Torsional shear stress with arbitrary cross-sections in homogeneous isotropic elastic material using finite element method

A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 409 TORSIONAL SHEAR STRESS WITH ARBITRARY CROSS- SECTIONS IN HOMOGENEOUS ISOTROPIC ELASTIC MATERIAL USING FINITE ELEMENT METHOD Dang-Bao T an1,2 1. VSB–Technical Uni e si y o Os a a, Facul y o Ci il Enginee ing, Depa men o S uc u es, Lud íka Podéš ě 1875/17, 708 00 Os a a, Czech Republic; dang.bao. an@ sb.cz 2. Thu Dau Mo Uni e si y, Facul y o A chi ec u e, Depa men o Ci il Enginee ing, T an Van On 06, 75000 Binh Duong P o ince, Vie nam;bao d@ dmu.edu. n ABSTRACT De e mining he shea s ess o a s uc u al elemen caused by o sion is a i al p oblem. The analy ical solu ion o he Sain -Venan o sion is only sui able o simple c oss-sec ions. The nume ical me hods o e alua e he shea s ess due o o sion o complica ed c oss-sec ions is indispensable. Many scien is s ha e s udied he o sion p oblem wi h a ious nume ical me hods. This pape aims o p esen an e icien ini e elemen me hod o assessing he shea s ess wi h a bi a y c oss-sec ions in homogeneous iso opic elas ic ma e ial due o o sion. MATLAB is he language o p og amming he nume ical me hod. The alida ion examples we e pe o med o show he eliabili y and e iciency o he au ho ’s nume ical me hod. KEYWORDS To sional shea s ess, Isopa ame ic eigh -noded quad ila e al elemen s, Sain -Venan o sion, Wa ping unc ion INTRODUCTION To sion in he s uc u e occu s due o asymme ical loads, ei he by geome ic dimensions o by in e connec ions be ween membe s. In many cases, o sion can be go e ning design ac o s. I is, he e o e, essen ial o accu a ely de e mine he shea s ess caused by o sion. Sain -Venan analysed he o sion p oblem using he semi-in e se me hod, assuming an unknown displacemen o sa is y he equilib ium equa ions and bounda y condi ions. P and l hen in oduced he s ess unc ion o he Sain -Venan o sion and he me hod o memb ane analogy [1]. Howe e , de e mining he shea s ess due o o sion by he analy ic solu ion is complica ed o a complex single connec ed domain o mul iply connec ed domain. A single connec ed domain is a domain whe e he c oss- sec ion is bounded by a closed. The mul iply connec ed domain is he domain whe e he c oss- sec ion is bo de ed by se e al closed. Nowadays, s uc u al elemen s such as beams and columns a e mo e and mo e complica ed in hei shapes. Hence he use o nume ical me hods o de e mine shea s ess due o o sion is indispensable. Va ious nume ical me hods o assess o sional shea s ess ha e been pe o med by many esea che s [2-22]. Ely, J. F., and Zienkiewicz, O. C. [2] i s sol ed Poisson’s equa ion o P and l’s s ess unc ion using he ini e di e ence me hod and in es iga ed he ec angula sec ion wi h and wi hou holes. He mann, L. R. [3] u ilized he ini e elemen me hod o calcula e he wa ping unc ion o he o sion o i egula sec ional shapes. Based on he Hellinge –Reissne p inciple, Xiao, Q. Z., e al. [4] de eloped a 4-node elemen wi h ou s ess pa ame e s o de e mine he shea A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 410 s ess o he polygonal sec ion. G u mann, F. e al. [5] used he ini e elemen me hod o e alua e shea s ess using he wa ping unc ion, which is mo e con enien han using P and l’s s ess unc ion when conside ing a mul iply connec ed domain. G u mann’s nume ical me hod has been implemen ed in o an enhanced e sion o he p og am FEAP [6], which used isopa ame ic ou - noded quad ila e al elemen s. Fialko, S.Y. e al. [7] de eloped a nume ical me hod using cons an iangula elemen s o sol e he Sain Venan p oblem o o sion, and o sionless bending o p isma ic ba s is ealized in he SCAD so wa e [8]. Jog, S. e al. [9] p esen ed a ini e-elemen o mula ion o Sain Venan o sion wi h a mul iply-connec ed domain in an aniso opic ma e ial. Recen ly, Behesh i, A. [10] de i ed a ini e elemen om s ain-g adien elas ici y o o sion o p isma ic ba s wi h minimal dimensions. Besides he ini e di e ence and ini e elemen me hod, many au ho s ha e applied o he nume ical me hods such as he bounda y elemen me hod (BEM) [11-16], line elemen -less me hod (LEM) [17-18], a null- ield in eg al echnique [19], and ini e- olume me hod [20-21] o analyse he o sion p oblem. Ka sikadelis, J. T. e al. [11] examined he o sion o gene al composi e ba s by BEM while Gaspa i, D. e al. [12] ackled o ho opic beams wi h a polygonal c oss-sec ion. Ba one, G. e al. [13] implemen ed he Complex Va iable Bounda y Elemen Me hod o examine he o sion p oblem in he single connec ed domain. Lee, J.W. e al. [14] ob ained a new BEM om he gene al Cauchy in eg al o mula de i ed om he Bo el–Pompeiu o mula o analyse he o sion p oblem. Pa adiso, M. e al. [15] achie ed an e icien me hod o de e mine he wa ping unc ion pa ame e wi h he gene al c oss-sec ion. Chen, K. H. e al. [16] in oduced a new e o es ima ion echnique in BEM o op imise he o sion p oblem wi h a mul iply-connec ed domain. Di Paola, M. e al. [17] p oposed LEM o deal wi h shea s ess in o sion p oblem wi h iso opic ma e ial and a bi a y c oss- sec ion. San o o, R. [18] handled he Sain Venan o sion p oblem o o ho opic beams wi h a gene al c oss-sec ion by LEM. Chen, J-T. e al. [19] in oduced he null- ield in eg al echnique o analyse he o sion p oblem o ci cula c oss-sec ions wi h ound holes. Chen, H. e al. [20-21] de eloped a new ini e- olume based on Bansal and Pinde a’s wo k o in es iga e Sain Venan ’s o sion p oblems o homogeneous and composi e p isma ic ba s wi h mul iply connec ed domain. In summa y, all o he wo ks show he easibili y and e ec i eness in academia. Cu en ly, o he au ho ’s knowledge, G u mann’s me hod [5] has been de eloped in o he FEAP p og am [6 ] o he Uni e si y o Cali o nia, Be keley, and Allplan B idge [22]. Fialko’s me hod [7] is simila o G u mann, de eloped as a module in he SCAD comme cial so wa e [8]. I means he me hods o G u mann and Fialko a e p ac ical. Howe e , due o he use o he ou -noded quad ila e al elemen in FEAP and he cons an s ain iangle elemen in SCAD, o achie e high accu acy, i is necessa y o mesh e y smoo hly, which a ec s he calcula ion speed. So, his esea ch aims o de elop a new nume ical me hod (NMB) based on he wo k o G u mann by using he isopa ame ic eigh -noded quad ila e al elemen . The alida ion examples we e pe o med o show he eliabili y and e iciency o NMB. FINITE ELEMENT METHOD PROCEDURE Figu e 1 shows he a bi a y c oss-sec ion o p isma ic beam in homogeneous iso opic elas ic ma e ial, he longi udinal axis is he x-axis, he c oss-sec ion deno ed  is in yz plane. The mul iply connec ed domain  is bounded by 1 2 1 , ,..., , nn     . S is he cen e o g a i y. On 1 2 1 , ,..., , nn     he igh -handed o hogonal basis sys em is de ined wi h angen ec o and ou wa d no mal ec o [ , ]T yz nnn . Wi h he o ien a ion o he associa ed coo dina e s is uniquely de ined. The displacemen ield x y z [u ,u ,u ]T u is exp essed as [1] T x u   , yx uz   , , zx uy   (1) A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 411 whe e x  : o sion angle, x d dx    , ( , ) Tyz  : wa ping unc ion o o sion. He e, he cons ain o he wa ping o o sion is 0. TdA     (2) Fig. 1 - C oss-sec ion o a p isma ic ba . The shea s esses a e gi en as T xy Gz y         , . T xz Gy z         (3) The pola second momen o a ea can be w i en as . TT T I y y z z dA zy                         (4) The go e ning di e en ial equa ion o he Sain -Venan o sion is exp essed as 22 22 0. TT in yz       (5) Meanwhile, he bounda y condi ion is gi en as ( 1,2,..., ), TT y z y z i n n n z n y on i n yz         (6) whe e y dz nds  , . z dy nds  (7) The go e ning di e en ial equa ion (5) is ans o med in o he weak o m by using he Gale kin’s me hod as below ( , ) ( ) 0, TT Tyz G dA n z n y ds y y z z                           (8) wi h es unc ion 1()H   . A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 412 The equa ion (8) is app oxima ed by using he ini e elemen me hod. G u mann’s me hod used isopa ame ic ou -noded quad ila e al elemen s. To imp o e he compu a ion speed, NMB used isopa ame ic eigh -noded quad ila e al elemen s. The [ , ]T yzx , he unknown unc ion T  and he es unc ion  a e in e pola ed in he local coo dina e sys em as ollows   8 1 , hii i N    xx ,   8 1 ( ) , T h T ii i N       ,   8 1 ( ) , , hii i N       (9) whe e h deno es he app oxima e solu ion o he ini e elemen me hod,   , i N  deno es he shape unc ion o he elemen . Figu e 2 shows he isopa ame ic eigh -noded quad ila e al elemen used in NMB. The shape unc ions o his elemen can be desc ibed as ollows [23, 24] 12 34 22 56 2 78 11 ( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )( 1 ), 44 11 ( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )( 1 ), 44 11 ( , ) (1 )(1 ), ( , ) (1 )(1 ), 22 11 ( , ) (1 )(1 ), ( , ) (1 )(1 22 NN NN NN NN x h x h x h x h x h x h x h x h x h x h x h x h x h x h x h x h x h x h x h x = - - - - - = + - - + - = + - - + + = - + - + + = - - = + - = - + = - - 2).h (10) Fig 2 -. Isopa ame ic eigh -noded quad ila e al elemen . Inse ing he de i a i es o () Th  and h  in o he equa ion (8) eads as 88 11 1 ( ) 0. numel e T e ij j i ij e KF      (11) He e, deno es he assembly ope a o wi h numel he o al numbe ini e elemen s. The s i ness pa e ij K o he nodes i and j and he igh hand e i F ead 11 11 11 , ( , ) ( , ) ( , ) ( , ) ( , ) . e j j j j ei i i i ij e Q Pjj eii ij p q p q p q p q p q p q pq N N N N N N N N K dA d d y y z z y y z z NN NN K w w y y z z                                                              J J (12) A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 413 11 11 11 , ( , ) ( , ) ( , ) . e ei i i i ie Q P eii i p q p q p q p q pq N N N N F z y dA z y d d y z y z NN F w w z y yz                                              J J (13) whe e J deno ed as Jacobian ma ix de ined as , yz yz Jxx hh éů ¶¶ ęú ęú ¶¶ ęú =ęú ¶¶ ęú ęú ¶¶ ëű (14) p w and q w a e he weigh s and p  and q  a e he in eg a ion poin s o he Gaussian in eg a ion echnique. NMB uses 3 x 3 Gauss quad a u e de i ed om he 1D case, whe e he quad a u e poin s a e loca ed a 3/5 , 0 and 3/ 5 , and he co esponding weigh s a e equal o 5/9, 8/9, and 5/9, espec i ely (see [23, 24]). The alue T i  o one a bi a y nodal poin i has o be alue 0. VALIDATION EXAMPLES The objec i e o his sec ion is o demons a e he eliabili y and e ec i eness o NMB. Fo his pu pose, ou alida ion examples de i ed om [1, 5, 9] we e examined, and hei esul s a e compa ed wi h hose analyzed by NMB implemen ed in MATLAB R2015a. A ba o squa e c oss-sec ion subjec ed o he o sion momen 1 T M kN.m wi h he leng h o he edge 2 m is analysed in he i s example. Jog, C.S. e al. [9] in es iga ed his p oblem using 16 isopa ame ic nine-noded quad ila e al elemen s. To achie e con e gence alues o he shea s ess and he pola second momen o a ea, he squa e c oss-sec ion is di ided in o 16, 64, 256 elemen s wi h uni o m mesh by NMB, espec i ely. Figu e 3 shows he disc e iza ion o he squa e c oss-sec ion wi h 16 eigh -noded quad ila e al elemen s. The poin s, A, B, C, D, co espond wi h he coo dina es (0, 1), (2, 1), (0, 0), (0, 2), espec i ely. The analy ical esul s o maximum shea s ess and pola second momen o a ea ob ained om [1, 9] a e max 0.592   kPa, 2.24923 T I m4, espec i ely. Table 1 shows he compa ison o he esul s o he maximum shea s ess, he pola second momen o a ea be ween he me hods. The esul s o NMB-16 elemen s and Jog, C.S. e al. [9] a e he same. When he squa e c oss-sec ion is e ined in o 256 elemen s, NMB is in good ag eemen wi h analy ical solu ions. Tab. 1 - The maximum shea s ess max  and he pola second momen o a ea T I o squa e c oss- sec ion Fac o s Analy ical solu ions Jog, C.S. e al. [9] NMB- 16 elemen s NMB- 256 elemen s () max a  [kPa] 0.592 0.6173 0.619174 0.601461 ()b T I [m4] 2.24923 2.2519 2.25187 2.24925 E o (a), (%) - 4.273 4.590 1.598 E o (b), (%) - 0.1187 0.1173 0.000 Figu e 4 depic s he dis ibu ion o shea s ess xz  o squa e c oss-sec ion, which dec eases he magni ude g adually om he midpoin o he edge o he cen e o g a i y along he y-axis [1]. A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 414 Figu e 5 and 6 p esen he a ia ion o he shea s ess xz  along he AB and CD segmen , espec i ely. Wi h 16 elemen s, NMB and Jog, C.S. e al. [9] canno cap u e p ecisely he shea s ess dis ibu ion along he CD segmen . When he mesh is ine enough, in his case, 256 elemen s, he esul s o NMB and heo y ha monize e y well. Fig. 3 - Disc e iza ion o he squa e c oss- sec ion wi h 16 elemen s. Fig. 4 - Dis ibu ed shea s esses xz  o squa e c oss-sec ion. Fig. 5 - The a ia ion o shea s ess xz  along he AB segmen . Fig. 6 - The a ia ion o shea s ess xz  along he CD segmen . The second example is an equila e al iangle subjec ed o he o sion momen 1 T M kN.m wi h he heigh o he iangle 0.2 m. The analy ical esul s o maximum shea s ess max  and pola second momen o a ea T I a e 1.62380 MPa and 6158.40 cm4, espec i ely [1]. The iangula c oss- sec ion was disc e ized 4, 8, 14, 37, 57, 109, 658 eigh -noded quad ila e al elemen s wi h non- uni o m mesh o ob ain he con e gence esul s. A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 415 Tab. 2 - The maximum shea s ess max  and he pola second momen o a ea T I o iangula c oss- sec ion Fac o s Analy ical solu ion NMB- 658 elemen s E o , (%) max  [MPa] 1.62380 1.6247 0.055 T I [cm4] 6158.40 6158.40 0.000 I is clea om Table 2 ha he esul s o he maximum shea s ess, he pola second momen o a ea ob ained om NMB a e in good ag eemen wi h he heo e ical solu ion [1]. Figu e 7 shows he iangula c oss-sec ion di ided by 14 eigh -noded quad ila e al elemen s. The poin s, E, F, co espond wi h he coo dina es (0, 0.2 3 ), (0.2, 0.2 3 ), espec i ely. Figu e 8 depic s he s ess dis ibu ion xy  o iangula c oss-sec ion, whe e magni ude alues do no exis along wi h segmen EF [1]. Figu e 9 p esen s he s ess dis ibu ion xz  o iangula c oss-sec ion. I can obse e om Figu e 10 ha he esul o he a ia ion o shea s ess xz  along he EF segmen s o NMB and analy ical me hod is e y well ma ched. Fig. 7 - Disc e iza ion o he iangula c oss- sec ion wi h 14 elemen s. Fig. 8 - Dis ibu ed shea s esses xy  o iangula c oss-sec ion. Fig. 9 - Dis ibu ed shea s esses xz  o iangula c oss-sec ion. Fig. 10 - The a ia ion o shea s ess xz  along he EF segmen wi h NMB-109 elemen . A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 416 The hi d example conce ns a ba o squa e c oss-sec ion wi h hole applied o he o sion momen 1 T M kN.m. Figu e 11 shows he geome ical dimensions and he meshing o 96 isopa ame ic eigh -noded quad ila e al elemen s by NMB o he squa e c oss-sec ion wi h hole. The poin s, G, H, I, J, co espond wi h he coo dina es (0, 1), (1.6, 1), (0, 0), (0, 2), espec i ely. Jog, C.S. e al. [9] analysed his p oblem using 96, 1536, 6144 isopa ame ic nine-noded quad ila e al elemen s, espec i ely. Figu e 12 depic s he a ia ion o pola second momen o a ea co esponding o 96, 1536, 6144 disc e iza ion elemen s in es iga ed by NMB. In Table 3, he con e gence o he pola second momen o a ea alue by NMB is p esen ed as a compa ison wi h ha ob ained om Jog, C.S. e al. [9]. Figu e 13 and 14 p esen he s ess dis ibu ion xy  , xz  o squa e c oss-sec ion wi h hole, espec i ely. Figu e 15 and 16 plo he a ia ion o shea s ess xz  along he segmen s, GH, IJ, espec i ely, e alua ed wi h he wo me hods, Jog, C.S. e al. [9], and NMB. F om he igu es and able men ioned abo e, he eliabili y o he NMB is once mo e e i ied. Fig. 11 - The dimension and he disc e iza ion o 96 elemen s o he squa e c oss-sec ion wi h hole. Fig. 12 - The a ia ion o pola second momen o a ea co esponding o disc e iza ion elemen s. Tab. 3 - The pola second momen o a ea o squa e c oss-sec ion wi h hole Fac o s Jog, C.S. e al.- 6144 elemen s [9] NMB- 6144 elemen s E o , (%) T I [m4] 1.707 1.70718 0.01054 A icle no. 30 THE CIVIL ENGINEERING JOURNAL 2-2021 DOI 10.14311/CEJ.2021.02.0030 417 Fig. 13 - Dis ibu ed shea s esses xy  o he squa e c oss-sec ion wi h hole. Fig. 14 - Dis ibu ed shea s esses xz  o he squa e c oss-sec ion wi h hole. Fig. 15 - The a ia ion o shea s ess xz  along he GH segmen . Fig. 16 - The a ia ion o shea s ess xz  along he IJ segmen . Acco ding o Figu e 17, he b idge c oss-sec ion is an example o mul iply connec ed domains was conside ed he ou h example. Wi h his example, he pola second momen o a ea o he b idge c oss-sec ions neglec ed he can ile e pa ob ained by he analy ical solu ion is 40.0 m4 [5]. A compa ison wi h a heo e ical explana ion is no possible. The compa ison be ween NMB and FEAP [5,6], implemen ed G u mann’s nume ical me hod by using isopa ame ic ou -noded quad ila e al elemen , was pe o med on he con e gence speed. The b idge c oss-sec ion was di ided in o 50000 nodes wi h a uni o m mesh by FEAP o ob ain he con e gence o he esul s. In NMB, he b idge c oss-sec ion is meshed wi h 80, 406, 1892, 4634, 11414 nodes o in es iga e he con e gence esul . Figu e 18 shows he b idge c oss-sec ion di ided in o 80 nodes by NMB. Figu e 19 depic s he con e gence o he pola second momen o a ea ob ained by NMB wi h 4634 nodes.