Negative Shear Lag Effect in Continuous Double T Beam by FEM

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 1 NEGATIVE SHEAR LAG EFFECT IN CONTINUOUS DOUBLE T BEAM BY 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-2021-0005 Abs ac . This pape p esen s he use o a ini e elemen me hod (FEM) o analyze he nega i e shea lag e ec on p es essed con inuous double T beam. Howe e , he nume ical me hod can be applied o i) any c oss-sec ion, ii). he mos common ypes o suppo s, such as ixed, pinned, olle , iii) and any applied load, concen a ed, o dis ibu ed passed h ough he shea cen e o he c oss- sec ion. The cha ac e is ics 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 a linea isopa ame ic elemen , is de eloped o compu e he de lec ion, o a ion angle, bending wa ping pa ame e , 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. Keywo ds Shea lag, es ained wa ping, p es essing, load, ini e elemen me hod 1. In oduc ion Acco ding o he elemen a y beam heo y, when he beam elemen is unde load, he longi udinal no mal s esses a e assumed o be p opo ional o he dis ance om he neu al axis. Howe e , in p ac ice, hese s esses a e nonuni o mly dis ibu ed o e he wid h o he c oss-sec ion. This phenomenon is called shea lag. Many au ho s ha e esea ched he shea lag phenomenon. Reissne , E. [1] es ablished a displacemen ield along he axis o he beam, aking in o accoun he e ec o shea lag, which is exp essed by a pa abolic unc ion, and used he p inciple o minimum po en ial ene gy o ob ain he lange s ess a he c oss-sec ion. Du ing he pas se e al yea s, many au ho s [2-15] based on hinned wall beam heo y [16] imp o ed a spanwise displacemen [1], which conside s he wa ping o langes, o analyze shea lag due o lexu e. Mos s udies ocus on concen a ed and uni o mly dis ibu ed loads. The e ec o p es essed load on shea lag has only been pe o med in a ew s udies [7, 11]. Chang, S. T. [7] de i ed an analy ical me hod om [2] and [1] o calcula e he shea lag o simply suppo ed p es essed conc e e. Zhou, S. J. [11] p oposed a new FEM o analyze he shea lag in p es essed conc e e box gi de s. The p ac ical design o he beam s uc u e equi es as e and easie pa ame e adjus men han he 3D modelling using shell o solid. Some au ho s [17-21] ha e ans o med he 3D analysis o he shea lag phenomenon in o sepa a ed 2D c oss-sec ional and 1D modelling. El Fa mi, R. [17, 18] de i ed he nonuni o m shea and o sional wa ping o a bi a y homogeneous c oss-sec ion om he Sain Venan beam p oblem ex ension. Le Co ec, V. and Filippou, F. C. [19] de ined he axial displacemen due o wa ping by in e pola ion a wa ping deg ees o eedom numbe on he c oss-sec ion o es ablish FEM o mula ion o shea o sional wa ping in he elas ic and elas oplas ic analysis o beams. Fe adi, M.K. e al. [20] ob ained FEM ha accu a ely cap u es no mal s ess due o es ained wa ping by ep oducing c oss-sec ional wa ping as a linea combina ion o wa ping modes. Dika os, I. C. and Sapoun zakis, E. J. [21] p oposed a heo y o de e mine nonuni o m wa ping due o lexu e o composi e beam wi h a bi a y c oss-sec ion using he Bounda y Elemen Me hod (BEM). The pu pose o his pape is o in es iga e he nega i e shea lag e ec due o p es essing load in con inuous double T beam by es ablishing a nume ical me hod using FEM, based on displacemen and s ain ields de i ed om Dika os, I. C. and Sapoun zakis, E. J. [21]. A 2D FEM based on he Gale kin app oach is ob ained om compu ing he wa ping unc ion o he co esponding ellip ic di e en ial equa ions. The o he kinema ical SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 2 a iables o he beam a e calcula ed om he p inciple o i ual wo k by de eloping a 1D FEM. 2. Fo mula ion he p oblem 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 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. The coo dina e sys em is Sxyz h ough he shea cen e o he c oss-sec ion S. CXYZ is he pa allel sys em wi h Sxyz h ough he cen e o g a i y C. Fig. 1: C oss-sec ion o a p isma ic beam. 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 exposed o he a bi a y dis ibu ed o concen a ed loads, ans e se loading pz(x) along he z-di ec ion, bending momen mY(x), and wa ping momen () P CY mx ϕ along wi h he Y di ec ion. The c oss- sec ion is assumed wi h no dis o ion. The geome ic cons an s o he beam a e de ined as 2 2 , , , , PP CY CY YY ZZ PP CY CY Ad IZd IYd I d ϕϕ ϕϕ Ω Ω Ω Ω =Ω =Ω =Ω =Ω     (1) whe e A is he a ea o c oss-sec ion, , YY I Z Z I a e he second momen s o a ea wi h espec o Z, Y-axis, espec i ely, P P CY CY I ϕϕ is he wa ping cons an . P CY ϕ de ines he shea wa ping unc ion wi h espec o cen e o g a i y C, which is ob ained om , PP CY CY Z ϕφ =− (2) whe e (,) P CY yz is de e mined om he ellip ical di e en ial equa ions 2in 0on , P PZ CY YY PP CY CY yz n AZ I nn yz φ φφ  ∇=− Ω    ∂∂ +=Γ ∂∂ (3) whe e P ZZ A kA= is de ined as he p ima y shea a ea, kZ is he shea co ec ion ac o ob ained om he de ini ion gi en in [22] In addi ion, he e alua ed wa ping unc ion P CY φ om (3) con ains an in eg a ion cs, which can be ob ained om G u mann, F. [23] 1. sP CY cdA A φ Ω =−  (4) The displacemen ield is exp essed as [21] (, ,) (, ,) (, ,) () () (,), (, ,) 0, (, ,) (), PS P YYCY uxyz u xyz u xyz x Zxyz xyz wxyz wx θηϕ =+ =+ = = (5) whe e ,,u w a e he o al displacemen s co esponding wi h he axis x, y, z ,(,,) PS uuxyz a e he p ima y and seconda y axial displacemen , espec i ely, () Y x θ is he angle o o a ion o he sec ion abou he Y axis, Y η is he bending wa ping pa ame e . The s ess ield ob ained om he heo y o elas ici y as [21] () ()() ()() ,, ,, , ,, , , , , PS xx xx PS xy xy PS xz xz P xx xx Y x Y x CY PP SP xy xy z y CY y z CY y PP SP xz xz z z CY z z CY z EEZE GGZ G GGZ G σσ ττ ττ σεθ ηϕ τγγ ϕ γϕ τγγ ϕ γϕ == + == + + == + + 1424314243 1444244431442 443 144424443 14243 (6) whe e , P Z xY w γ θ =+ , , S Z YxY w γ ηθ =− − a e de ined as he p ima y and seconda y shea s ains, espec i ely. I is emphasized ha he s ess componen x x σ is composed o i) he classic no mal s ess, P x x σ , de e mined om he enginee ing beam heo y and ii) he wa ping no mal s ess, S x x σ , caused by wa ping o he c oss-sec ion is he p ima y eason o he shea lag phenomenon. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 3 The bending momen , he wa ping momen , he p ima y shea o ce, he seconda y shea o ce a e deno ed , Y M , P CY M ϕ , P z Q S z Q, espec i ely ob ained as [21] ()() () () ()() ()() , , ,, 22 ,, ,, 22 ,, , , , . PPP CY CY CY Yxx YYYx P xx CY Y x PPP ZxyCYyxzCYz P ZCYyCYz SPP ZxyCYyxzCYz SP P ZCYy CYz MZdEI MdEI Qd G Qd Gd ϕϕϕ σθ σϕ η τφ τφ γφ φ τϕ τϕ γϕ ϕ Ω Ω Ω Ω Ω Ω =Ω= =Ω=  =+Ω   =+    =+Ω   =− + Ω         (7) 3. FEM p ocedu es 3.1. De lec ion wx, o a ion angle 𝜽𝒀 and bending wa ping pa ame e 𝜼𝒀 The p inciple o i ual wo k igno ing olume ic o ces is used o es ablish he s i ness ma ix o he 1D beam elemen . Assume he beam elemen includes wo end nodes, 1, 2, The symbol (.) δ deno es he i ual quan i ies. The in e nal i ual wo k is () , ixxxxxyxyxzxz V WdV σδε τδγ τδγ =++  (8) whe e V is he olume o a p isma ic beam. Subs i u ion he Eq. (6) and Eq. (7) o Eq. (8), he in e nal i ual wo k can be ew i en as ()() () ,, ,, 00 PP CY CY LL iYY YxYx YxYx WEI dxEI dx ϕϕ δθ θ δη η =+ +  ()() 0 L PPP ZZZ GA dx δγ γ +  ()() 0 () L PSS ZZZ GA A dx δγ γ − (9) The ex e nal i ual wo k is () 0() P CY L ezxYY Y Span Wpxwmmdx ϕ δδθδη =+++  14444444244444443 ,1 () xyz end node u wd δδδ Ω++ Ω  14444244443 + ,2 (), xyz end node u wd δδδ Ω++ Ω  14444244443 (10) whe e x, y, z a e he componen s o ac ion ec o applied on he la e al su ace o he beam which is ela ed o he end nodes- ex e nal loads as [21] () ˆˆ () , () , ˆ, P CY zzYx P xCY px d mx Zd m d ϕ ϕ ΩΩ Ω =Ω = Ω =Ω   (11) Using he exp ession Eq. (11), he Eq. (10) can be ep esen ed as e W=000 () P Cy LLL zx YY Y pxwdx m dx m dx ϕ δδθδη ++ +  222 111 ˆˆˆ . P CY zi xi Yi Yi Yi i iii pw m m ϕ δδθδη === ++  (12) To de i e he elemen s i ness ma ix, he a iables wx, 𝜃, and 𝜂 need o be in e pola ed wi hin each elemen . wx, 𝜃 and 𝜂 a e independen a iables. As a esul , any kind o 𝐻 shape unc ion can be used o he p esen beam. We use 1D linea isopa ame ic shape unc ion o bo h a iables [24, 25]. Tha is, [] [] [] 1 12 2 1 12 2 1 12 2 () () , () () , () () , x x x Y Y Y Y Y Y w wN N w NN NN ξξ θ θξξ θ η ηξξ η  =   =   =  (13) whe e 1, x w 2, x w 1, Y θ 2, Y θ Y1, η Y2 η a e he nodal displacemen s, o a ion angle, bending wa ping pa ame e a he beam end nodes (1) and (2), espec i ely () 1 11, 2 N ξ =− () 2 11. 2 N ξ =+ (14) The co esponding elemen nodal deg ees o eedom is 11Y1 2 2Y2 {,,,,,}. T xY x Y ww θη θη =d (15) Subs i u ing Eq. (13-14) o Eq. (9) and Eq. (12) leads o he elemen s i ness ma ix and elemen o ce ec o as 1234 , eeeee =+++KKKKK (16) 2 123 ()()() , zY P CY e eee e e ip im im ϕ =+++ + +F FFFF F F (17) whe e 1 e=K1 11 1 T YY E IJd ξ − BB () () 11 , T YY i i i i E Iw J ξξ =BB (18) () () 1 222 1 22 , PP CY CY PP CY CY eT T ii i i EI Jd EI w J ϕϕ ϕϕ ξ ξξ − = =   KBB BB (19) () () 1 333 1 33 , ePT Z PT Zi i i i GA Jd GA w J ξ ξξ − = =   KBB BB (20) SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 4 () () 1 444 1 44 () () , ePT Z PT Zii i i GA A Jd GA A w J ξ ξξ − =− =−   KBB BB (21) 1 11 1 eT Z pJd ξ − = FN () 1, T iiZ i wpJ ξ =N (22) 1 22 1 eT Y mJd ξ − = FN () 2, T iiY i wmJ ξ =N (23) 1 33 1P CY eT mJd ϕ ξ − = FN () 3, P CY T ii i wmJ ϕ ξ =N (24) () , z e ip F() , Y e im F() P CY e im ϕ F a e he ans e se o ces, bending momen s, wa ping momen s a he end nodes i, espec i ely, , 2 L J= (25) [] [] [] 11, 2, 21,2, 31,1 2,2 41,112,22 11 2 21 2 312 0000, 00 00 , 00, , 00 00, 0000, 00 00 , xx xx xx xx NN NN NN NN NNNNNN NN NN NN  =  =  =  =− − −  = = = B B B B N N N (26) , i wi ξ a e he weigh s and he coo dina e o in eg a ion poin s o he Gaussian in eg a ion echnique. In he p esen s udy, o a oid he shea locking, we use one-poin Gauss quad a u e (2, i w=0) i ξ =. Assembling he elemen s i ness ma ix and load ec o s in he sys em ma ix equa ion gi en below ..=Kd F (27) 3.2. Wa ping unc ion, P CY φ , P CY ϕ 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 ion (3) is ans o med o weak o m as 0. PP P CY CY Z YY A dd yy zz I φφ ηη η ΩΩ   ∂∂ ∂∂ +Ω−Ω=     ∂∂ ∂∂    (28) The wa ping unc ion alue, P CY φ , in Eq. (28) is app oxima ed by he FEM, which is p esen ed in [22]. Finally, he alue P CY ϕ is calcula ed om Eq. (2). 4. Valida ion example In his sec ion, a compu e code is de eloped in he MATLAB R2015 based on he o mula ions desc ibed in he p e ious sec ions. A p es essed con inuous beam wi h a double T c oss-sec ion was analyzed using his code. The beam's geome ical cha ac e is ics, he bounda y and loading condi ions a e shown in Fig. 2. Poin s B1, B2, and B3, a e on he uppe pla e and ha e he y coo dina es -1.55 m, -0.825 m, and 0, espec i ely. The s aigh endons a e posi ioned uni o mly a he cen e o he slab. On he plan, endons a e a anged a he in e nal suppo wi h a o al leng h o 2.8 m. Assume he e a e six endons, each o which is s essed wi h a o ce o 140 kN. Modulus o elas ici y E= 30 x 103 MPa, and he Poisson a io 𝜈= 0.2. Fig. 2: Con inuous beam, he load, and he geome y o he c oss-sec ion, uni s [mm]. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 5 Fig. 3: The dis ibu ion o he classic no mal s ess in c oss-sec ion a x=14 m, uni s [kPa]. Fig. 4: The dis ibu ion o he wa ping no mal s ess in c oss-sec ion a x=14 m, uni s [kPa]. Fig. 5: The dis ibu ion o he o al no mal s ess 𝜎 in c oss-sec ion a x=14 m, uni s [kPa]. This example was analyzed by employing 840 axial elemen s and 126 elemen s in he c oss-sec ion. The geome ic cons an o he c oss-sec ion de e mined om he p esen s udy a e as A= 0.287 m2, Z k= 0.2606, YY I= 0.00977 m4, P P CY CY I ϕϕ = 0.00248 m4. The p es essed load applied o he beam is ans o med in o wo equi alen loads, Px= 840 kN, and My= 115.9 kNm, shown in Fig 2. ETABS 2018 [26] is used o simula e 3D model included beam, shell, and endon elemen s. The shell and beam elemen s a e di ided in o 9856, 700 elemen s, espec i ely. Fig. 3, Fig. 4, Fig.5 show he con ou plo o he classic s ess, he wa ping no mal s ess, and he o al no mal s ess 𝜎 on he c oss-sec ion p edic ed by he p esen s udy a he posi ions x= 14m, espec i ely. In Fig. 6, he a ia ion o he o al no mal s ess 𝜎 along he wid h o he uppe lange o he c oss-sec ion a he posi ion x= 14 m is shown and compa ed be ween he esul om i) Enginee ing beam heo y, ii) 3D simula ion ETABS 2018. In Table 1, he no mal s ess x x σ in he poin s B1, B2, and B3 p edic ed by he p esen s udy a e gi en and compa ed wi h he esul s om 3D simula ion. F om he abo e Figu es and Tables, he in luence o he shea lag phenomenon is appa en , and he p esen s udy's accu acy can be e i ied. Fig. 6: The a ia ion o he o al no mal s ess 𝜎 along he wid h in he uppe lange a in e nal suppo . Tab.1: Compa ison o he no mal s ess 𝜎 [MPa] a he poin s B1, B2, B3 Me hods Poin B1 Poin B2 Poin B3 P esen s udy 3.9924 3.9487 4.0066 ETABS 2018 [26] 3.96 3.93 4.025 E o (%) 0.818 0.475 0.457 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 6 5. Conclusions In his pape , FEM is de eloped o analyze he nega i e shea lag e ec due o p es essing in con inuous double T beam. Howe e , he nume ical me hod can be used o a bi a y c oss-sec ions wi h mos bounda y condi ions and load ypes. A h ee-span con inuous beam subjec ed o he p es essing load a he in e nal suppo was analyzed and compa ed wi h he esul s om 3D simula ion so wa e. I was obse ed ha he p esen s udy could p edic he nega i e shea lag phenomenon accu a ely due o p es essing load. Acknowledgemen 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] REISSNER, E., 1946. Analysis o shea lag in box beams by he p inciple o minimum po en ial ene gy. Qua e ly o Applied Ma hema ics. 1946. Vol. 4, no. 3, pp. 268–278. ISSN 0033-569X. DOI: 10.1090/qam/17176. [2] CHANG, S. T. and F. Z. ZHENG, 1987. Nega i e Shea Lag in Can ile e Box Gi de wi h Cons an Dep h. Jou nal o S uc u al Enginee ing. 1987. Vol. 113, no. 1, pp. 20–35. ISSN 0733-9445. DOI: 10.1061/(asce)0733- 9445(1987)113:1(20). [3] LUO, Q. Z., Q. S. LI and J. TANG, 2002. Shea lag in box gi de b idges. 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