Print in ret Beas, ertamon Pes le A VLASOV BEAM ELEMENT EDUARDO N. DvoRKIN, Disco CeLENTAl Instituto de Materiales y Estructuras, Facult (NO, ALBERTO CurTINo and Gustavo G101A iad de Ingenieria, Universidad de Buenos Aires, Buenos Aires, Argentina (Received Abstract—A finite element for the analysis of clement is based on Viasov’s beam theory. In rot require the use of numerical integration and ‘nonuniform torsion, 1. INTRODUCTION When formulating the mathematical model_to analyze the stress state ofa solid, the analyst may find it convenient to introduce several assumptions ac- cording to the geometrical and loading characteristics of the solid under analysis. These assumptions lead to one of the following mathematical models: (a) three-dimensional continuum; (b) plate or shell; (©) solid section beam; (4) thin-walled open section beam. The theory used to analyze thin-walled open section beams was developed by Viasov[l] and many researchers have investigated its numerical implementation in linear and nonlinear finite element analysis{2-9, 16), In this paper we presenY/a finite element implemen- tation of Viasov's theory. The most relevant aspects of the Vlasov beam elefnent that we have developed are: + it does not require the use of numerical inte- gration; + only one element can exactly model simple cases of nonuniform torsion. The numerical results that we present in Sec. 3 demonstrate the effectiveness of the element in linear analyses of straight and curved spatial beams struc- tures. ‘We believe that the present formulation is a robust basis for the future development of a nonlinear Vlasov beam element. In [16] Krajcinovic presented a formulation very close to the one we present here. 2. OUR VLASOV BEAM ELEMENT Based on Viasov’s beam theory, we developed a two node beam element. 8 July 1988) led open section beam structures is presented. The analyses the element is very effective since it does only one element can exactly model simple cases of In the local coordinate system (3, f, 4) of Fig. | the cross-section principal planes are (%,2) and (/, 2), and the nodal degrees of freedom for nodes 1 and 2 (superindex *k’), are W, 6 and Ww, the displacements of the cross- section centroid; 64 and 64, the local rotations around the cross- section centroid (6% and 65 |, because Viasov’s theory neglects shear deformations); 64, the local rotation around the cross-section shear center (torsional rotation); and 6;|,, the derivative of the torsional rotation with respect to the beam axial coordinate. Hence our element has seven d.o.f. per node. For the axial displacement, vi, we use a linear interpolation and for the transversal displacements, @ and 6, we use the classical hermitian interp- olationf10}. For the torsional angle a polynomial interpolation hhas been used by most authors. However, in order to obtain a more powerful element, we use the follow- ing interpolation based on the analytical solution to some simple nonuniform torsion problems (16): G.= Aye + Aye + A + Ay (ta) (Gia inl (ab) where E is the material Young modulus, G is the ‘material shear modulus, /d is the moment of inertia, for pure torsion as calculated for a thin-walled folded cross-sectionfI] and Iw ig the sectorial moment of inertia of the cross-section} The most relevant features of the interpolation presented in eqns (1) are that it can represent: + rigid body rotations ‘ the exact solution to some simple torsional prob- lems{1}. 187188, EDUARDO N. DVORKIN ef al. ‘A nonconventional feature of the proposed tor My = le" + AL) + MAL — NYA sional interpolation is the use of material and cross- section properties in it Q= “ya The four constants in eqn (1a) can be written in terms of the nodal d.o.f. by solving: Dy =(-2AL +e —e MYA (3b) roo oI fA 6 and A -2 10] | a, 64, , , eo ee cillal~|é@ Aa ib + e#QL —2)—e MAL +2), Be) det de® 1 OJ | ae Gils For any point on the beam element axis, with an (2) axial coordinate 2, ‘The solution to eqn (2) is: (2) = AGE)O. @ ‘Ay 2); My Oe] FO In the above, A D2, OO, 2, sh oa | _ | % M22 Os Om h Wa eo 6 41] 0, Oyo ou! | a | oO ‘ 4 Ae My My M% Oy} [BI 06307030 0,003 W200 026%) where and 2, = 401 eya bed : ‘ A= " (a) We + ALA eome oo Om —l+e “ya : iat where hy = (2G /LY — 3G /LP +1: 2/L? = 227) + ~22/L) + HE/LP PL? — 2/1] hy = [22 /LY — 32/LYP + 1: —25/L? + 28/L — 2: —22/L) + 3 /LP: -2/L? + FL] hy=[-2/L + 12/2] he Ley +e ADs) + 205, +.Oy FeO +e + 20 +. Ds + Day + 205 + Quy! #4 + 0 HM + 104+ QJ. (5b) Qe (-1 FAL +e-*YIA My, =A = eV 2, Dy = Ae — 1A [-1 +e Anya y= (14 AL eid Dy, = He — eA y= AQ —e*— eV/A Dy = He — eA y= 42-0 — ety 24 AID + eM(AL 1) e+ ALVA When the beam element is under the action of external loads. its equilibrium configuration is, defined by the Principle of Minimum Potential Energy10, 11], 5(u—v)=0. © where w is the elastic energy stored in the element and v is the potential of the conservative external loads. In the case of a Vlasov beam, the elastic energyf! 1} is = 5 [BAGH + El6") + Ella’ + Gld(6;' + Elw(62?)\dz (1)‘A Vlasov beam element 189 Fig. |. Vlasov beam element. where Ji, and J, are the section principal moments of where inertia. In (7), the first term on the r.h.s. is the extensional strain energy, the second and third terms represent Fly 0 0 0 0 bending strain energie, the fourth term is the strain 0 Ely 0 0 0 energy associated to St. Venant’s torsion andthe fifth term is the strain energy association to nonuniform ec re et) torsion] 0 0 0 Guo From eqn (4) we obtain 0 0 0 0 Ew i Le 0 where 4 =area of the cross-section. oh 0 oO ‘The external loads acting on the beam element =/0 0 mh 0} 0=80. () are 000m ooo distributed loads per unit length distributed moments per unit length Replacing the above in eqn (7), we obtain b distributed bimoment{1) per unit length Bi, Pi, Py concentrated loads 50-60) [i renee jo ) MoMj,¥1, concentrated moments ; B concentrated bimomentst}190 EDUARDO N. DVORKIN et al. Hence, the potential energy of the external loads{11] is [i [awa sae - +mb+ oo; Jae +E [Fu + Pi, + Pla! — AT + sa +, +a] ay From eqn (4), 5a] [hy 000 68 ob 0 0 oon 0 =| 0 bh 0 0 | 60=660. (12) h 0 0 0 56. 00 0 hk 60! 00 0 hy Hence, bu =o0"{['Drpoe + EOP where B= Be. By f. Sting tise, B] Pra (Pi, Py, B and we define Replacing eqns (9) and (13) in (6) we obtain RO=R (14) where R [ference (1) In eqn (14), R is the stiffness matrix of the Vlasov ‘beam element in local coordinates, and R is the vector of nodal loads equivalent to the acting external loads, also in local coordinates. It is very important to notice that the integrals in eqns (13) and (15) can be evaluated in closed form and therefore we do not need to perform any numer- ical integration. In order to be able to assemble a structural model, we transform eqn (14) to the system of global coordi- nates (x, J R=TR (16a) u=T0 (16b) K=TRT" (16) where U is the vector of the global d.o.f., U=[U'V'W7'010}0'0;), 0272702030784) It is important to notice that we assume 6; to be unique for all elements sharing a node. This assump- tion implies a ‘warping continuity’ and has been introduced by BaZant{3, 12) The transformation matrix T is easily formed taking into account that for any vector X x) fx Xx, |=S| x, (7a) cl Le where (a7) The ¢, are the base vectors of the global coordinate system and the @, are the base vectors of the local coordinate system. 3. NUMERICAL EXAMPLES, In order to investigate the performance of the Vlasov beam element in linear analyses we solved: (a) simple models of one and two elements; (b) spatial beam structures. 3.1. One- and two-element models The interpolation function given in eqn (1) was constructed in order to be able to exactly represent simple cases of nonuniform torsion with only one element, To demonstrate this capability we modeled, using one element, the cases shown in Fig. 2(a) and, using two elements, the cases shown in Fig. 2(b) For all those cases the finite clement results (rota~ tions, torsional moments, St. Venant torsional moments and bimoments) were identical to the ana- lytical results presented by Vlasov in [1] 3.2. Spatial beam structures 3.2.1. Curved beam. In Fig. 3(a) we show a curved I-beam under six different loading and boundary conditions, For the first three cases we solved using, for each of them, four, eight and 12 equal elements. We‘A Viasov beam element vt \ 2; —— to ‘ \ 2 2 cme 8.Cot nde 2 | See : 3-0 2 coe | acorn! cot maee 3 1 went 4 2 ano" Fig. 2. One- and two-element models. (a) One-element models; (b) two-clement models, ‘can see in Fig. 3(b,c, d) the convergence of the finite element solution towards the analytical solution given by Heins and Spates{13, 14] In Fig. 3(¢,f,g) we compare, for the last three cases, our finite element solution obtained using 12 equal elements against the analytical solution. In the drawings we call ‘warping moment” the difference between the values of the total torsional ‘moment in a section and the value of the St. Venant torsional moment in it, 3.2.2. Tapered beam. In Fig. 4(a) we show a tapered beam in torsion, with its left end fixed (@ =30000K1eS Gzi2000 1S ‘Section IWF ‘eam Length (L ‘Mig rads =240" in torsion and warping. The analytical solution to this problem was obtained by Kitipornchai and ‘Trahair{15). In Fig. 4(b) we compare our finite ele- ment solution obtained using four equal elements against their analytical solution. 3.2.3. Curved beam with variable cross-section. In Fig. 5(a) we show a curved I-beam with variable cross-section, and in Fig. 5(b) we compare our solu- tion, obtained using 16 unequal elements, against the finite difference solution presented by Heins(13]. In points where the solution is discontinuous we indicate average values, ES case | Trsent Loosing | outt tone Loang 1 p= [Fe lap 2 aT iP PH [oF 3 p14 |p—*# 4 | ===> | == 5 P om om 20 Ton peOT (P) "wWRDG-] PALIND “¢ “Ha @e Ba We Ba 52 oz le vs q 59, Js (arp voisaen 2% (a9 vors304 ]? ° (uyuossio, ‘uii0n yuu AS bud.044 vs eee, suewqez! o \y (PHED yeonhvouy — J syousp zi ° 6) av (up uonoayeq ‘und 40 190,‘A Vlasov beam element 19s [lacen 75rmn| 4g Poesémm = 2 dalemente = tritici) rod) Torsional ange ( sv Fig. 4. Tapered beam, § 3 GEES 's Be Bee ay 20178 20%158"__20°x78" £ se 60 WEB i 22°x118 22x88 22°x158" é Feb S © IG elements Sa ee ee es 2 AF = Fite diferences (3) T=08K-in/in = a | é ae e Fig. S(a and 6). Curved beam with variable cross-section.196 [EDUARDO N. DVORKIN ef al. 4. CONCLUSIONS AA finite element based on Vlasov’s theory of thin- walled open section beams was presented. The main difference between the present element and most of the existing Vlasov beam elements lies in the torsional interpolation [sce eqn (1)}, which was designed so as to be able to represent (16}: ‘rigid body rotations + the exact solution to simple cases of nonuniform torsion using only one element, ‘The element was implemented for linear analysis and some important features are that ‘© the use of numerical integration is not required ‘ the clement proved to be very effective in the analyses of straight and curved beam structures The torsional interpolation that we present in this paper for linear analysis can also be used in nonlinear analysis, and we are presently working towards the development of a geometrically nonlinear Vlasov beam element based on it REFERENCES, LV. Z. Vlasov, Thin-Walled Elastic Beams. 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