Wrinkling Limit in Tube Bending

Transcription

1 Xi Wang Jian Cao Depatment of Mechanical Engineeing, Nothwesten Univesity, Evanston, IL 6008 Winkling Limit in Tube Bending Thin-walled tube bending has found many of its applications in the automobile and aeospace industies. This pape pesents an enegy appoach to povide the minimum bending adius, which does not yield winkling in the bending pocess, as a function of tube and tooling geomety and mateial popeties. A doubly-cuved sheet model is established following the defomation theoy. This appoach povides a pedictive tool in designing/optimizing the tooling paametes in tube bending. DOI: / Intoduction A tube possesses a combination of light weight and high stiffness, which has attacted many applications in the aeospace, automobile, oil and vaious othe industies. In the auto industy, fo example, tube hydofoming has been identified as one of the key technologies to educe vehicle weight while inceasing the stiffness and integity of automobiles 1. A cucial challenge in hydofoming is the optimization of tool and pocess conditions. A typical tube hydofoming pocess, like manufactuing exhaust manifolds, engine cadles and fames, involves bending of tubes as the fist step followed by hydofoming in a closed die. Duing the otay-daw tube bending shown in Fig. 1 Stelson and Lou, the clamp die holds the tube which is bent tightly against the bending die at the leading end of the bende. The pessue die defoms the tube into the desied shape. A mandel is used to suppot the inside of the tube and pevent the collapse of the tube Li et al. 3. As a esult of bending, axial compessive stess is developed in the tube section nea the bending die. Thus, local winkles may be initiated, as winkling is a phenomenon of compessive instability. The defomation and spingback in otaydaw tube bending has been studied by Li and Stelson 3, Stelson and Lou, etc. Howeve, the winkling poblem in tube bending has not eceived much attention. Cuent industial pactice is to add enough design toleance into the poblem so that no winkles will be fomed in tube bending. With the eve-inceasing demand of low-cost manufactuing and design fo manufactuability, a eliable pediction tool is needed. Reseach effots on the pediction of winkling have been made in the last fifty yeas. The analytical solution can povide a global view in tems of the geneal tendency and the effect of individual paametes on the onset of winkling and can be achieved in an almost negligible computational time. Howeve, past analytical wok has concentated on some elatively simple poblems such as a column unde axial loading Shanley 4, a cicula ing unde inwad tension, and an annula plate unde bending with a conical punch at the cente, etc. Plastic bifucation analysis is one of the most widely used analytical appoaches to pedict the onset of winkling. Hutchinson and Neale 5 and Neale and Tuǧcu 6 studied bifucation phenomenon of doubly cuved sheet metal by adopting Donnell-Mushtai-Vlasov DMV shell appoximations. The investigation was applicable to the egions of the sheet which ae fee of any suface contact. Howeve, all the above analyses neglected the bounday condition along the edge of the egion being examined fo winkling. Fatnassi et al. 7 pefomed a theoetical investigation to pedict the nonaxisymmetic buckling in the thoat of cicula elastic-plastic tubes subjected to a nosing opeation along a fictionless conical die. The buckling point and associated modes ae detemined by Hill s bifucation theoy in conjunction with a nonaxisymmetic buckling mode. Pue bending of inelastic cylindical shells have been caied out expeimentally and numeically by Coona and Kyiakides 8, Kyiakides and Ju 9, etc. A thinne shell was found to develop shot wavelength peiodic ipples on the compessed side of the shell as shown in Fig. and the shell buckled locally and collapsed soon afte the appeaance of the winkles. Theefoe, thin-walled shell bending is limited by vaious shell buckling modes in addition to the limit load instability. Ju and Kyiakides 10 numeically investigated the instabilities of long cicula shell bending, which involves bifucation into shot wavelength ipples, localization following a natual limit load and inteaction of the two. They used Sandes shell kinematics and the pinciple of vitual wok to simulate the gowth of axial ipples and the localization phenomena. Coona and Vaze 11 addessed the esponse, which was buckling and collapse of long thin-walled squae tubes unde pue bending, analytically and expeimentally. Rayleigh- Ritz type fomulations wee developed based on the pinciple of vitual wok to pedict the esponse of the tubes and the citical cuvatue at which the ipples appea. As shown in Fig. 1, the bent tube is esticted by tooling and it is found that the bounday estictions ove the examined aea fo winkling ae vey citical to the onset of winkles in the pevious wok Wang and Cao Howeve, these above studies ae concened with long shells without any bounday estiction at the ends. Contibuted by the Mateials Division fo publication in the JOURNAL OF ENGI- NEERING MATERIALS AND TECHNOLOGY. Manuscipt eceived by the Mateials Division July 5, 000. Guests Editos: Jian Cao and Z. Cedic Xia. Fig. 1 Rotay-daw bending with mandel Stelson 430 Õ Vol. 13, OCTOBER 001 Copyight 001 by ASME Tansactions of the ASME

2 employed as detailed in Hutchinson and Neale 5 and Neale and Tuǧcu 6. At the instant of buckling, the in-plane components of Lagangian stain tenso at a distance x 3 fom the middle suface of the cuved sheet can be appoximated by E x 3 () whee the Geek indices ange fom 1 to, E and epesent the stetching and bending stain which ae given by Fig. Winkling on compession side of shell in pue bending Ju and Kyiakides 9 The enegy method has been anothe appoach to analytically investigate the buckling poblem such as flange winkling in Cao and Wang 15, Wang and Cao 16, etc. Recently, Wang and Cao 1 14 poposed a winkling citeion fo sheet winkling without nomal constaints by using an enegy appoach consideing the bounday estictions of the examined aea. They applied this appoach to the cases of a staight side-wall and a cuved wall with an infinite cuvatue in anothe diection in sheet metal foming pocesses. The pedictions ageed vey well with expeimental esults in all the cases examined. In this pape, we aim to extend this appoach to the sheet with double cuvatues, i.e., the winkling poblem in tube bending. In the pesent pape, an enegy appoach is established to povide a stess-based citeion fo the geneal double-cuvatue sheet unde compession. The effective dimensions ove the egion undegoing compessive hoop stess ae intoduced as dimension paametes. The citical buckling stess is obtained as a function of local cuvatues, mateial popeties, geometical dimensions, and stess atio. This citeion is then used to pedict the onset of winkling in tube bending. The effects of tube thickness, tube adius and mateial popeties on the minimum bending adius without the occuence of winkling ae discussed. Geneal Winkling Citeion This section aims to establish a geneal winkling citeion fo the onset of winkling in the doubly cuved sheet unde a plane stess condition. In the following analysis, the pebuckling stess state in the sheet ove the egion examined fo winkling is assumed to be at membane state, and thus, the shea stains and stesses ae ignoed. All the fomulations ae developed within the context of thin plate and shell theoy, theefoe, the thickness of the sheet and all the stess states though the thickness ae assumed to be unifom befoe buckling. Stains ae expected to be small and the chaacteistic wavelength is lage compaed to sheet thickness and yet small compaed to the adii of the cuvatues of the sheet such that the stain measues given by Donnell- Mushtai-Vlasov DMV appoximations can be adopted. Defomation theoy is employed in the analysis since popotional loading befoe buckling is assumed. The enegy method has been extensively employed in Timoshenko 17 to study the elastic buckling of thin plates and shells with vaious bounday conditions. In his enegy appoach, a deflected fom may be assumed fo the plate and the citical buckling condition can be assessed by equating the intenal enegy of the buckled plate, U, and the wok done by the in-plane membane foces, T. If the intenal enegy fo evey possible assumed deflection is lage than the wok poduced by membane foces, the sheet is consideed unde a stable equilibium condition. Hence, the stability condition can be expessed as TU (1) To obtain the intenal enegy fo evey possible assumed deflection, the fomulations fo a geneal doubly cuved sheet ae E 1 u,u, b w w, whee a comma denotes covaiant diffeentiation with espect to in-plane coodinates (x 1,x ), u and w ae the displacements in the in-plane diection (x 1,x ) and the buckling deflection nomal to the middle suface of the sheet, b is the cuvatue tenso of the middle suface in the pebuckling state. If a 3-D constitutive law with the fom of L is adopted, whee the moduli L ae defined in the Appendix, the elationships fo the membane stess esultants can be given by t/ N t/ and the bending moments ae given by t/ (3) dx 3 tl E (4) M x 3 dx 3 t3 t/ 1 L (5) The intenal enegy unde the assumption of DMV thin plate and shell theoy can be obtained as US M d N de ds (6) whee S is the egion of the sheet s middle suface ove which the winkles occu. By assuming the vitual displacements u 0 and using the defomation theoy, Eq. 6 can be simplified as U t3 w, w, ds t L b 4SL b w S ds (7) The extenal wok done by the membane foces acting in the middle plane of the sheet is epesented as T 1 SN 11 w,1 N w, ds (8) Fo a thin cuved sheet, the bounday condition o continuity condition along the edges of the egion being examined fo winkling stongly affects the citical buckling condition since the admissible deflection mode will be diffeent. By appopiately choosing the deflection fom to eflect the bounday estiction and equating the enegy UT, the citical conditions can be calculated analytically as a function of in-plane stess, mateial popeties, and geomety paametes. Geneally, the enegy equality is consideed ove the entie egion being examined and the stess field befoe winkling is assumed to be unifom ove the entie egion. Howeve, fom ou pevious wok on flange winkling and side-wall winkling in sheet metal foming, it is demonstated that the dimensions of the effective compessive aea ae citical to the initiation of winkles. Theefoe, this effective aea will be implemented in the above enegy integation in Eqs Buckling Condition in Tube Bending A typical tube used in tube bending, as shown in Fig. 1, has a atio of tube diamete () to tube thickness t geate than 10 and is bent aound the bending die with the adius of R d. Hence, Jounal of Engineeing Mateials and Technology OCTOBER 001, Vol. 13 Õ 431

3 Fig. 3 the thin-shell assumption employed in the pevious section is suitable. The thin-walled sheet in the bending aea is fee of intenal pessue leading to the assumption of unde plane stess defomation. The mandel acts to povide a nea-pefect clamping condition at the end and theefoe, consideing its effective compessive aea, the tube is simplified as a clamped cuved sheet within ( 1, ) and ( 1, ) in two cuvilinea diections as shown in Fig. 3. The nomal deflection, w, satisfies the following bounday conditions w0, w 0, at 1, (9) w0, w 0, at 1, Thus, the deflection of the plate is assumed to be of the fom ww 0 1cos m 1 11cos n 1 1(10) whee m is the wave numbe along the cicumfeential diection of the tube and n is the wave numbe in the diection. Having Eq. 10 as the assumed deflection fom, the intenal enegy U becomes U t3 4 1 L 33 1 L w w cos w R w L 33 1 w R 1 t L L w R cos w 4L 33 w L 3333 w R 1 R cos w w RRdd (11) whee R(R d sin ) With the stess esultants N 11 t and N t, the extenal wok done by the membane foces acting in the middle plane yields T t 1 Schematic of cuved sheet model in tube bending 1 1 w 1 w R Rdd (1) By substituting the assumed deflection in Eq. 10 into Eqs. 11 and 1, the enegy equality TU yields the solution fo the citical condition on the onset of winkling. In the case that T is always less than U fo all the admissable deflection foms, no winkling would develop at that bending condition, i.e., bending adius R d. Numeical Analysis and Results Citical Die Radius. The analytical model fo the bent tube section between the clamping die and pessue die o mandel as shown in Fig. 1 can be teated as a pue bending poblem. Fo simplicity, the thickness vaiation is ignoed, i.e., the out-of-plane stain z 0. Theefoe, fo the lage defomation of the bent tube, we have stain in the bending diection and cicumfeential stain as lnr/r d (13) The mateial is chaacteized as an elastic-plastic mateial following Swift s law K( 0 ) n. Hill s 1948 yield citeion is employed to descibe the nomal anisotopic behavio of the mateial and the anisotopy paamete R is defined as the atio of plastic stain in the width diection to plastic stain though thickness in a uniaxial tensile test. Thus, the stess distibution can be deived as 1R K 1R n 1R lnr/r d 0 (14) Fom Eq. 13, it can be seen that the ange of in the effective compessive aea is between 0,. By choosing 1 0, we have, which is the bending angle. In addition, it is found numeically that the minimum stess exists when the wave numbe in the cicumfeential diection m equals to 1. By substituting these simplifications into Eqs. 11 1, we obtain the enegy functional FUTI n4 n 4 I n 1 t I n I n0 1 t I 0 (15) whee 6R sin d (16a) I n 0 I n0 0 I 0 I 0 I n4 0 8 R 1cos d R 3 3 1cos L 3333d 8 4 (16b) (16c) cos R 1cos sin cos L R cos 1cos L 33 8 R sin cos 1cos R L 33 d (16d) 4R 3 cos 18R t 1cos L 6R sin cos 3 1R cos sin cos t R 18 t 1cos L 33 L 3333 d (16e) The enegy function in Eq. 15 is a function of die adius R d, wave numbe n as well as mateial popeties, thickness and ge- 43 Õ Vol. 13, OCTOBER 001 Tansactions of the ASME

4 Fig. 6 Nomalized citical die adius vesus nomalized tube adius fo AL6061-T6 and 304 stainless steel Fig. 4 Pofile of citical die adius fo AA6061-T6 omety paametes such as local cuvatues. The citical condition on the onset of winkling means that the enegy equality function in Eq. 15 has a nontivial solution fo wave numbe n, which yields I n 1 t I 4*I n4 * I n0 1 t I 0 (17) The Eq. 17 unde 0 gives the minimum bending adius R dc at which the winkles would not be initiated duing the tube bending, which is defined as the citical die adius. The coesponding wave numbe is calculated as the citical wave numbe n c I n 1 t I /I n4 (18) Effects of Geomety and Mateial Popeties. Fom the above analysis it appeas that the citical conditions at which winkling occus in the bent tube, i.e., the citical die adius and citical wavelength, ae affected by the tube adius, tube thickness and mateial popeties. The undestanding of these elationships will be helpful to pevent winkling duing tube bending pocesses. Figues 4 and 5 show the pofiles of citical die adius vesus tube adius and sheet thickness fo diffeent mateials. The mateial used in Fig. 4 is AA6061-T6 with K410 MPa, n0.085, and 0 0 while the mateial used in Fig. 5 is 304 stainless steel with K1356 MPa, n0.549, and It is illustated in both mateials that the minimum bending adius inceases with an incease in tube adius and a decease in tube thickness. In Li and Stelson 3, they did not obseve the winkles in the bending of a tube with the tube thickness of 0.73 mm and tube adius of 4.4 mm. The die adius they used is 8.7 mm. In ou pediction, fo such a case, the minimum bending adius without winkles should appoximate 13.0 mm, which fits to thei expeimental obsevation. Figue 6 displays the dependence of nomalized citical die adius, R d /t, on nomalized tube adius, /t fo AA6061-T6 and 304 stainless steel. Fo both mateials, the nomalized citical die adius tends to incease with tube adius. When the nomalized tube adius /t is small, i.e., close to 10, thee is no distinguishable diffeence between two mateials. Howeve, with inceased /t, the steel shows lowe citical die adius than AA6061-T6, i.e., has bette fomability in tube bending. Figue 7 epesents the effect of the stain hadening exponent n on the citical die adius. In Fig. 7, the paametes used ae t 0.7 mm, 5 mm and K800 MPa. It is seen that when the stain hadening exponent n is in the ange between , which ae typical values fo most cabon steel and aluminum mateials, citical die adius changes little with n. When n inceases beyond 0.5, the citical die adius deceases damatically. This also illustates that the stainless steel with n0.549 has a lowe citical die adius than aluminum as shown in Fig. 6. Figue 8 investigates the effect of the stength coefficient K on the citical die adius. In Fig. 8, the paametes used ae 5 mm, t0.7 mm, and n0.. The change of citical die adius with mateial stength coefficient K is elatively unnoticeable fo mateial with a highe stength, i.e., when K is geate than aound Fig. 5 Pofile of citical die adius fo 304 stainless steel Fig. 7 Effect of stain hadening exponent n on citical die adius fo tä0.7 mm, Ä5 mm, and KÄ800 MPa Jounal of Engineeing Mateials and Technology OCTOBER 001, Vol. 13 Õ 433

5 eliminate the costly tial-and-eo appoach. Expeiments on otay-daw bending will be conducted to veify the pesent appoach in the nea futue. Acknowledgments The financial suppot fo this wok is fom NSF gant No. DMI and is geatly appeciated. Fig. 8 Effect of stength coefficient K on citical die adius fo Ä5 mm, tä0.7 mm, and nä0. Fig. 9 Citical wavelength vesus tube thickness unde Ä5 mm fo AL6061-T6 and 304 stainless steel 600 MPa. Only when K is lowe than aound 600 MPa, it can be seen that a highe K yields a highe citical die adius. It appeas that the effect of stength coefficient K on citical die adius is elatively insignificant fo most high stength mateials. It is found that the citical wavelength is almost independent of die adius fo both AA6061-T6 and 304 stainless steel. Howeve, the citical wavelength lagely depends on tube thickness as shown in Fig. 9, whee tube adius is 5 mm. The citical wavelength inceases with thickness and AA6061-T6 has a little highe wavelength than stainless steel. Conclusions The pediction of winkling in tube bending pocesses has been a challenging topic. The analysis of a cuved tube can be chaacteized as buckling of a sheet with double cuvatues. The analytical model fo the onset of the winkling of an elastic-plastic doubly-cuved sheet is developed in this pape using the enegy method and the effective compessive aea, which is the actual aea unde compession in the tube obtained fom stess analysis eithe analytically o numeically. By defining the appopiate bounday conditions, the citical conditions can be obtained though enegy equality. A simple analytical model is established fo the tube bending poblem and the citical condition is obtained in tems of die adius and wavelength. It is found that the effects of tube adius and tube thickness ae significant on the minimum die adius without the occuence of winkling. The minimum bending adius inceases with an incease in tube adius and a decease in tube thickness. In addition, fo most high stength mateial with a stain hadening exponent n lowe than 0.5, the dependence of a minimum bending adius on mateial popeties is elatively mino. This appoach will povide a pedictive tool in designing/ optimizing the tooling and foming paametes and theefoe may Nomeclatue A, B, C paametes in the Voce s law E elastic modulus E s, s secant modulus and equivalent Poisson s atio E stetching stains I ni, I, I paametes (i0,,4) L tangential moduli L tangential moduli fo plane stess condition M bending moments N membane stess esultants R anisotopy paamete R d bending die adius b cuvatue tenso of the middle suface m, n wave numbe in the hoop diection and lateal/ adial diection adius of tube t thickness of the plate s ij stess deviato, i, j1, x, y, x i coodinates, i1,,3 u displacements in the in-plane diections w nomal deflection w 0 deflection amplitude T extenal wok done by membane foces U bending enegy stess components,,1, stain components ij Konecke delta bending stains y0 initial yield stess x, y stess components effective stess effective stain bending angle i, i span angle, stess components in cuvilinea coodinates Poisson s atio Appendix A Fo a 3-D constitutive law with the fom of L, the tangential moduli in the J defomation theoy ae given by L ijkl E s 1 s 1 ik jl il jk s 1 ij kl 1 s q s ijs kl (A1) whee s ij is the stess deviato and ij denotes the Konecke delta, the secant modulus E s /, and the equivalent Poisson s atio s is obtained fom s E s E E s E (A) The paamete q is given as E t q1 3E s E t 1 (A3) 3 The incemental moduli fo plane stess condition become L L L 33L 33 L 3333 (A4) 434 Õ Vol. 13, OCTOBER 001 Tansactions of the ASME

6 The covaiant deivative of the deflection is w, w, w, 0 w, 1 w Refeences w, 1 R w w, 1 R cos w w cos w R (A5) (A6) (A7) (A8) 1 Altan, T., et al., 1999, Fomability and Design Issues in Tube Hydofoming, Hydofoming of Tube, Extusions and Sheet Metal, Vol.1,K.Sieget, ed., pp Stelson, K. A., and Lou, H., 1995, Toleance Analysis of Thee-dimensional Tube Bending: Wost Case and Statistical Methods, Tansactions of NAMRI, SME, XXIII, pp Li, H. Z., Fageson, R., and Stelson, K. A., 1994, A Method of Adaptive Contol of Rotay-daw Thin-walled Tube Bending with Spingback Compensation, Tansactions of NAMRI, SME, XXII, pp Shanley, F. R., 1947, Inelastic column theoy, J. Aeonaut. Sci., 14, pp Hutchinson, J. W., and Neale, K. W., 1985, Winkling of Cuved Thin Sheet Metal, Plastic Instability, Pesses Ponts et Chaussées, Pais, pp Neale, K. W., and Tuǧcu, P., 1990, A Numeical Analysis of Winkle Fomation Tendencies in Sheet Metals, Int. J. Nume. Methods Eng., 30, pp Fatnassi, A., Tomita, Y., and Shindo, A., 1985, Non-axisymmetic Buckling Behavio of Elastic-plastic Cicula Tubes subjected to a Nosing Opeation, Int. J. Mech. Sci., 7, pp Coona, E., and Kyiakids, S., 1988, On the collapse of inelastic tubes unde combined bending and pessue, Int. J. Solids Stuct., 4, pp Ju, G. T., and Kyiakides, S., 199, Bifucation and localization instabilities in cylindical shells unde bending -II: pediction, Int. J. Solids Stuct., 9, No. 9, pp Kyiakides, S., and Ju, G. T., 199, Bifucation and localization instabilities in cylindical shells unde bending -I: expeiments, Int. J. Solids Stuct., 9, No. 9, pp Coona, E., and Vaze, S. P., 1996, Buckling of elastic-plastic squae tubes unde bending, Int. J. Mech. Sci., 38, No. 7, pp Wang, X., and Cao, J., 1999, On the onset of winkling of sheet with in-plane cuvatues and without nomal constaint, Tansaction of Noth Ameican Manufactuing Reseach Confeence, SME, XXVII, pp Wang, X., and Cao, J., 000, On the pediction of side-wall winkling in sheet metal foming pocesses, Int. J. Mech. Sci., 4, No. 1, pp Wang, X., and Cao, J., 001, Winkling analysis in shink flanging, to appea J. Manuf. Sci. Eng., August. 15 Cao, J., and Wang, X., 1999, An Analytical Model fo Plate Winkling unde Ti-axial Loading and Its Application, Int. J. Mech. Sci., 4, No. 3, pp Wang, X., and Cao, J., 000, An analytical model fo flange winkling in sheet metal foming, Jounal of Manufactuing Pocesses,, No., pp Timoshenko, S., 1961, Theoy of Elastic Stability, McGaw-Hill, New Yok. Jounal of Engineeing Mateials and Technology OCTOBER 001, Vol. 13 Õ 435