Modeling of Fillets in Thin-walled Beams Using Shell/Plate and Beam Finite. Elements
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1 Modelng of Fllets n Thn-walled Beams Usng Shell/Plate and Beam Fnte Elements K. He Graduate Research Assstant W.D. Zhu Professor and Correspondng Author Emal: wzhu@umbc.edu; Tel: Department of Mechancal Engneerng Unversty of Maryland, Baltmore County 1000 Hlltop Crcle Baltmore, MD 150 ABSTRACT Fllets are commonly found n thn-walled beams. Ignorng the presence of a fllet n a fnte element (FE) model of a thn-walled beam can sgnfcantly change the natural frequences and mode shapes of the structure. A large number of sold elements are requred to accurately represent the shape and the stffness of a fllet n a FE model, whch makes the sze of the FE model unnecessarly large for global dynamc and statc analyses. In ths work the equvalent stffness effects of a fllet n a thn-walled beam are decomposed nto n-plane and out-of-plane effects. The n-plane effects of a fllet are analyzed usng the wde-beam and curved-beam theores, and the out-of-plane effects of the fllet are analyzed by modelng the whole fllet secton as a slender bar wth an rregular cross-secton. A smple shell/plate and beam element model s developed to capture the n-plane and out-of-plane effects of a fllet on a thn-walled beam. The natural frequences and mode shapes of a thn-walled L-shaped beam specmen calculated usng the new methodology are compared wth ts expermental results for 8 modes. The maxmum error between the calculated and measured natural frequences for all the modes s less than % and the assocated modal assurance crteron values are all above 95%. The methodology s also appled to other thn-walled beams, and excellent agreement s acheved between the natural frequences from the shell/plate and beam element models and those from the sold element models. Whle the shell/plate and beam element models
2 provde the same level of accuracy as the ntensve sold element models, the degrees of freedom of the shell/plate and beam element models of the thn-walled beams are only about 10% or less of those of the sold element models. 1. INTRODUCTION Thn-walled beams wth fllets are wdely used n engneerng structures. To create an accurate fnte element (FE) model of a thn-walled beam, the effects of a fllet cannot always be neglected. Fgure 1(a,b) show two ntensve sold element models of two thn-walled beams wth an L-shaped cross-secton and free boundary condtons. All the sold element models n ths work are created usng ABAQUS 6.5. The two beams n Fg. 1 have the same materal propertes: E υ ρ 9 3 = GPa, = 0.33, = kg/m (1) where E s the elastc modulus, υ s the Posson s rato, and ρ s the mass densty. For all the cases n ths work, the materal propertes used are the same as those n Eq. (1). The only dfference between the two beams n Fg. 1 s the one n Fg. 1(a) has a fllet between the flanges and the other n Fg. 1 does not have t. Whle the cross-sectons of the L-shaped beams consdered n ths work are symmetrc about ts axs of symmetry (see Fg. ), the analyss can be readly appled to thn-walled beams wth dfferent wall wdths (see Fg. 9). The radus of the fllet between the flanges n Fg. 1(a) and the thckness of the flanges are m, the wdth of the flanges s b = 0.06 m, and the length of the beams s l = 0.08 m (see Fg. 1(c)); the radus of the fllet at the outer end of each flange of the beams n Fg. 1 s r out = m. The calculated natural frequences of the frst elastc modes of the two beams have a 19.41% dfference, and the assocated modal assurance crteron (MAC) value, calculated usng the dsplacements that are perpendcular to the flange surfaces at sx unformly spaced locatons at the outer end of each flange and the mddle edge of the beam (see Fg. 1(c)), s only 15.93%; note that the dsplacements perpendcular to both flange surfaces are used at the locatons along the mddle edge of the beam. The natural frequency of the beam n Fg. 1(a) s hgher than that n Fg. 1 because gnorng the presence of a fllet leads to an under-predcted stffness of the structure and the effect of the stffness ncrease due to the presence of the fllet s larger than that of the mass ncrease. (a) (c) Fg. 1 The frst elastc mode shapes of two L-shaped beams wth (a) and wthout a fllet between the flanges
3 from the sold element models. The natural frequency of the beam n (a) s 306.7Hz and that n s 439. Hz. The locatons that are used for calculatng the assocated MAC value are shown as dots n (c). Thn-walled beams can always be modeled usng sold elements; ths can requre a dense mesh at a fllet due to stress concentraton and the shape of the fllet [1], whch can result n large memory occupaton and computaton tme consumpton. When we model relatvely large structures or create FE models that requre model updatng and optmzaton, the use of sold elements can be computatonally expensve. Usng shell/plate elements to model thn-walled beams s the most effcent approach because the shell/plate elements can accurately model these structures wth a much smaller number of degrees of freedom (DOF). However, how to model a fllet usng shell/plate elements remans a challengng task. The flanges of a thn-walled L-shaped beam can be easly modeled usng shell/plate elements wth unform thckness. It s not dffcult ether to model the fllet at the outer end of each flange (see Secton 3..), where the stress gradent s not hgh and there s no need to have a dense mesh. The fllet between the flanges has to be modeled separately because the thckness at the fllet regon s non-unform and the stress gradent s hgh there. A straghtforward approach s to use a seres of shell/plate elements wth varable thckness to model a half fllet (see Fg. ). The resultng model can approxmate the shape of the fllet by ncreasng the number of shell/plate elements along the x and y axes at the fllet regon. However, f we ncrease the number of elements along the x and y axes (see Fg. 1(c)), the number of elements along the z axs has to be ncreased at the same tme because the shape of each shell/plate element cannot be too spndly [,3]. Hence the DOF of the FE model can stll be very large. Moreover, t s dffcult to model the nteracton between the two seres of shell/plate elements by varyng only the thckness of shell/plate elements at the fllet regon. Fg. The cross-secton of an L-shaped beam wth the fllet between the flanges modeled by two seres of shell/plate elements wth varable thckness. Another possble approach to model a thn-walled beam s to use shell/plate elements to model the walls and sold elements to model a fllet (Fg. 3(a)). Ths approach has no problem to model the fllet, but the shape functons of
4 the dfferent types of elements are ncompatble at the connectng edges. The dsplacements at these connectng edges wll lose ther contnutes n ths sold and shell/plate element model. A fllet can also be modeled as webs usng shell/plate elements that are perpendcular to the walls (see Fg. 3). A web-lke model can accurately represent the shape of the fllet, but the materal s dscontnuous along the z axs (see Fg. 1(c)) of the beam [3]. Whle the plate theory can be used to analyze the walls of a flleted thn-walled beam, t s dffcult to model the boundary condtons of the walls at a fllet [4]. (a) Fg. 3 (a) Usng sold elements to model the fllet and shell/plate elements to model the flanges of a thn-walled L-shaped beam; and usng shell/plate elements that are perpendcular to the flanges to model the fllet. In most global dynamc and statc analyses, we are more nterested n the stffness and mass effects of a fllet on the global behavor of a thn-walled beam; the stress and mass dstrbutons at the fllet have less mportance. If a fllet can be modeled separately and represented by some equvalent stffness and mass, we do not have to create an accurate model to capture the shape of the fllet and the stress dstrbuton at the fllet n a global analyss. The equvalent mass can be easly calculated f we know the radus of the fllet and the densty of the materal, but t s very dffcult to calculate the equvalent stffness by drectly usng the elastcty theory. Brown and Seuglng [3] nvestgated the effects of a fllet n the cross-sectonal plane. The fllet was treated as a straght wde beam wth non-unform rectangular cross-secton and a fxed boundary n the analytcal model. In the FE model, the fllet was modeled as a brdge that connects the tangent sectons usng shell/plate elements (see Fg. 4). By matchng the rotatonal dsplacements of the tangent sectons, whch were derved from the analytcal and FE models usng the wde-beam theory [5], the elastc modulus and the thckness of the shell elements used at the brdge were determned [3].
5 Fg. 4 The brdge model of a fllet used n Ref. [3] Only the n-plane effects were consdered n Ref. [3]; the out-of-plane effects of a fllet on the bendng and torsonal stffnesses of a thn-walled beam were not consdered. Fgure 5(a,b) show the presence of a fllet between the flanges of an L-shaped beam can sgnfcantly change the natural frequency of the frst torsonal mode of the beam. The frst torsonal natural frequences of the L-shaped beams wth and wthout the fllet, calculated from the sold element models, have an 8.3% dfference, and the assocated MAC value, calculated usng the dsplacements that are perpendcular to the flange surfaces at sx unformly spaced locatons at the outer end of each flange (see Fg. 6(c)), s 76%. A more comprehensve model that can accurately capture both the n-plane and out-of-plane effects of a fllet s needed. (a) (c) Fg. 5 The frst torsonal modes of two L-shaped beams wth (a) and wthout a fllet between the flanges from the sold element models. The radus of the fllet between the flanges and the thckness of the flanges are m, r = m, b = 0.03 m, and l = 0.15 m. The frst torsonal natural frequency of the beam n (a) s 543 out Hz and that n s 50 Hz. The locatons at the outer ends of the two flanges, whch are used for calculatng the assocated MAC value, are shown as dots n (c).
6 There are two common types of fllets: a sngle fllet shown n Fg. 6 and double fllets shown n Fg. 7(a). The double fllets were dscussed n Ref. [3]; the dsplacements of the top tangent secton were calculated by modelng the upper fllet regon as a straght wde beam wth a fxed boundary (see Fg. 7). For a sngle-fllet regon shown n Fg. 6, usng the straght wde-beam model to calculate the dsplacements of a tangent secton s not always accurate, snce the centrodal axs of the wde beam cannot always be assumed to be a straght lne. Usng the straght-beam theory to solve a curved-beam problem can lead to a large error [6]. A more accurate method can be developed to analyze the n-plane effects of a sngle fllet. Fg. 6 A sngle fllet (a)
7 Fg. 7 (a) Double fllets; and the straght wde-beam model of the upper fllet regon of the double fllets n Ref. [3]. To resolve the problems mentoned above, we have developed a new method that uses shell/plate and beam elements to model a fllet n a thn-walled beam. Both the n-plane and out-of-plane effects of a fllet are taken nto account and they are calculated separately. By matchng the dsplacements at the fllet under gven appled loads and the bendng stffnesses of the cross-secton of the fllet, whch are derved from the analytcal and FE models, the equvalent stffnesses of the fllet are obtaned. The accuracy n calculatng the n-plane dsplacements of a tangent secton s mproved by usng the curved-beam theory. The method s used to model two thn-walled L-shaped beams. The calculated natural frequences and mode shapes of an alumnum L-shaped beam specmen are compared wth ts expermental results for 8 modes, and excellent agreement s acheved. The methodology s also appled to thn-walled beams wth other types of cross-sectons, ncludng an I-shaped beam and a box beam.. METHODOLOGY.1 A Sngle Fllet The fllet between the flanges of a thn-walled L-shaped beam, as shown n Fg. 8, s analyzed. The equvalent stffness of the whole fllet secton from the L-shaped beam, as shown n Fg. 8(a), can be decomposed nto two parts: the n-plane stffness and the out-of-plane stffness. When the cross-sectonal dmensons of the fllet secton are much smaller than (e.g., less than one-tenth of) the length of the L-shaped beam, we can assume that the two stffness effects are uncoupled. In calculatng the n-plane stffness, we analyze a fllet secton wth unt length (see Fg. 9) and treat t as a plane stran problem. The elastc modulus for the plane stran problem s taken to be E 1 * E = accordng to the wde-beam theory [5]. The out-of-plane stffness s the extra bendng υ and torsonal stffnesses of the L-shaped beam due to the whole fllet secton when t s n bendng and torson wth the beam. Because the cross-sectonal area of the fllet secton s much smaller than that of the beam, we can neglect the cross-sectonal deformaton of the fllet secton when the beam s n bendng and torson.
8 Fg. 8 An L-shaped beam wth a fllet between the flanges: (a) the whole fllet secton from the L-shaped beam; and a fllet secton wth unt length..1.1 In-plane effects We consder a perpendcular fllet wth the connectng flanges perpendcular to each other, as shown n Fg. 9, though the analyss s not restrcted to perpendcular fllets. Snce the stffness of a lnear structure s ndependent of the loads, we apply the moments M, the normal forces N, and the shear forces P, as shown n Fg. 9, on the two tangent sectons of the fllet, where a tangent secton s perpendcular to a connectng flange and passes through the tangent ponts. Snce the symmetrc axs has no deformaton, we can analyze the upper half fllet wth the symmetrc axs used as the fxed boundary. Fg. 9 The sngle-fllet regon wth the appled moments and forces on the tangent sectons
9 The equvalent n-plane stffness of the half fllet n Fg. 9 s related to the appled loads: θ k11 k1 k13 θ M K u = k k k u = N 1 3 v k31 k3 k 33 v P where K s the stffness matrx, whose entres are shown n Eq. (); and θ, u, and v are the rotatonal and translatonal dsplacements of the centrod B of the tangent secton, correspondng to the appled moment M and forces N and P, respectvely, as shown n Fg. 10. Equaton () can be wrtten as 1 θ M c11 c1 c13 M u = D N = c c c N 1 3 v P c31 c3 c 33 P where D = K s the flexblty matrx, whose entres are shown n Eq. (3). By matchng the dsplacements derved from the analytcal model to those from the FE model under the same appled loads, we can determne the flexblty matrx of the FE model and the n-plane equvalent stffness of the fllet. Due to symmetry there are sx ndependent entres n the D matrx; to match all of them requres us to develop a complcated FE model that has sx ndependent parameters to be determned. Based on the results n Ref. [3], the rotatonal dsplacement affects more the behavor of the flanges than the translatonal dsplacements, and we only need to match θ from the analytcal and FE models. () (3) The Castglano s theorem for a curved beam s employed to solve for the rotatonal dsplacement of B analytcally. The half fllet n Fg. 9 s modeled as a curved beam wth non-unform cross-secton and a fxed boundary (see Fg. 10). The center of the nner arc O s used as the orgn of the polar coordnates. The centrodal axs of the curved beam s assumed to be the locus of the centrods of the cross-sectons passng through O. The radal coordnate of a pont on the centrodal axs s r, and the angular coordnate of the pont, measured from the tangent secton, s α. The centrodal axs can be expressed as r t+ r r cosα 0 0 = + (4) where t s the thckness of the flanges, and r0 s the radus of the fllet.
10 Fg. 10 The dsplacements of the centrod B of the tangent secton and the polar coordnates of the curved-beam fllet model Shown n Fg. 11 s a typcal curved beam wth constant curvature of the centrodal axs and unform cross-secton. Wth the same appled loads as those n Fg. 10, the total stran energy of the curved beam s [7] s M N ap MN U= ( + + ) ds (5) EArh EA GA EAr 0 where E G= (1 +υ) s the shear modulus, r s the radus of curvature of the centrodal axs, h s the dstance between the centrodal axs and the neutral axs, A s the cross-sectonal area, and a s a correcton coeffcent that depends on the shape of the cross-secton wth a =1. for a rectangular cross-secton. The deflectons of the curved beam, whch are the same as those shown n Fg. 10, can be calculated by dfferentatng Eq. (5) wth respect to the appled loads [8] U U U θ =, u =, v= (6) M N P r Note that Eq. (5) s applcable when 0.6 < < 8, where C s the dstance between the centrodal axs and the C outer edge of the curved beam (see Fg. 11) [8].
11 Fg. 11 A curved beam wth constant curvature and unform cross-secton Equaton (5) s used for the curved-beam fllet model n Fg. 10, where r, C, and h are not constants due to the varable cross-sectons. Followng Young [5], the ntegral n Eq. (5) s evaluated by dvdng the curved beam n Fg. 10 nto n small sectons wth equal angle d α along the span, as shown n Fg. 1(a), so that r, C, and h can be assumed to be constants n each secton. In ths work, we use n = 500. The total stran energy U s the sum of the stran energes of all the sectons. The radal and angular coordnates of the th ( = 1,,, n) secton are r and α, respectvely, where π 0 < α < 4 secton are denoted by M, N, and P, as shown n Fg. 1(a), and ther expressons are derved from the. The nternal moment and forces actng on the th moment and force balances (see Fg. 1 ): t M =M + Prsn α + N( + r0 rcos α ) P=Pcosα + Nsnα N=N cosα Psnα The radus of curvature of the centrodal axs at the th secton s where R = r 3 dr + [ r ( ) ] dα dr dr + ( ) r( ) dα dα (7) (8) s the radal coordnate of the centrodal axs at the r t+ r r 0 0 = + (9) cosα th secton, and dr t+ r0 d r ( t r0) 3 ( ) (sec )(tan ), ( ) [(sec ) sec (tan ) ] d α α + = = α dα α + α α (10) are the frst and the second dervatves of r n Eq. (4) wth respect to α evaluated at α = α, respectvely. Due to the unt wdth (see Fg. 8) of the curved beam, the cross-sectonal area at the th secton s
12 A = C (11) where C t+ r r 0 0 = (1) cosα s the dstance between the centrodal axs and the outer edge of the curved beam at the dstance between the centrodal and neutral axes at the th secton s [5] C h = R R + C ln( ) R C The length of the centrodal axs of the th secton s th secton. The (13) dr ds = dα r + ( ) (14) dα Usng Eqs. (7)-(14) n Eq. (5) and usng the frst equaton n Eq. (6) yelds n U M M N N ap P M N N M θ = = + + * * * * ds M = 1 EARh M EA M GA M EAR M EAR M M N P = C * mθ + C * nθ + C * pθ E E E where (15) C C C mθ nθ pθ = = = n = 1 n 0 = 1 n 1 ds ARh t r cosα r h cosα ds (16) ARh rsnα hsnα ds ARh = 1 are the coeffcents correspondng to the appled moment M and forces N and n Eq. (5) s replaced by r functon of 0 t, where r0 P, respectvely; note that * E n Eq. (15). Fgure 13 compares the values of the three coeffcents n Eq. (16) as a vares from m to 0.05 m. Snce Cn θ s much smaller than Cm θ and E C pθ, the normal force N has much smaller contrbuton to the rotatonal dsplacement θ than the moment M and the shear force P. Hence the term correspondng to N n Eq. (15) can be neglected, and Eq. (15) becomes M P θ = C * mθ + C (17) * pθ E E
13 (a) Fg. 1 (a) The th secton of the curved beam n Fg. 11; and the free body dagram for calculatng ts nternal moment and forces
14 (a) Fg. 13 Comparsons of the three coeffcents ( C θ, C θ, and m n C θ ) (a) and of the two coeffcents ( C θ and p n C pθ ) as a functon of r 0 t In our FE model, a shell/plate element wth a fxed boundary s used to model the n-plane stffness effects of the half fllet (see Fg. 14); a rgd lnk s used to connect the shell/plate element and the centrod of the fllet regon O 1 on the fxed boundary, where a beam element s added to model the out-of-plane stffness effects of the fllet n Secton.1.. The wdth of the shell/plate element s one (see Fg. 8); the length L and the thckness s of E the element need to be determned. The shell/plate element s treated as a wde beam by settng E beam = ; 1 υ note that the elastc modulus and the Posson s rato of the shell/plate element are the same as those of the materal. For ths cantlever wde-beam element, the stffness matrx condton, s [9] f K, after mposng the fxed boundary 4EbeamI 6EbeamI f L L K = (18) 6EbeamI 1EbeamI 3 L L where 3 s I = s the cross-sectonal area moment of nerta of the wde beam, and thedsplacements at th 1 e free end of the wde beam are
15 L L 1L 6L 3 3 θ E f 1 beami EbeamI Ebeams Ebeams M M M ( ) 3 3 v = K = = P L L P 6L 4L P 3 3 EbeamI 3EbeamI Ebeams Ebeams Comparng the coeffcents of the M and P (19) terms n Eq. (17) wth those n the frst equaton n Eq. (19) and notng that * E = E beam yelds 1L 6L C = mθ, C 3 pθ 3 s = s (0) The length and the thckness of the shell/plate element can then be determned from Eq. (0). Fg. 14 The FE model for calculatng the equvalent n-plane stffness of the fllet.1. Out-of-plane effects The out-of-plane effects of the fllet are analyzed by modelng the whole fllet secton (see Fg. 8(a)) as a slender bar wth an rregular cross-secton that s n bendng (see Fg. 15(a)) and torson wth the L-shaped beam. The bendng and torsonal stffnesses of the whole fllet secton can be obtaned separately by calculatng the area moments of nerta and the torsonal stffness factor of the rregular cross-secton of the fllet n Fg. 15, respectvely. The warpng of the fllet secton s not consdered snce the dmensons of ts cross-secton are much smaller than ts length. x1 y1 Two parallel Cartesan coordnate systems, the x y coordnate system wth the orgn O and the coordnate system wth the orgn at the centrod O1 of the fllet cross-secton, are shown n Fg. 15; the x and
16 y axes are perpendcular to a tangent secton and pass through ts centrod. The area moments of nerta of the cross-secton about the x 1 and y 1 axes are determned from I y da I I x y d (1) x = = y, 1 x1y = A1 A1 A1 where x 1 and y1 are the coordnates of a small area 1 n the da x1 y1 coordnate system, and A 1 s the cross-sectonal area; I I due to symmetry of the cross-secton. The coordnates of the centrod O n the y 1 = x 1 1 x y coordnate system are xda 1 1 A1 A1 x =, y = A yda A 1 1 () where x and y are the coordnates of a small area da1 n the x y coordnate system. The torsonal stffness factor of the cross-secton, whch s not equal to the polar area moment of nerta becuase the cross-secton s not a crcle, s [5] where K K K D 4 = α (3) 1 t t K ( r t) t { 0.1 [1 ]} K = r0 + t 1( r0 + t) 1 t t [ (1 )] 4 3 = rt r0 19r0 r0 α = t D= t+ r r + t [ 3 0 ( 0 ) ] (4) (a) Fg. 15 (a) The whole fllet secton n bendng; and ts cross-secton wth the orentatons of the prncpal axes shown n dashed-dotted lnes.
17 Our fllet FE model uses a combnaton of shell/plate and beam elements; rgd lnks are used to connect the shell/plate elements and the beam elements (see Fg. 16(a,b)). The shell/plate elements are manly used to capture the n-plane effects of the fllet, and the beam elements are used to compensate for the out-of-plane stffness of the fllet that cannot be fully modeled by the shell/plate elements. The beam elements are located at the centrod of the cross-secton n Fg. 16. The area moments of nerta of the cross-secton n the FE model, whch nclude the area moments of nerta of two shell/plate elements and a beam element, can be calculated usng the parallel-axs theorems (see Fg. 16) [7] I =I +I +A dy +I +A dy beam shell 1 shell x1 x1 x1 shell 1 x1 shell I =I + A dy dy beam xy 1 1 xy 1 1 shell 1 (5) where A =Ls s the area of the shell/plate elements, dy1 and dy are the dstances between the centers of shell 3 3 shell 1 s L shell L s the shell/plate elements and the x 1 and y 1 axes, and I x' = and I x' = are the area moments 1 1 nerta of the two shell/plate elements. By matchng the area moments of nerta of the cross-secton that are derved from the analytcal and FE models, the area moments of nerta of the beam element beam I x 1 and I can beam x1y1 be determned. Due to symmetry, I beam = I. The prncpal area moments of nerta of the beam element I1 beam y1 x1 I beam beam and can be determned from I, I, and x1 y1 I beam x1y1 ; the prncpal axes of the beam element are selected to be the same as those of the cross-secton (see Fg. 15). (a)
18 (c) Fg. 16 (a) The FE model of a flleted L-shaped beam; the front vew of the FE model of the fllet n (a); and (c) the free body dagram of the beam element wth the rgd lnks. It s dffcult to drectly match the torsonal stffness factors of the cross-secton of the fllet from the analytcal and FE models, because n the FE model, the torsonal stffness factor cannot be wrtten as a sum of the torsonal stffness factors of the shell/plate elements and the beam element. Instead of matchng the torsonal stffness factors, we can match the torsonal dsplacement of the cross-secton from the analytcal model wth that of the beam element wth the rgd lnks. We apply two shear forces P on the two tangent sectons, as shown n Fg. 16. The torque T appled at the cross-secton s T = Pd (6) where d s the dstance between the forces P and the centrod of the cross-secton. The torsonal dsplacement ntroduced by T from the analytcal model s T Pd ϕ = = (7) GK GK In the FE model, a shear force P ntroduces a moment M r = PL and a force P at the outer end of a rgd lnk (see Fg. 16(c)). Snce the moment M r tends to bend the correspondng rgd lnk and does not affect the torson of the beam element, the torque appled on the beam element wth the rgd lnks s T = P( d L) (8) beam The torsonal dsplacement of the beam element s Tbeam Pd ( L) ϕ = = (9) GK GK Comparng Eq. (7) and Eq. (9), we obtan the torsonal stffness factor of the beam element beam beam
19 K beam d L = K (30) d The cross-sectonal area of the beam element A beam s obtaned by subtractng the areas of the shell/plate elements from the cross-sectonal area of the fllet. Wth all the parameters, ncludng L, s, x, y, I1, I, Kbeam, and A beam, determned above, the FE model of the fllet can be created. The shell/plate and beam elements along wth the rgd lnks n the fllet cross-secton (see Fg. 16) have a total of 18 DOF. Whle the same number of DOF s used n the brdge model n Ref. [3], the shell/plate and beam element model can capture both the n-plane and out-of-plane effects of the fllet. Dependng on the calculated length of the shell elements, a shell/plate element can be dvded nto two or more shell/plate elements.. Double Fllets To model the n-plane effects of a double-fllet regon, as shown n Fgs. 7(a) and 17(a), we need to calculate the rotatonal dsplacements of three tangent sectons JZ, FW, and HV. Fgure 18 shows the contours of the rotatonal dsplacements wthn a double-fllet regon, calculated from two densely meshed plane stran FE models, created usng ABAQUS 6.5, for two dfferent cases. In the frst case the fxed boundary s at the lower secton and the top tangent secton s subjected to a moment M = 10,000 Nm and a dstrbuted shear force p = 10,000 N/m (Fg. 18(a)). In the second case the fxed boundary s at the top tangent secton and the two sde tangent sectons are subjected to a moment M = 10,000 Nm and a dstrubted shear force p = 10,000 N/m (Fg. 18). The dmensons of the fllet regon are shown n Fg. 18(a). As shown n Fg. 18(a), the dsplacements around the FH secton s small and the contours are symmetrc about the secton BQ (see Fg. 17(a)). Hence t s reasonable to select the FH secton as a fxed boundary and model the upper fllet regon JZHF as a straght wde beam wth non-unform cross-secton (see Fg. 17) n calculatng the rotatonal dsplacement of the tangent secton JZ. In calculatng the rotatonal dsplacements of the tangent sectons FW and HV (see Fg. 17(a)), we can use the curved-beam model dscussed n Secton.1.1, snce the dstrbuton of the rotatonal dsplacements n Fg. 18 s smlar to that wthn a sngle-fllet regon, as shown n Fg. 19.
20 (a) Fg. 17 A double-fllet regon (a) can be modeled as a straght beam wth non-unform cross-secton and a fxed boundary. (a) Fg. 18 The dstrbutons of the rotatonal dsplacements wthn a double-fllet regon for two cases: (a) the fxed boundary s at the lower secton and the loads are appled on the top tangent secton; and the fxed boundary s at the top tangent secton and the loads are appled on the two sde tangent sectons.
21 Fg. 19 The dstrbuton of the rotatonal dsplacements wthn a sngle-fllet regon shown n Fg. 6. The fxed boundary s at the lower secton wth the top tangent secton subjected to a moment M = 10,000 Nm and a dstrubted shear force p = 10,000 N/m. The dmensons of the fllet regon are shown n the fgure. The rotatonal dsplacemet of the tangent secton JZ s calculated usng the Castglano s theorem (see Fg. 17) [7] [ M + P( r y)] ap = + EI AG 0 U dy dy r0 r0 Pr ( 0 y) dy dy C mθ + C pθ 0 0 U M M P θ = = + = M EI EI E E (31) where U s the total stran energy of the straght wde beam, r0 s the radus of the fllets, and I s the cross-sectonal area moment of nerta of the straght wde beam. The rotatonal dsplacements of the tangent sectons FW and HV are calculated by treatng both the FKGW and HXSV regons n Fg. 17(a) as a half sngle fllet and usng the approach dscussed n Secton.1.1. Analyss of the out-of-plane effects of the whole fllet secton s smlar to that dscussed n Secton.1.. The area moments of nerta of the fllet cross-secton n Fg. 17(a) can be calculated as descrbed n Secton.1.. The torsonal stffness factor of the cross-secton n Fg. 17(a) s [5] where K K K D 4 = α (3)
22 1 t t K ( r t ) t [ 0.1 (1 )] K = r0 + t1 1( r0 + t1) 1 t t [ (1 )] = rt r0 19r0 r0 α = c( ) t t ( t+ r ) + rt + D = 4 r + t (33) n whch t and t are the thcknesses of the central wall and the sde walls, respectvely, and 1 c t, f t < t 1 t1 = t1, f t > t1 t (34) The FE model of the double fllets uses a combnaton of shell/plate and beam elements; rgd lnks are used to connect the shell/plate elements and the beam elements (see Fg. 0). The method descrbed n Secton.1.1 can be used to determne the thcknesses and the lengths of the shell/plate elements n the fllet FE model. The area moments of nerta and the torsonal stffness factor of the beam elements can also be determned as dscussed n Secton.1.. The shell/plate and beam elements along wth the rgd lnks n the fllet cross-secton have 4 DOF. The same number of DOF s used n the brdge model n Ref. [3], but the shell/plate and beam element model here can capture both the n-plane and out-of-plane effects of the double fllets. A shell/plate element can be dvded nto two or more shell/plate elements. (a) Fg. 0 (a) The shell/plate and beam element model of the double fllets wth parts of connectng walls; and the front vew of the FE model of the double fllets n (a).
23 3. RESULTS AND DISCUSSION 3.1 Valdaton of Curved-beam Models of Sngle Fllets An mportant step n our analyss s to fnd a good approxmaton for the centrodal axs n the curved-beam fllet model and for the fxed boundary. We demonstrate here the accuracy of the curved-beam model wth the selected centrodal axs. We create a fnely meshed plate element model for a half sngle fllet wth unt wdth (see Fg. 8) usng ABAQUS 6.5, as shown n Fg. 1, to check the accuracy of our curved-beam model. The radus of the fllet and the thckness of the walls are m. A dstrbuted shear force p =10,000 N/m acts on the tangent secton of the half fllet. The rotatonal dsplacement of the centrod B of the tangent secton, calculated from the plate element model, s rad, and that from the curved-beam model s rad; the two dsplacements have only a 1.17% dfference. When we replace the shear force by a moment of 10,000 Nm n the clockwse drecton, the rotatonal dsplacement of B calculated from the FE model s rad, and that from the curved-beam model s rad; the two dsplacements have a dfference of.56%. (a) Fg. 1 (a) The plane stran FE model of a half sngle fllet subjected to a dstrbuted shear force on the tangent secton; and the deformaton of the half fllet n (a). h Note that for a curved beam wth 0< < 0.5, the dfference between ARh and I can be neglected [5]. R I Fgure shows the values of ARh and h R versus α, for fve dfferent values of r 0 between 0.3 and 5. We t I notce the varaton of ARh s very small (from to 1) n the range of 0.3 r0 < < t 5 h, and all of the R values are less than Hence ARh can be replaced by I n calculatng the dsplacements of the tangent secton, especally for a fllet wth a large value of r 0 t ; ths can sgnfcantly smplfy the analyss.
24 (a) Fg. (a) I ARh versus α π r0 for varous t ; and h R versus α π r0 for varous t. 3. Numercal and Expermental Results for L-shaped Beams The methodology developed n Secton.1 s appled to two thn-walled L-shaped beams wth free boundary condtons. In Secton 3..1, the natural frequences of an L-shaped beam calculated from the shell/plate and beam element model are compared wth those from the ntensve sold element model. All the shell/plate and beam element models n ths work are created usng SDTools [10]. In Secton 3.., the natural frequences and mode shapes of an alumnum L-shaped beam specmen calculated from the shell/plate and beam element model are compared wth ts expermental results. For all the thn-walled beams n ths work, a wall s modeled by shell/plate elements whose thckness equals the wall thckness (see Fgs. 16(a), 0(a), and 5) Comparson between the shell/plate and beam element model and the sold element model The dmensons of the L-shaped beam are t = m, r = m, l = 0.8 m, b= 0.06 m (35) 0 There s no fllet at the outer end of each flange. The calculated parameters for the shell/plate and beam element model of the fllet are: L = m, s = m, x = y = m, I 1 = m, I = m, = m, and K beam A beam = m 5. The shell/plate and beam element model of the L-shaped beam has 17,98 DOF, whch s only 3.7% of those of the sold element model, and the DOF of the fllet n the shell/plate and beam element model are only.40% of those n the sold element model. The natural frequences of the frst 19 elastc modes
25 calculated from the two FE models are compared n Table 1; the maxmum dfference s 1.83% for all the modes. The mode shapes of the L-shaped beam from the two models are essentally the same, wth the 17th elastc mode shapes from the two models shown n Fg. 3. Table 1 The natural frequences of the frst 19 elastc modes of the L-shaped beam calculated by the shell/plate and beam element model and the sold element model Mode Shell/plate Sold Dff. Mode Shell/plate Sold Dff. number and beam element model (Hz) element model (Hz) number and beam element model (Hz) element model (Hz) % % % % % % % % % % % % % % % % % % % (a) Fg. 3 The 17th elastc mode shapes of the L-shaped beam calculated from the shell/plate and beam element model (a) and the sold element model
26 3.. Expermental valdaton Most dmensons of the L-shaped beam specmen n Fg. 4(a) are shown n Fg. 4 and t r0 r out = m, = m, = m (36) where r and r were measured from enlarged photographs of the fllets. There are a total of 18 holes wth a 0 out radus of m on the two flanges of the specmen. The specmen was placed on two soft foams at the two ends of the beam to smulate the free-free boundary condtons (see Fg. 4(a)). Modal testng was conducted on the specmen; the rovng hammer method was used. The specmen was excted at selected ponts on each flange surface, as shown n Fg. 4(a), usng a PCB 086D80 mpact hammer. The response of the specmen was measured usng a Polytec OFV 353 laser vbrometer wth an OFV 3001 controller to avod mass loadng. The exctaton and measurement drectons are perpendcular to the flange surfaces. Two measurement ponts shown n Fg. 4(a) were used to capture all the possble modes and to dstngush the 14th and 15th modes, whose natural frequences are very close (see Table ); the 14th mode has a relatvely large dsplacement at the left measurement pont and the 15th mode has a relatvely large dsplacement at the rght measurement pont. The expermental data were collected usng a 36-channel LMS spectrum analyzer. Three mpact tests were averaged at every exctaton pont to ensure repeatable results wth a good coherence. The natural frequences and mode shapes of the specmen were extracted from the measured frequency response functons usng LMS Test.Lab. The soft foams are consdered to be a vald approxmaton of the free boundary condtons because the hghest measured rgd-body mode natural frequency s 1%, whch s wthn the 10-0% range defned n Ref. [11], of the frst elastc mode natural frequency. (a)
27 Fg. 4 (a) An alumnum L-shaped beam specmen restng on two soft foams at the two ends, wth the exctaton and measurement ponts marked; and ts dmensons. The fllet at the outer end of each flange s modeled by shell/plate elements (see Fg. 5); ther equvalent thckness, whch s m, s so found that the mass of the specmen at the outer end of the flange s modeled correctly. The calculated parameters for the shell/plate and beam element model of the fllet between the flanges are : L= m, s= m, x = y = m, I 1 = m, I = m, K beam = m , and = m. The shell/plate and beam element model of the A beam specmen has 54,5 DOF. The measured and calculated natural frequences for 8 elastc modes are shown n Table, and the assocated MAC matrx s shown n Table 3. Note that the 1st elastc mode of the L-shaped beam s not ncluded n the comparson because t s a longtudnal mode, whch depends only on the cross-sectonal areas of the L-shaped beam and s not affected by the fllet modelng technque dscussed n ths work. The maxmum error between the measured and calculated natural frequences for all the modes s less than 1.93%, and the assocated MAC values are all above 95%. Note that the sold element model of the specmen s not created because over 700,000 DOF are needed to accurately model the 8 elastc modes, whch s out of the computatonal capacty of a regular personal computer. Note also that the DOF of the shell/plate and beam element model can be sgnfcantly reduced f the number of modeled modes and the requred accuracy are reduced. For nstance, we can use 5,754 DOF to model the frst 10 elastc modes, wth the maxmum error between the measured and calculated natural frequences less than % and the assocated MAC values all above 94%. The measured and calculated mode shapes of the 4th elastc mode are shown n Fg. 6. Note that some off-dagonal entres n the MAC matrx are hgher than 0% because usng 44 exctaton ponts cannot fully capture the dfferences between the measured mode shapes, especally of the hgher modes. For nstance, Fg. 7 shows the measured and calculated mode shapes of the nd and 9th elastc modes. Whle we can dstngush the two calculated mode shapes, t s dffcult to dstngush the two measured mode shapes, whch s the reason why the off-dagonal entres assocated wth the two modes are about 90%. If we gnore the presence of the fllet
28 between the flanges, the errors between the measured and calculated natural frequences of the 1st and 16th elastc modes can be 10.88% and 14.93%, respectvely. The mode shapes of the 1th and 18th elastc modes, calculated from the models that consder and gnore the fllet between the flanges of the L-shaped beam, as shown n Fg. 8, are completely dfferent. Because the natural frequences of the 1th and 13th elastc modes are very close (see Table ), the two mode shapes are shfted when we gnore the presence of the fllet. The same stuaton occurs for the 18th and 19th modes, whch have close natural frequences. (a) Fg. 5 (a) The shell/plate and beam element model of the alumnum L-shaped beam; and an enlarged vew of the FE model. Table The measured and calculated natural frequences (NF) of the alumnum L-shaped beam Mode Measured Calculated Error Mode Measured Calculated Error number NF (Hz) NF (Hz) number NF (Hz) NF (Hz) % % % % % % % % % % % % % % % %
29 % % % % % % % % % % % % Table 3 Entres of the MAC matrx n percent correspondng to the 8 measured and calculated mode shapes of the alumnum L-shaped beam; the horzontal and vertcal mode numbers correspond to the measured and calculated modes, respectvely.
30 (a) Fg. 6 The measured (a) and calculated mode shapes of the 4th elastc mode (a) Fg. 7 The measured and calculated mode shapes of the nd (a) and 9th elastc modes
31 (a) Fg. 8 The mode shapes of the 1th (a) and 18th elastc modes calculated from the models that consder (left) and gnore (rght) the fllet between the flanges of the L-shaped beam 3.3 Numercal Results for Thn-walled Beams wth Other Types of Cross-sectons The methodology developed s also appled thn-walled beams wth other types of cross-sectons and free boundary condtons. Fgure 9(a) shows the densely meshed sold element model of a thn-walled beam wth a rectangular hollow secton (.e., a box beam). The length of the box beam s 0.9 m, and the cross-sectonal dmensons of the beam are shown n Fg. 9. (a) Fg. 9 (a) The ntensve sold element model of the box beam; and ts cross-sectonal dmensons.
32 The calculated parameters for the shell/plate and beam element model of the box beam are: L = m, s = m, x = y = m, I 1 = m, I = m, K beam = m, and = m. The shell/plate and beam element model has 58,176 A beam DOF, whch s 9.47% of that of the sold element model; the DOF of the fllet n the shell/plate and beam element model are 5.74% of that n the sold element model. The natural frequences of the frst 19 elastc modes, whch are calculated from the shell/plate and beam element model and the sold element model, are shown n Table 4; the maxmum dfference s 1.95%. The mode shapes of the box beam from the two models are essentally the same, and those of the 18th elastc mode are shown n Fg. 30. Table 4 The natural frequences of the thn-walled box beam calculated from the sold element model and the shell/plate and beam element model Mode Shell/plate Sold Dff. Mode Shell/plate Sold Dff. number and beam element model (Hz) element model (Hz) number and beam element model (Hz) element model (Hz) % % % % % % % % % % % % % % % % % % % (a) Fg. 30 The mode shapes of the 18th elastc mode of the box beam calculated from the shell/plate and beam element model (a) and the sold element model
33 The last thn-walled beam we have modeled usng the shell/plate and beam element model s a thn-walled beam wth an I-shaped cross-secton (see Fg. 31). The length of the I-shaped beam s 0.8 m, and the cross-sectonal dmensons of the beam are shown n Fg. 31. There are two sets of double fllets for the I-shaped beam. The calculated parameters for the shell/plate elements that connect to the sde walls of the beam, and for the beam elements, n the shell/plate and beam element model are: L = m, s = m, x = y = m, I 1 = m, I = m, = m, K beam and A beam = m. The thckness and the length of the shell/plate elements that connect to the central wall of the beam are calculated to be m and m, respectvely. (a) Fg. 31 (a) The ntensve sold element model of the I-shaped beam; and ts cross-sectonal dmensons. The natural frequences of the frst 17 elastc modes, whch are calculated from the shell/plate and beam element model and the sold element model, are shown n Table 5; the maxmum dfference s 1.99%. The DOF of the shell/plate and beam element model and the sold element model are 64,84 and 607,818, respectvely; the DOF of the I-shaped beam and the double fllets are reduced by 89.33% and 93.8%, respectvely, by usng the shell/plate and beam element model. The mode shapes of the I-shaped beam from the two models are essentally the same, and those of the 14th elastc mode are shown n Fg. 3. Table 5 The natural frequences of the thn-walled I-shaped beam calculated from the sold element model and the shell/plate and beam element model Mode Shell/plate Sold Dff. Mode Shell/plate Sold Dff. number and beam element model (Hz) element model (Hz) number and beam element model (Hz) element model (Hz) % % % %
34 % % % % % % % % % % % % % (a) Fg. 3 The mode shapes of the 14th elastc mode of the I-shaped beam calculated from the shell/plate and beam element model (a) and the sold element model 3.4 Remarks The key step to creatng the fllet model s to calculate ts equvalent stffness. Assumng the equvalent stffness can be decomposed nto the n-plane and out-of-plane effects allows us to develop an accurate analytcal model of the fllet. Wth the analytcal equvalent stffness, we can desgn dfferent FE models for the fllet. We use the shell/plate and beam elements here for two reasons. Frst, the shell/plate and beam element model has a smple stffness matrx, and the parameters of the model can be easly determned. Second, the combnaton of shell/plate and beam elements provdes enough adjustable parameters to model both the n-plane and out-of plane effects of the fllet; more adjustable parameters can be ntroduced f more dsplacements need to be matched. For nstance, n analyzng the n-plane effects of a sngle fllet, we can change the orentaton of the shell/plate element to match a translatonal dsplacement of the tangent secton; the rotatonal dsplacement wll not be affected by changng the orentaton of the shell/plate element. 4. CONCLUSION Ignorng the presence of a fllet n a thn-walled beam can sgnfcantly change ts natural frequences and mode shapes. A new and accurate method that uses shell/plate and beam elements model a fllet n a thn-walled beam s developed for ts global dynamc and statc analyses. Wth a small number of DOF n the fllet cross-secton, the
35 model can capture both the n-plane and out-of-plate effects of the fllet on the thn-walled beam. The numercal results for dfferent types of thn-walled beams show that the shell/plate and beam element models can provde the same level of accuracy as the ntensve sold element models, but can reduce the DOF of the fllets by over 93%. The calculated natural frequences of an alumnum L-shaped beam specmen from the shell/plate and beam element model are wthn a % error from the measured ones for 8 modes, and the assocated MAC values are all above 95%. The methodology can be used to create an accurate FE model of a large structure consstng of flleted thn-walled beams, wth an acceptable computatonal cost. ACKNOWLEDGEMENT Ths work s supported by the Natonal Scence Foundaton through Grant No. CMS and the Amercan Socety for Nondestructve Testng (ASNT) through the 007 ASNT Fellowshp Award. REFERENCES [1] Wllam, B. B., 1994, A Frst Course n the Fnte Element Method, nd Edton, Rchard D. Irwn, Inc., pp [] Seuglng, R. M. and Brown, A. M., 001, Modelng of fllets n thn-walled structures for dynamc analyss, Proceedngs of the 19th Internatonal Model Analyss Conference, Kssmmee, FL, pp [3] Brown, A. M. and Seuglng, R. M., 004, Usng plate fnte elements for modelng fllets n global response analyss, Fnte Elements n Analyss and Desgn, Vol. 40, pp [4] Frswell, M. I. and Mottershead, J. E., 1995, Fnte Element Model Updatng n Structural Dynamcs, Kluwer Academc Publshers, Dordrecht, The Netherlands, pp [5] Young, W. C., 1989, Roark s Formulas for Stress and Stran, 6th Edton, McGraw-Hll, Inc., New York. [6] Bores, A. P., Schmdt, R. J. and Sdebottom, O. M., 1993, Advanced Mechancs of Materals, 5ht Edton, John Wley & Sons, Inc., New York, p [7] Ugural, A. C. and Fenster, S. K., 1995, Advanced Strength and Appled Elastcty, 3rd Edton, Prentce-Hall, Inc., Upper Saddle Rver, New Jersey. [8] Ragab, A. R. and Bayoum, S. E., 1998, Engneerng Sold Mechancs, CRC Press LLC, New York, pp [9] Weaver, W. Jr., Tmoshenko, S. P., and Young, D. H., 1990, Vbraton problem n Engneerng, 5th Edton, John Wley & Sons, Inc., Sngapore, pp [10] Bálmes, E., 006, Structural Dynamcs Toolbox, Users Gude, Verson 5.3, Scentfc Software Group. [11] Ewns, D. J., 000, Modal Testng: Theory, Practce and Applcaton, nd Edton, Research Studes Press Ltd., Baldock, Hertfordshre, UK, pp
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