n se up he igh -handed sys em. The beam is en o ced o he a bi a y loads, dis ibu ed o que m (x), concen a ed o que M i(x), concen a ed wa ping momen Mwi(x). The c oss-sec ion is assumed wi h no dis o ion. Fig. 1: C oss-sec ion o he p isma ic beam The displacemen ield is exp essed as [17] (,,)uxyz ='(,) (,), PS xS S yz yz θφ φ + (1) (,) (), x xz z x θ =− (2) (, ) (), x wxy y x θ = (3) whe e u, , w a e he axial and ans e se displacemen s o he beam wi h espec o Sxyz x θ is he angle o wis (,) P Syz φ is he p ima y wa ping unc ion wi h espec o shea cen e S de i ed om 20in on . P S PP SS yzyzn nnznyn yz φ φφ ∇= Ω ∂∂ +=− Γ ∂∂ (4) Mo eo e , he es ained wa ping unc ion can be compu ed as ollows by using he ans o ma ion o coo dina es [18] 1 (,) (,) (,) . PP P SOSS O yz yz yZ zY yzdA A φφ φ Ω =+−−  (5) whe e (,) P Oyz φ is he wa ping unc ion wi h espec o 𝑂𝑦𝑧 sys em coo dina es, A is he a ea o he c oss-sec ion S S φ is he seconda y wa ping unc ion wi h espec o he shea cen e S, gi en as ''' 2() in 0on , SP x SS SS SS yz n Ex G nn yz θ φφ φφ ∇=− Ω ∂∂ +=Γ ∂∂ (6) The s ess ields ob ained om he heo y o elas ici y as [17] "() (,), P xx x S Ex yz σθφ = (7) xy τ ='() P xy P S x Gx z y τ φ θ  ∂−+   ∂  144424443 , S xy S S Gy τ φ ∂ ∂ 123 (8) xz τ ='() P xz P S x Gx z z τ φ θ  ∂++   ∂  144424443 , S xz S S Gz τ φ ∂ ∂ 123 (9) In addi ion, he shea s ess in (8), (9) can be seen as composed o wo pa s: p ima y shea s ess and seconda y shea s ess. The wa ping momen , he o al wis ing momen , he p ima y wis ing momen , he seconda y wis ing momen a e deno ed by , w M , M , P MS M ob ained as [17] w M=P xx S d σφ Ω −Ω= '' , Sx EC θ − (10) ', PP PP P SS xy xz x M zyd yz GI φφ ττ θ Ω   ∂∂ =−++Ω    ∂∂    =  (11) S M= PP SS SS xy xz yz φφ ττ Ω  ∂∂ −− =   ∂∂  ''' (), S EC x θ − (12) , P S MM M=+ (13) whe e I is he o sional cons an , de e mined as I =22 , PP SS yzy z d zy φφ Ω  ∂∂ ++ − Ω   ∂∂   (14) S C a e he wa ping cons an de e mined by () 2. P SS Cd φ Ω =Ω  (15) The ela ion be ween he s ess esul an , o al wis momen M , and he dis ibu ed o que m (x) is exp essed as [17] (). Mmx x ∂=− ∂ (16) A e some algeb a, he equilib ium equa ion o he nonuni o m o sion p oblem o a homogeneous iso opic ba wi hou shea de o ma ion e ec is exp essed as 42 42 . xx S dd EC GI m dx dx θθ −= (17) SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 10 3. FEM p ocedu es 3.1. Angle wis , x θ Applying Gale kin’s me hod, one o he me hods o weigh ed esidual, he go e ning equa ion Eq. (17) is ans o med o weak o m as 42 42 0 0, Lxx S dd IEC GI mdx dx dx θθ η  =−−=     (18) whe e η is he es unc ion. The beam is disc e ized in o a numbe o ini e elemen s. A e some manipula ion, he weak o mula ion o Eq. (18) can be exp essed as 22 22 1ee nxx S LL i dd dd IEC dxGI dx dx dx dx θθ η η =  =+−     0 0, e L x w L ddd mdx GI M M dx dx dx θηη ηη  +− + − =     (19) whe e Le is an beam elemen domain n is he numbe o elemen s o he beam. The He mi ian polynomial in e pola ion unc ion, which achie es he co ec solu ion o he p oblem wi h e ining mesh, p o es mo e lexible han he Hype bolic in e pola ion unc ion, which gi es he mos accu a e esul s wi h he mos analy ical solu ion [19]. We choose cubic unc ions o he spa ial in e pola ion o he wis angle, x θ , in e ms o nodal a iables. To his end, we conside an elemen ha has wo nodes, one a each end. The wis angle can be exp essed as 12 112 3 24 () () () () , xx xx x dd Hx Hx Hx Hx dx dx θθ θθ θ =+ + + (20) whe e 23 23 12 23 3 23 23 34 23 23 32 2 () 1 , () , 32 () , () , x xxx Hx Hx x l ll l xx xx Hx Hx ll ll =− + = − + =− =−+ (21) 12 12 ,,, x x xx dd dx dx θθ θθ a e he nodal deg ees o eedom. Inse ing he Eq. (20), Eq. (21) o Eq. (19) esul s in he s i ness ma ix o he beam elemen 12 , ee =+KK K (22) whe e 111 0, e L eT S EC dx= KBB (23) 222 0, e L eT GI dx= KBB (24) whe e 2 22 2 3 12 4 12222 , dH dH dH dH dx dx dx dx  =   B (25) 3 12 4 2. dH dH dH dH dx dx dx dx  =  B (26) The co esponding elemen nodal deg ees o eedom is 12 12 ,,, . T xx xx dd dx dx θθ θθ  =  e d (27) The hi d e m in Eq.(19) esul s in he elemen o ce ec o . Fo a gene ally dis ibu ed o que, we need o compu e 1 2 03 4 () , e L H H mx dx H H    =     e F (28) whe e e F is he elemen o ce ec o . The las e m in Eq. (19) is he bounda y condi ions o he o al wis ing momen and he wa ping momen a he wo bounda y poin s, x = 0 and x = L, o he beam. I hese bounda y condi ions a e known, he known o al wis ing momen and wa ping momen a e included in he sys em o ce ec o a he wo bounda y nodes. O he wise, hey emain unknown. Howe e , he wis angle and he de i a i e o he wis angle a e known as geome ic o his case. Assembling he elemen s i ness ma ix and ec o s leads o he sys em ma ix equa ion gi en below .=Kd F (29) 3.2. Wa ping unc ion, , P S SS φφ Using Gale kin’s me hod, wi h es unc ion 1()H η ∈Ω and applying he Gauss-G een heo em, he go e ning equa ions Eq. (4), Eq. (6) a e ans o med o weak o m as PP SS d yy zz φφ ηη Ω  ∂∂ ∂∂ +Ω−   ∂∂ ∂∂  (𝑛𝑧−𝑛 𝑦)𝜂𝑑𝑠 =0. (30) ''' () 0. SS P SS x S Ex dd yy zz G φφ θ ηη φη ΩΩ  ∂∂ ∂∂ +Ω− Ω=   ∂∂ ∂∂   (31) The wa ping unc ion alues, P S φ , , S S φ in Eq. (30) and Eq. (31) a e app oxima ed by he FEM, which is p esen ed in [3, 4]. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 11 4. Valida ion examples In his sec ion, he accu acy o he p oposed FEM has been examined. A compu e code is de eloped in he MATLAB R2015a so wa e based on he o mula ions desc ibed in he p e ious sec ions. The ob ained esul s a e compa ed wi h he a ailable wo ks o he li e a u e. 4.1. Example 1 As a i s example, a hin-walled beam wi h an I-shaped c oss-sec ion shown in Fig. 2 is analyzed and compa ed wi h he esul s ob ained by T alli, A.M. [8] and Kim, N. I. e al. [11]. The I-sec ion beam is clamped– ee and he concen a ed o sional momen Mx = 25 kNm is applied a i s ee end. The leng h o he beam is 5 m. The Young’s modulus and he Poisson’s a io a e E = 2 x 106 MPa and 𝜈= 0.3 , espec i ely. The i s es was analyzed using 16 axial elemen s and 19 elemen s in he c oss-sec ion. The geome ic cons an s o he c oss-sec ion de e mined om he p esen s udy a e as ollows 76 2.1559 10 S Cm − =× , 64 2.8643 10 Im − =× . Fig. 2: A can ile e beam, he applied load, and he geome y o he I c oss-sec ion, uni s [mm]. Fig. 3: The a ia ion o wis angle along he x-axis. In Table 1, he angle o wis a he ee end p edic ed by he p esen s udy is gi en and compa ed wi h he esul s o o he esea che s. The a ia ions o he angle o wis along he leng h o he hin-walled beam a e shown in Fig. 3. I can be seen om Table 1 and Fig. 3 ha he wis angle esul s ob ained om he p esen s udy a e in excellen ag eemen wi h o he wo ks. Tab.1: Compa ison o he wis angle 𝜃 a he ee end calcula ed by di e en me hods Me hods 𝜃 [𝑟𝑎𝑑] a x = L P esen s udy 0.5171 T alli [8] 0.5183 Kim e al [11] 0.5173 Vlaso [11] 0.5167 Fig. 4: Dis ibu ion o he axial s ess 𝜎 in 2D c oss-sec ion a ix-end, uni MPa, Fig. 5: Maximum o no mal s ess 𝜎 along he axis x o he beam. Fig. 4 shows he con ou plo o he axial s ess 𝜎 a he clamped end o he I-sec ion beam. Fig. 5 depic s he a ia ion along he axis x o he no mal s ess 𝜎. Fig. 6 and Fig. 7 p esen he a ia ion along he axis x o he wa ping momen and he o sional momen s, espec i ely. I is obse ed om Fig. 4 ha he a ia ions o he axial s ess a e symme ic o he a ea cen e o he c oss-sec ion. In con as o he langes, no signi ican axial s ess appea s a he web o he I-sec ion beam. The maximum alue o 𝜎 = 515.911 MPa occu s a he co ne s o he SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 12 langes. In Fig. 5, he a ia ions o he maximum axial s ess 𝜎 a he ixed-end o he c oss-sec ion along he leng h o he beam a e shown and compa ed o he ones ob ained om [8, 11]. I can be seen ha he axial s ess p edic ed by he p esen s udy is in good ag eemen wi h he esul s o T alli, A. M. [8]. Fig. 6: The a ia ion o wa ping momen Mw along he axis x o he beam. Fig. 7: The a ia ion o o sional momen s along he axis x o he beam. 4.2. Example 2 A can ile e ed hin-walled beam wi h a channel c oss- sec ion depic ed in Fig.8 is analyzed in his sec ion. The dep h, wid h, and hickness o he c oss-sec ion a e h = 0.833 m, b = 0.917 m, = 1/6 m, espec i ely. The leng h o he beam is L = 4 m and, i is made o a ma e ial wi h he modulus o elas ici y E = 2 x 105 MPa and he shea modulus G = 0.77 x 105 MPa. The conside ed hin-walled beam is subjec ed o uni o mly dis ibu ed wis ing momen equal o mx = 4.07 x 103 kNm/m. The second example was analyzed using in he p esen s udy by employing 80 axial elemen s and 64 elemen s in he c oss- sec ion. The geome ic cons an s o he c oss-sec ion de e mined om he p esen s udy a e as ollows 6 0.0059 S Cm=, 4 0.00407889 Im=. Fig. 8: Can ile e beam, he applied load, and he geome y o he channel c oss-sec ion, uni s [mm]. Shakou zadeh, H. e al. [7] used FEM based on Bensco e 's model o sol e his p oblem and compa ed i wi h he esul s ob ained om Vlaso 's heo y, which neglec s he shea de o ma ion e ec . Meanwhile, Sapoun zakis, E. J. e al. [20] used Analog Equa ion Me hod (AEM) and AEM wi h isogeome ic analysis wi h 3 B-spline ypes: i) Quad a ic B-spline, ii) Cubic B-spline), iii) Quad a ic B-spline o in es iga e his p oblem. Fig. 9: The a ia ion o wis angle along he x-axis. Fig. 10: The a ia ion o de i a i e o he wis angle along he x-axis. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 13 Fig. 9 and Fig.10 show he a ia ion o he angle o wis and he de i a i e o wis angle along he x-axis, espec i ely. The alue o he angle o wis and he de i a i e o wis angle a he ee end, and he wa ping momen a he ixed end a e p esen ed in Table 2 and compa ed wi h he ones ob ained om Shakou zadeh, H. e al [7], Sapoun zakis, E. J. e al. [20]. Fig. 11: The a ia ion o wa ping momen along he x-axis. Fig. 12: The a ia ion o he o sional momen along he x-axis. Tab.2: Compa ison o he wis angle, he de i a i e o wis angle, and wa ping momen be ween he me hods Me hods 𝜃 [ ad] a x = L 𝜃 󰆒 [ ad/m] a x = L Wa ping momen Mw [N.m2] a x = 0 P esen s udy -0.0429 -0.0114 -19.083 x 106 Sain - Venan model [7] -0.103 - - Vlaso model [7] -0.045 -0.012 -18.33 x 106 FEM-Bensco e model [7] -0.045 -0.011 -18.17 x 106 AEM (50NP) [20] -0.050 -0.009 -16.75 x 106 AEM(Quad a ic B- spline) [20] -0.039 -0.006 -13.48 x 106 AEM(Cubic B-spline) [20] -0.061 -0.013 -18.29 x 106 AEM(Qua a ic B- spline) [20] -0.046 -0.008 -15.5 x 106 Fig. 11 and Fig.12 depic he a ia ion o he wa ping momen and he o sional momen s along he axis, espec i ely. I can be seen om Table 2 ha he wis angle analyzed by he p esen s udy is in good ag eemen wi h he esul s o Vlaso model [7], FEM-Bensco e model [7], and AEM(Cubic B-spline) [20]. The dis ibu ion o he axial s ess 𝜎 and shea s ess 𝜏 inside he channel c oss-sec ion a he posi ion x = 0.5 m is shown in Fig. 13. The maximum no mal s ess 𝜎 = 657.865 MPa occu s on he ip o he lange. As expec ed, he dis ibu ion o he shea s ess 𝜏, wi h he maximum alue 205.981 MPa, dec eases om he ou e lange edge o he inne edge [21]. (a) (b) (c) Fig. 13: The no mal s ess 𝜎 and b) he shea s ess 𝜏 b) he shea s ess 𝜏 a x = 0.5 m om he ixed-end, uni s [MPa]. 5. Conclusion In his pape , a ini e elemen me hod is de eloped o sol e nonuni o m o sion wi hou he shea de o ma ion e ec o he p isma ic beam wi h an a bi a y c oss-sec ion wi h homogeneous iso opic ma e ial. Two examples we e pe o med o alida ing he accu acy o he p esen s udy. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 14 The compa ison o he esul s om he p esen s udy wi h he co esponding esul s in he li e a u e shows ha he p esen s udy can p edic he esponses o he non-uni o m o sion wi h a bi a y c oss-sec ion wi hou shea de o ma ion e ec accu a ely. Linea analysis o nonuni o m o sion conside ing shea de o ma ion would be one o he u u e esea ch opics. Acknowledgmen s The wo ks we e suppo ed by he S uden G an Compe i ion VSB-TUO. The egis a ion numbe o he p ojec is SP2021/77 “Nonuni o m o sion in p isma ic beams wi h a bi a y c oss-sec ions using FEM”. Re e ences [1] TIMOSHENKO, S. P and J. N. GOODIER. Theo y o elas ici y. New Yo k, USA: McG aw-Hill Book Company, 1951. [2] PILKEY, W. D. Analysis and design o elas ic beams: Compu a ional me hods. New Yo k, USA: John Wiley & Sons, 2002. ISBN: 0-471-38152-7. [3] TRAN, D. B., J. 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B- splines in he Analog Equa ion Me hod o he gene alized beam analysis including wa ping e ec s. Compu e s and S uc u es. 2017. Vol. 180, pp. 60–73. ISSN 00457949. DOI: 10.1016/j.comps uc.2016.03.007. [21] GENOESE, A. e al. A mixed beam model wi h non- uni o m wa pings de i ed om he Sain Venàn od. Compu e s & S uc u es. 2013. Vol. 121, p. 87–98. ISSN 0045-7949. DOI: 10.1016/j.comps uc.2013.03.017. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 15