A THIN-WALLED COMPOSITE BEAM ELEMENT FOR OPEN AND CLOSED SECTIONS

Transcription

1 6 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A THIN-WALLED COMPOSITE BEAM ELEMENT FOR OPEN AND CLOSED SECTIONS A. H. Sheikh, O. T. Thomen Department of Mechanical Engineering, Aalborg Univerit, Denmark Keword: Thin-walled beam, Laminated compoite, Warping, Shear deformation, Croectional rigiditie, Finite element anali, Bending-torion-etenion-hear coupling Abtract An efficient beam element for the anali of thin-walled laminated compoite beam of open/cloed ection i propoed. The croectional tiffne of the beam i derived analticall where all poible coupling between torion, bending and aial deformation are conidered. In the finite element approimation of the beam, the torional deformation require a C formulation due to incorporation of warping deformation while the bending deformation require a C formulation due to the incorporation of hear deformation. The difficult of implementation of both thee formulation in the preent coupled problem i uccefull overcome b adopting an efficient approach for the finite element approimation of the bending deformation. The element i ued for the numerical eample of open ection I beam and cloed ection bo beam problem having different boundar and loading condition. The obtained reult are compared with analtical/numerical and eperimental reult available in literature to demontrate the performance of the propoed element. Introduction The modelling of thin-walled laminated compoite beam and beam like lender tructure, a an one dimenional condened beam element, ha drawn the attention of reearcher for quite ometime, and a number of invetigation have been carried out to tud the different apect of thi. A few repreentative tudie relevant to the preent contet are given in [-8]. One of the initial application of compoite beam theor wa found in the anali of helicopter rotor blade. It ha ubequentl been applied to the anali of pultruded compoite profile and other application, including the anali of long wind turbine blade made of compoite material. The invetigation carried out o far can be broadl divided into two group, baed on the approach followed to evaluate the contitutive matri of the beam element, defined a the beam cro-ection tiffne coefficient. The firt and mot common approach i baed on an analtical technique, while the other approach require a twodimenional finite element anali to obtain the cro-ection tiffne matri. Hodge and hi coworker [3] pioneered the econd approach, which i defined a the o-called variational amptotic beam ection anali (VABS). It i baed on a method known a the variational amptotic method (VAM) [9], where the three dimenional elaticit problem i tematicall divided into a two dimenional cro-ectional problem, and a one dimenional beam problem. VABS ha the advantage that beam having olid or thick-walled cro ection can be analed, where the three dimenional tree can be etracted in the pot proceing tage. Oppoed to thi, the full analtical approach ma be preferred pecificall for the anali of beam with thin walled cro ection in order to avoid the additional two-dimenional finite element anali required in VABS. In the preent tud, a full analtical (cloedform) approach i adopted for the derivation of the cro-ectional tiffne matri conidering different effect and their coupling to ield a ver general formulation, which include a torional warping moment apart from the uual de St. Venant torion contribution, aial force, bi-aial bending moment and tranvere hear force, thu ielding a 77 cro-ectional tiffne matri. All the element of thi matri are eplicitl derived for open I ection and cloed bo ection profile. For the contitutive equation of an beam wall, defined locall, proviion i kept to enable the pecification of either

2 A.H. SHEIKH, O.T. Thomen plane tre condition (ero normal tre along the wall profile) or plane train condition (ero normal train along the wall profile). In the one dimenional finite element approimation, the torional deformation require C continuit of the twiting rotation due to incorporation of the out of plane warping deformation. Thi requirement i atified with the ue of a Hermetian interpolation function conidering the twiting rotation and it derivative with repect to the length coordinate a the nodal unknown. The aociation of the derivative of the twiting rotation help to impoe warping retraint or warping free condition b contraining or releaing thi nodal unknown. At the ame time the bending deformation require C continuit of the tranvere diplacement due to the incorporation of the tranvere hear deformation of the beam wall. A reduced integration technique i required for the evaluation of the tiffne matri in order to avoid hear locking. A the bending deformation i not uncoupled from the other mode of deformation, including torion, it i difficult to implement C with a C formulation having different integration cheme. Lee [7] tried to olve the problem b an amended repreentation of the torional deformation, o a to model it with a C formulation like bending deformation, but thi involved a non-phical parameter in the formulation. Moreover, the C formulation with reduced integration technique i uceptible to dipla inherent numerical diturbance like the occurrence of puriou mode. Keeping thee apect in view, the finite element implementation of the bending deformation i carried out with a different approach baed on the concept of the firt author []. It doe not require a reduced integration technique, which effectivel eliminate the problem mention above. Baed on thi methodolog a three node beam element, a hown in Fig., ha been developed, where the node at the two end contain even degree of freedom (three tranlation, three rotation and the derivative of the twiting rotation), while the internal node contain five degree of freedom (three tranlation and two bending rotation). A computer code ha been written in FORTRAN for the implementation of the element, which ha been ued to olve numerical eample of compoite beam having open I and cloed bo ection. The reult obtained in the form of deflection, angle of twit, and bending lope are compared with analtical, eperimental and/or other finite element anali reult available in literature. The reult how a ver good performance of the propoed element in term of convergence and olution accurac. The developed element i alo utilied to derive ome new reult, which are preented for future reference. Formulation A portion of the beam hell wall, with it local coordinate tem --n and diplacement component, along with the global coordinate tem -- and diplacement component, of the beam i hown in Fig.. In Fig., O i the centroid, and P i the hear centre/pole of the beam ection. The diplacement component at mid-plane of the hell wall in the local coordinate tem (--n) ma be epreed in term of the global diplacement component of the beam [] a u = U ϕ () v = V co α W inα r () w = V in α W coα q (3) where ϕ i the warping function, = V Ψ ( V i the derivative of V with repect to and Ψ i the rotation of beam ection about for the tranvere hear deformation) i the bending rotation of the beam ection with repect to, and = W Ψ i the bending rotation of the beam ection with repect to. The correponding diplacement component at a point awa from the hell mid-plane ma be epreed a Fig.. Three node beam element w u u n ψ = n ()

3 AND CLOSED SECTIONS w v v n ψ = n (5) w = w (6) where ψ n and ψ n are the hear rotation of the hell ection about and, repectivel, for tranvere hear deformation. It i aumed that ψ = while ψ n ma be epreed in term of the correponding global beam parameter a n With the above equation, the diplacement component at an point within the hell wall ma be epreed in term of the global beam diplacement component a u = U ( ninα ) ( ncoα ) ( ϕ nq) ( r n) (8) v = V co α W inα (9) w = V in α W coα q () ψ n = Ψ inα Ψ coα (7) The train component at the correponding point in the local ai tem (--n) ma be epreed a u ε ε v εˆ = = ε u v () ε n ψ n ε n ψ n {} Auming the normal train component ε to be ero (plane train condition), or the normal tre component σ to be ero (plane tre condition), the reduced train vector ma be epreed in term of the global diplacement parameter with the help of the above equation (7-) a Fig.. Cro-ection of a portion of hell wall of the thin-walled beam with coordinated tem {} ε u ε u v = ε = ε n ψ n 3

4 A.H. SHEIKH, O.T. Thomen = U Ψ ( ninα ) ( n coα ) ( ϕ nq) ϕ coα Ψ inα n r Ψ inα Ψ coα () σ σ σ = 6 σ n σ n ε ε ε ε 5 n ε n () The above equation can further be rearranged in matri form a U = (3) {} ε [ H ] = [ H ]{} ε where [ H ] T V W ninα ncoα = ϕ nq ( n r ϕ ) coα inα, inα coα The tre train relationhip conidering the tranvere hear deformation of the laminated hell wall [] in the local coordinate tem (--n) ma be epreed a Auming ε = or σ = along with ψ n =, the above equation reduce to σ = σ σ n ε ε 55 ε n (5) where =, 6 = 6, 66 = 66 and 55 = 55 for plane train ( ε = ); and = /, 6 = 6 6 /, 66 = / and 55 = 55 for plane tre ( σ = ). Uing equation (3) and (5) the train energ of the tem can be written a = T T U {}{ ε σ} dv = {}[ ε D]{} ε d where T [ D] = [ H ] [ ][ H ] T = [ H ] [ ][ H ] ddn ( dn) d = [ C] d (6) All the element of the cro-ectional tiffne matri [D] are eplicitl derived for open I ection and cloed bo ection profile. For thi purpoe the warping function ϕ ued in the above equation are taken a ϕ = rd δ / δ (7) A c

5 AND CLOSED SECTIONS d d where δ =, δ = and A c i the croectional area encloed b the mid-plane contour (cloed ection). For an open ection profile, the warping function ma be impl obtained b dropping the econd term aociated with econdar warping, thu giving ϕ = rd. For the one dimenional finite element implementation, a quadratic approimation ha been adopted for the aial deformation, which follow a Lagrangian tpe formulation. The torional deformation i baed on a Hermetian formulation a mentioned earlier, where a cubic approimation ha been adopted. The approimation of the bending deformation coupled with tranvere hear deformation i baed on the concept propoed b Sheikh []. Baed on thi the field variable are approimated a follow U = a (8) a a3 V = a (9) 3 a5 a6 a7 W = a () 3 8 a9 a a Ψ = a a () 3 Ψ = a a () 5 (3) 3 = a6 a7 a8 a9 It hould be noted that Ψ and Ψ are taken a the field variable intead of and, which are uuall ued in a tpical C formulation. It hould be noted alo that and appear a nodal unknown intead of Ψ and Ψ. Now, with the help of equation (9-), the bending rotation and ma be epreed a The unknown ( a, a, a3, L a9 ) in the above equation (8-3) are epreed in term of the nodal diplacement vector { δ } after ubtitution of U (eq. (8)), V (eq. (9)), W (eq. ()), (eq. ()) and (eq. (5)) at all three node of the beam element (ee Fig. ); and (eq. (3)) and it derivative at the two eternal node a { δ } = [ R]{ a} or { } = [ R] {} δ a (6) where { a} [ a a a a a ] T 3 L 8 9 { δ } = [ U V W U V W =, U3 V3 W , and the matri [R] of order of 99 contain the element nodal coordinate. ε in equation The generalied train vector {} (3) ma be epreed in term of { a } uing equation (8-3) a U V W U V Ψ W Ψ Ψ Ψ {} ε = = = [ S]{} a ] T (7) where the matri [S] of order of 79 i a function of ε can be finall epreed in. The train vector { } term of the nodal diplacement vector { δ } uing equation (6) a = a a3 a5 a6 3a7 ( { ε } = [ S ][ R] { δ } = [ B]{} δ (8) = a a5 a9 a 3a (5) With the above equation, the train energ (eq. (6)) of the tem ma be epreed a 5

6 A.H. SHEIKH, O.T. Thomen T T T U = {} δ [ B] [ D][ B] d{} δ = {}[ δ K ]{} δ (9) where [K] i the element tiffne matri. Uing equation () and (6), the element load vector due to a ditributed tranvere load of intenit q acting in the direction of ma be epreed a T { P} [ R] q [ Sq ] = d T (3) where the row matri [ S q ] of 9 i function of. The integration involved in the evaluation of the element tiffne matri [K] and the load vector {P} are carried out numericall following the Gau quadrature technique. 3 Numerical Eample In thi ection numerical eample of I and bo beam are analed uing the propoed element, and the reult obtained are compared with analtical, eperimental and/or numerical reult available in literature for mot of the cae. The anali i uuall baed on plane tre condition, unle pecified otherwie. In all eample the beam wall are aumed to be contituted b identical laer of identical thickne, but the laer ma have different orientation. The geometr of the beam ection are defined in term of centre line dimenion. 3. Simpl upported I beam under uniforml ditributed load A.5m long open ection I beam impl upported at it two end, and ubjected to a uniforml ditributed tranvere load of kn/m along the web of the beam, i analed uing the propoed element. The beam ha a depth of 5mm, a flange width of 5mm and the ame thickne of.8mm for the flange and the web. The tud i made with different mmetrical tacking equence, where the flange and the web are having identical la-up for all cae. The material propertie of the laer are given in Table. The anali i carried out auming both plane tre and plane train condition. The value of deflection at the centre of the beam and the bending lope at it upport obtained in the preent anali are preented in Table and Table, repectivel. The reult for the deflection are compared with thoe of Lee [7] and Lee and Lee [] in Table. Lee [7] conidered the effect of tranvere hear deformation, wherea the tud of Lee and Lee [] i baed on claical laminate theor. In both the tudie [7, ], one dimenional finite element anali ha been applied after obtaining the cro-ectional tiffne matri analticall. For the validation of the reult in [7,], the beam tructure wa analed uing ABAUS [3] where the S9R5 hell element wa ued to model the beam. The reult produced b ABAUS [3] are alo included in Table. The table how an ecellent agreement of the preent reult with the other reult, epeciall the reult of Lee [7]. Moreover, the preent reult baed on plane tre condition are found to be cloer to the reult produced b ABAUS [3]. It i alo oberved that onl two element could attain the convergence in all the cae. Table. Deflection w (cm) at the centre of the impl upported I beam under uniform ditributed loading (E = 53.78GPa, E = 7.93GPa, G = 8.96GPa, G 3 = 8.96GPa, G 3 = 3.5GPa, ν =.5) Stacking equence [/]S [5/-5]S [3/-3]S Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Lee [7] ( σ = ) Lee [7] ( ε = ) Lee and Lee [] ( σ = ) Lee and Lee [] ( ε = ) ABAUS [3] Stacking equence [5/-5]S [6/-6]S [75/-75]S Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Lee [7] ( σ = ) Lee [7] ( ε = ) Lee and Lee [] ( σ = ) Lee and Lee [] ( ε = ) ABAUS [3] Number of element 6

7 AND CLOSED SECTIONS Table. Bending rotation (rad) at the upport of the impl upported I beam under uniform ditributed loading Stacking equence [/]S [5/-5]S [3/-3]S Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Stacking equence [5/-5]S [6/-6]S [75/-75]S Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Clamped bo beam under uniforml ditributed load A bo beam clamped at both the end, and ubjected to uniforml ditributed tranvere load of 6.5kN/m along the mid-plane of one of the web, i analed uing the propoed element auming both plane tre and plane train condition. The beam i aumed to be free from aial and warping retraint at the upport. The beam i. m long, 7 mm deep and 5 mm wide, where all the beam wall are mm thick having a tacking equence of (5/- 5) /(/) 6 /(5/-5). The material propertie aumed for all laer are: E = 8. GPa, E = 9.65 GPa, G = G 3 = G 3 =.55 GPa, ν =.3. The value of the deflection and angle of twit at the centre of the beam obtained uing the preent anali are preented in Table 3, along with reult obtained b Kollar and Springer [5] auming plane train condition, and Vo and Lee [] auming both plane tre and plane train condition. Kollar and Springer [5] olved the problem in cloed form (analticall), while Vo and Lee [] applied one dimenional finite element anali after obtaining the cro-ectional tiffne matri analticall. A both of the tudie [5,] did not conider the effect of tranvere hear deformation, the preent anali wa alo carried out with a high value of tranvere hear rigidit (G 3 = G 3 = G 6 ), and the reult obtained are included in Table 3 (marked ). The reult how a ignificant effect of the tranvere hear deformation. The table how that the convergence of the preent finite element formulation i good. It alo how a good agreement between the reult obtained b the different technique having imilar bai. Table 3. Deflection and angle of twit at the centre of the clamped bo beam under uniform ditributed loading w (m) Θ 3 (rad) Preent ( σ = ) Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Preent ( ε = ) Preent ( σ = ) Preent ( σ = ) Preent ( σ = ) Preent ( ε = ) Preent ( ε = ) Preent ( ε = ) Vo and Lee [] ( σ = ) Vo and Lee [] ( ε = ) Kollar and Springer [5] ( ε = ) G3 = G, G3 = G3 = G, Cantilever I beam under tip load A 3 inch long cantilever I beam ubjected to a tranvere unit load (. lb) at the free end i analed uing the propoed element auming retrained warping at both end. The beam ha a depth of.5 inch, a flange width of. inch. and ame thickne (. inch) i aumed for the flange and the web. The tacking equence of the flange i /9//9/9//5/5, while that of the web i /9//9/9//9/. The aumed material propertie are: E =.59 6 Pi, E =. 6 Pi, G = G 3 = G 3 =.89 6 Pi, ν =.. The variation of deflection, bending lope and twiting rotation along the length of the beam are plotted in Fig. 3, Fig. and Fig. 5, repectivel. The reult for the bending lope and twiting rotation are compared with the numerical reult of Jung et al. [] and the eperimental reult of Chandra and Chopra [5] in Fig. and Fig. 5. 7

8 A.H. SHEIKH, O.T. Thomen Bending lope (rad) Deflection (inch) Ditance from the upport (inch) Fig. 3. Variation of deflection of the I beam Preent Mied FEM Diplacement FEM Eperimental Ditance from the upport (inch) Fig.. Variation of bending lope of the I beam Jung et al. [] ha produced reult baed on a mied formulation, a well a on the diplacement formulation of Smith and Chopra [6], where one dimenional finite element ha been applied after getting the cro-ectional tiffne matri analticall. The figure how a ver good agreement between the reult. 3. Cantilever bo beam under tip load/twiting moment A 3 inch long cantilever bo beam having a depth of.5 inch, a width of.93 inch and auming the ame thickne (.3 inch) for all the wall coniting of 6 laer i analed uing the propoed element. Bending lope (rad) Preent Analtical FEM Eperimental Ditance from the upport (inch) Fig. 6. Variation of bending lope of the bo beam under a tranvere load at the tip 35 Preent Mied FEM Diplacement FEM Eperimental 8 6 Preent Analtical FEM Eperimental Twiting rotation (rad) Ditance from the upport (inch) Fig. 5. Variation of twiting rotation of the I beam Twiting rotation (rad) Ditance from the upport (inch) Fig. 7. Variation of twiting rotation of the bo beam under a tranvere load at the tip 8

9 AND CLOSED SECTIONS The aumed loading i either a unit tranvere load (. lb) applied at the free end, or a unit twiting moment (. lb-inch) applied at the free end. The tacking equence of the top and bottom wall i (5/5) 3, while that of the left and right wall i (5/-5) 3. The material propertie of all laer are aumed to be identical and equal to thoe adopted in the previou eample. The variation of the bending lope and twiting rotation along the length of the beam are plotted in Fig. 6 to Fig. 9 along with the analtical and eperimental reult of Chandra et al. [], and the finite element reult of Stemple and Lee [7]. The reult of the different method are found to be in ver good agreement. Bending lope (rad) 8 6 Preent Analtical FEM Eperimental Ditance from the upport (inch) Fig. 8. Variation of bending lope of the bo beam under a twiting moment at the tip Twiting rotation (rad) Preent Analtical FEM Eperimental Ditance from the upport (inch) Fig. 9. Variation of twiting rotation of the bo beam under a twiting moment at the tip Concluion A full coupled beam element ha been developed for the anali of thin-walled laminated compoite beam of open and cloed cro ection including aial diplacement, torion, out of plane warping, bi-aial bending and tranvere hear deformation. The contitutive equation of the beam element are derived analticall conidering the coupling of all thee mode of deformation. The reulting compoite beam theor i applied to open I ection and cloed bo ection beam. The incorporation of tranvere hear deformation demand a C formulation for the one dimenional finite element approimation of the bending deformation, while the torional deformation demand a C formulation for the incorporation of out of plane warping. The difficult in implementing both formulation in the preent coupled problem i uccefull overcome b adopting an efficient approach for the finite element approimation of the bending deformation. Numerical eample of compoite open and cloed ection beam having different load and boundar condition are analed uing the propoed element. The reult obtained are compared with analtical, eperimental and/or other finite element reult available in literature, and the comparion how a ver good performance of the propoed full coupled beam element. Some new reult are alo preented for future reference. Acknowledgement The work preented wa carried out a part the Innovation Conortium Integrated Deign and Proceing of Lightweight Compoite and Sandwich Structure (abbreviated KOMPO-SAND ) funded b the Danih Minitr of Science, Technolog and Innovation and the indutrial partner Compohield A/S, DIAB ApS (DIAB Group), Fiberline Compoite A/S, LM Glafiber A/S and Veta Wind Stem A/S. The upport received i gratefull acknowledged. Reference [] Bauld N.R. and Teng L.S. A Vlaov theor for fiber-reinforced beam with thin-walled open cro ection. International Journal of Solid and Structure, Vol., No. 3, pp 77-97, 98. [] Chandra R., Stemple A.D. and Chopra I. Thinwalled compoite beam under bending, torion, and etenional load. AIAA Journal, Vol. 7, No. 7, pp 69-66, 99. 9

10 A.H. SHEIKH, O.T. Thomen [3] Cenik C.E.S. and Hodge D.H. VABS: A new concept for compoite rotor blade cro-ection modeling. Journal of the American Helicopter Societ, Vol., No., pp 7-38, 997. [] Jung S.N., Nagaraj V.T. and Chopra I. Refined tructural model for thin and thick walled compoite rotor blade. AIAA Journal, Vol., No., pp 5-6,. [5] Kollar L.P. and Springer G.S. Mechanic of compoite tructure. t edition, Cambridge Univerit Pre, 3. [6] Salim H.A., and Devalo J.F. Torion of open and cloed thin-walled laminated compoite ection. Journal of compoite material, Vol. 39, No. 6, pp 97-5, 5. [7] Lee J. Fleural anali of thin-walled compoite beam uing hear-deformable beam theor. Compoite Structure, Vol. 7, No., pp -, 5. [8] Librecu L. and Song O. Thin-walled compoite beam. t edition, Springer, 6. [9] Berdichevk V.L. Variational-amptotic method of contructing a theor of hell. PMM, Vol. 3, No., pp , 979. [] Sheikh A.H. A new concept to include hear deformation in a curved beam element. Journal of Structural Engineering, ASCE, Vol. 8, No. 3, pp 6-,. [] Redd J.N. Mechanic of laminated compoite plateand hell : theor and anali. nd edition, CRC Pre,. [] Lee J. and Lee S. Fleural-torional behavior of thin-walled compoite beam. Thin-Walled Structure, Vol., No. 9, pp 93-35,. [3] ABAUS/Standard Uer' Manual, Verion 6., Hibbit, Kalon & Sorenen Inc., 3. [] Vo T.P. and Lee J. Fleural-torional behavior of thin-walled cloed-ection compoite bo beam. Engineering Structure, To be publihed in 7 (available online). [5] Chandra R. and Chopra I. Eperimental and theoretical anali of compoite I beam with elatic coupling. AIAA Journal, Vol. 9, No., pp 97-6, 99. [6] Smith E.C. and Chopra I. Formulation and evaluation of an analtical model for compoite bobeam. Journal of the American Helicopter Societ, Vol. 36, No. 3, pp 3-35, 99. [7] Stemple A.D. and Lee S.W. A finite element model for compoite beam with arbitrar cro-ectional warping. AIAA Journal, Vol. 6, No., pp 5-5, 988.