Dynamic Progressive Buckling of Square Tubes

Transcription

1 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber Dynai rogressive Bukling of Square Tubes Chih-Cheng Yang Departent of Autoation Engineering Kao Yuan Institute of Tehnology Lu-Chu, Kaohsiung, Taiwan 8, Republi of China Assistant rofessor Abstrat The low veloity axial ipat of thin-walle square tubes is taken as quasi-stati an the influene of inertia fores is, therefore, ignore. The wrinkles evelop progressively an the phenoenon is known as ynai progressive bukling. In the present stuy, ost of the aluiniu tubes suffer syetri rushing an soe of the il steel tubes suffer extensional rushing. Many rushe il steel tubes have a ixe type of rushing whih onsists of type I an type II oes an a general ixe ollapse oe is stuie. For low veloity ipat with large striking asses, the theoretial preitions for ynai progressive axial rushing of thin-walle square tubes gives a reasonable agreeent with the orresponing experiental results provie the effetive rushing istane is reognize an, for il steel, the influene of aterial strain rate sensitivity is retaine for the ynai behavior. However, for the ases of high veloity ipat with sall striking asses, the inertia effet is signifiant an the theoretial preition using a quasistati etho is unsuitable. Key wors: thin-walle square tube, quasi-stati, ynai progressive bukling, general ixe ollapse oe.. Introution As in the ase of the experiental stuies on the axial ipat of thin-walle square tubes in the previous stuy [,], the rushing oes uring low an high veloity ipats are ifferent. The axial ipat of thin-walle square tubes with low veloities, in whih the strain istribution is unifor [,4], is taken as quasi-stati, an the influene of inertia fores is, therefore, ignore [5]. The wrinkles evelop progressively an the phenoenon is known as ynai progressive bukling. The eforation oe is the sae as for stati loaing an is a quasi-stati response, sine inertia fores are neglete. However, these two types of responses are ifferent beause the influene of aterial strain rate sensitivity in ynai progressive bukling is iportant for ertain aterials, even when inertia effets are not [5]. This aspet is stuie theoretially. There are any stuies on the ynai bukling of thin-walle square tubes subjete to axial loas. A theoretial analysis using the basi ollapse eleents evelope by Hayuk an Wierzbiki [6] an Wierzbiki an Abraowiz [7] is reporte by Abraowiz an Jones [8] for the progressive bukling of thin-walle square box oluns subjete to axial loas, whih onsiers the effetive rushing istane, together with the influene of aterial strain rate sensitivity. It transpires that the four ifferent eforation oes [8,9], are ae up of one or two types of basi ollapse eleents shown in Figure, govern the behavior for ifferent ranges of the iensionless paraeter /h (h is wall thikness an is sie length of a square rosssetion, see Figure ). However, the ollapse oes of a square tube in the experiental stuy of Referene [,] are not always the sae with those entione in Referenes [8,9]. Most of aluiniu tubes have syetri rushing an soe of il steel tubes have extensional rushing. But, any rushe il steel tubes have ixe types whih onsist neither of six type I an two type II (asyetri oe A [8]) nor of seven type I an one type II (asyetri oe B [8]). Therefore, a general ixe ollapse oe is stuie. The theoretial ean ynai rushing loa is also opare with the experiental result.. Basi ollapse eleents The behavior of ynai progressive bukling on a square tube is often iealize as rigi-plasti sine the energy absorbe elastially is usually not signifiant an ynai (inertia) effets are neglete an the proble is treate as quasi-stati. The aterial is regare as rigiperfetly plasti with a onstant value of the flow stress σ. Sine large plasti strains present in areas of high urvature, elasti effets an be neglete an σ an be onsiere as an average flow stress. A kineatially aissible etho of analysis is use to evelop approxiate theoretial preitions with two basi ollapse eleents [8], as shown in Figure. The energy absorbe in the type I basi foling eleent is [8] Hb H E = M 6I + π + 4I, () h b where I =.555, I =.48, H is efine in Figure an b is the raius of the toroial shell eleent in the kineatially aissible veloity fiel [7,8,]. 47

2 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber The energy absorbe in the type II basi foling eleent is also exaine in Referene [8], but the onsieration of an iprove kineati ehanis in Referene [9] gives H E = M π + π + πh h whih is slightly ifferent to equation () in Referene [8]. / b =.7 (5) h h to preit / = 8., (6) M h where M = σ h /4. 4. Extensional ollapse oe The square tube whih suffere the extensional oe rushing is isplaye in Figure (b). The extensional ollapse oe onsists of one layer with four type II eleents shown in Figure. Therefore, by equating the internal energy absorption to the external work 4 H E 4 M H H h = = π + π + π, (7) an iniize with respet to H [9] / H =, (8) h h leas to ean stati rushing loa [9] / = 8π + π. (9) M h 5. General ixe ollapse oe Figure () shows a general ixe oe of a square tube. A general ixe oe onsists of n type I an n type II basi foling eleents, as shown in Figure, with a total initial height (n +n )H/. Thus, equating the internal energy absorption (n E + n E ) to the external work [ (n + n )H/] gives the ean stati rushing loa ( ) b 6n I + ( n + n ) π +, () h H = () M n + n H H 4n I + n π + n π b h when assuing that H an b are the sae for the syetri an asyetri layers present in the general. Syetri ollapse oe eforation oe. Equation () an also give the ean The iealize syetri ollapse oe, as shown in stati rushing loas ( ) of the four ollapse oes in Figure (a), for a square tube with wall thikness h Referenes [8,9] when set (n = 4, n = ), (n =, n = onsists of four type I eleents eah of whih absorbs the 4), (n = 6, n = ) an (n = 7, n = ), respetively for energy preite by equation (). The internal energy syetri, extensional, asyetri A an asyetri B equates to the external work oes. When iniizing equation () with b gives 4 Hb H E 4 M 6 I I H h 4 = = b + π +,() H 4I b =, () h I h where is ean stati rushing loa an H is the an with respet to H gives initial axial separation between the loations of the plasti hinges at the top an botto eges of a basi n ( ) I + nπ n + n π =. () foling eleent as iniate in Figure. Then, equation b h H () is iniize with respet to b an H [8] By substituting () into () leas to 4 / H b I b n + n I =.99 (4) n + n = h h h π h 6. () I h an 48 When setting n = 4 an n =, equations (5) an (4) an be eue fro equations () an (), respetively, for the syetri oe. Also, when setting n = an n = 4, equation (8) an be eue fro equations () together with () for the extensional oe. However, if we onsier /h in equation () as a variable, it is iffiult to obtain an analytial solution when n n. A etho of suessive approxiations is use to gain approxiate solutions in Referene [9] for the asyetri ases of (n = 6, n = ) an (n = 7, n = ). In orer to obtain a ore aurate solution of equation (), /h is onsiere as a known onstant. Therefore, the last ter on the left-han sie of equation () is a onstant when n an n are known an the solution is / b = A, (4) h h where A is a oeffiient whih epens on /h an also varies with various sets of n an n. Then, equation () is given as where H h = A h 4I I A /, (5) A =. (6) These two oeffiients, A an A, are slightly ifferent to those in equations (7), (8), () an () in Referene [9] for the asyetri ases of (n = 6, n = ) an (n = 7, n = ) beause 5 /h [9].

3 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber Finally, iniization of equation () gives / = B B B + π M h h +, (7) where 64n 4π B = IA +, (8) n + n A an B B 4n n + n A / = π (9) n n + n =. () 6. Effetive rushing istane A siplifie etho for the eterination of the effetive rushing istane of opresse box oluns is introue in Referene [] by using the geoetry of the opletely rushe basi foling ehaniss. It is shown in Referene [8] that the effetive rushing istane is δ =.7 H for the syetri ollapse oe an n B 6 n + n = 8.. () 7. Material strain rate sensitivity For ynai ipat tests, the influene of strain rate sensitivity shoul be retaine in the theoretial analysis for il steel square tubes beause its iportane [8]. The epirial Cowper-Syons uniaxial onstitutive equation / q σ & ε = + s () σ D is wiely use to assess aterial strain rate effets in strutures [5], where the supersripts an s enote ynai an stati behavior, respetively. The two aterial onstants, D = 7 s - an q = 7, are obtaine in Referene [] fro experiental tests on speiens for the ultiate stress il steel. The average strain rate in an axially rushe square tubes is estiate as [8] hv & () ε =, () b δ where δ is the effetive rushing istane an V is the δ e ean veloity of the striking ass uring rushing [8,9] =.77 () H V V =, () for the extensional an the asyetri ixe ollapse oes. It is transpires fro Referenes [7] an [] that V is the initial veloity at ipat. The final rolling raius the ifferenes in the rushing istane o not alter the b f is assoiate with the toroial eforation surfae an right-han sies of equations () an () [9]. Therefore, is erive in Appenix of Referene [8]. When using assuing [9] (4) an (5), it is shown that δ = H () / b f =.5 (4) an using equations (6) an () gives a iensionless h h ean rushing loa an / / = 5. (4) b f M h = A (5) b h h for a square tube with the syetri oe of for the syetri ollapse oe an general ixe eforation [8,9]. Siilarly, [9] ollapse oe, respetively, where A δ = H (5) b =.5 A. e an equations (9), (7) an () preit ean rushing Equation () together with equations (), (4), (5), loas [9] (), (), (4) an (5) preits / V = (6) ε& =. (6) M h an for the extensional ollapse oe, an V / / = B + B + B M h h ε& = A s (7) (7) for the syetri ollapse oe an general ixe for a square tube with a general ixe ollapse oe, ollapse oe, respetively, where A s =.6/A A b. It where ay be shown that [8,9] n V B = 8. IA + 6., (8) ε&.5 (8) n + n A n for the extensional ollapse oe. B = 6. A (9) n + n When replaing σ in equations (4), (6) an (7) an (where M = σ h /4) by σ given by equation () if σ Y 49 f

4 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber σ u an equations (6)-(8), the ean ynai rushing loas are / q / V = (9) M D h / q / V = (4) M D h M = + As V D / q (4) / / B + B + B h h for the syetri, extensional an general ixe ollapse oes, respetively, an where D = 7 s - an q = 7 aoring to the aterial test results in Referene [] on il steel speiens for the ultiate stress. 8. Mean plasti flow stress σ The plasti flow stress in the expression for the fully plasti oent M = σ h /4 ay be efine as [9] σ ( ε ) = ε σ ( ε ) ε ε (4) to preit a ore realisti value of average stresses uring the rushing proess. However, aoring to the experiental stuy for the stati tests on square tubes, the experiental ean rushing fore of a il steel tube is lose to the theoretial preition (UTS) with σ = σ UTS [], but, for an aluiniu tube, is lose to the theoretial preition (.%p) when σ = σ.% [4]. The flow stress oesn t reah the ultiate stress when the present aluiniu tube is opresse to the unstable state, instea, the flow stress reahes the.% proof stress, approxiately, then the tube bukles. While, when the il steel tube is opresse to the unstable state, the flow stress has passe through the.% proof stress an the axiu stress alost equals to the ultiate stress. Therefore, the ean flow stress σ for aluiniu square tubes is equal to the.% proof stress, approxiately, while for il steel square tubes, σ σ UTS. The two approxiations, σ σ y an σ σ UTS, have been use alreay in Referene [9]. 9. Coparison between theoretial an experiental results Soe experiental results an theoretial preitions for the ean ynai rushing loas are liste in Tables an for il an aluiniu speiens, respetively. The subsript LDV enotes the easureent with the ai of a laser Doppler veloieter syste. The eforation of a speien, L LDV, is given by integrating the laser Doppler veloieter signal. The liits of the integral were taken at initial ipat an at the first zero veloity, 5 an therefore, the laser Doppler veloieter result also inlues any elasti eforation of the speien. The absorbe energy E a is alulate fro initial ipat to the first zero veloity an E is the total kineti energy. The experiental ean ynai rushing fore an is obtaine by = L LDV ( x) x. (4) L LDV The theoretial preition of the ean ynai rushing fore is given by using Equations (9)-(4) for il steel tubes an Equations (4), (6) an (7) for aluiniu tubes. The theoretial preitions for ynai progressive axial rushing of thin-walle square tubes gives a reasonable agreeent with the orresponing experiental results provie the effetive rushing istane is reognize an, for il steel ases, the influene of aterial strain rate sensitivity is retaine for the ynai behavior. The influene of aterial strain rate sensitivity is iportant for il steel sine, as (n) isplaye in Table, where is the theoretial preition without strain rate effet whih is not as aurate as the theoretial preition with strain rate effet when opare to the experiental results. However, when inreasing the ipat veloity, the inertia effet beoes iportant an the ifferene between the experiental results an theoretial preitions inreases although any regular wrinkles are still fore on the tubes. This phenoenon an be observe in Figures 4 an 5 for il steel an aluiniu speiens, respetively. The orresponing experiental results are liste in Tables an 4 for the il steel an aluiniu speiens, respetively. The ratios of the experiental results to theoretial preitions inrease as an inrease of ipat veloity. When the ipat veloities are low with large striking asses, the ratios are lose to, i.e. a goo agreeent between theoretial preitions an experiental results, exept for the ase with only a few regular oplete wrinkles on a tube when ean rushing loa is strongly influene by the first peak loa [,4]. For high veloity ipats with sall striking asses, the inertia effet is iportant an, in aition, the strain istribution is not unifor all over a tube [,4], therefore the ifferene between experiental result an theoretial preition by using a quasi-stati etho inreases, as isplaye in Figures 4 an 5. As isusse in Referene [5], the transverse inertia effets shoul be iportant in general for higher veloity ipat. The oe of eforation of the struture uner ynai loaing ay be signifiantly ifferent fro the quasi-stati oe on aount of inertia fores evelope within the struture by the rapi aeleration whih parts of it experiene uring the ipat. Ta an

5 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber Callaine [6] onute ore etaile experiental on a siple plate-struture an pointe out that inertia is the oinant fator in the first phase (pure opression phase) of the ynai response to ipat, while, in ontrast, the seon phase, in whih erergy is issipate by the rotation of plasti hinges, is ore sensitive to strain-rate effets. However, aoring to the stuy on the energy-absorbing strutures by Su, Yu an Rei [7], the inertia effet appears to be the oinant fator in the entire eforation, rather than only in the first phase as suggeste in Referene [6], an the strain-rate effet enhanes the loa-arrying apaity of these types of strutures uring their entire ynai eforation. The obine effets of strain-rate an inertia ake the peak loa uh higher an the final isplaeent uh saller. When the ipat veloities are low with large striking asses, the ratios of the experiental results to theoretial preitions erease with an inrease of the ipat veloity beause the influene of the first peak loa is not signifiant when ore oplete wrinkles are fore. This phenoenon also ours in the test groups of saller striking asses, as shown in Figures 4 an 5 for the il steel an aluiniu tubes, respetively. However, sine, when inreasing the ipat veloity, the iportane of the inertia effet inreases whih akes the first peak loa higher, also, the oplete wrinkles in the tubes of sall striking ass test group are less than those of large striking ass test group when in the sae levels of the striking energy, as investigate in Referenes [,], the ratios for high ipat veloities (sall striking asses) are larger than for low ipat veloities (large striking asses) an the slopes of ratio to ipat veloity for the sae striking ass groups ten to erease at the higher veloities, as shown in the figures. That is, for the high veloity ipats with a sae sall striking ass, the influene of the first peak loa on the ean ynai rushing loa is signifiant when the eforation is sall (saller ipat veloity), while, although higher ipat veloity ay ause ore eforation whih ay reue the influene of the first peak loa on the ean rushing loa, the rising peak loa whih is ause by the inertia effet for higher veloity ipat inreases the ean rushing loa, therefore, the ratios o not vary signifiantly when inreasing the ipat veloity with a sall striking ass. In aition, although the strain rate effet is not signifiant for the present aluiniu tubes, exept for the inertia effet, the strain-harening effet is iportant for high spee ipat whih auses large strains in the tube [].. Conlusion The low veloity axial ipat of thin-walle square tubes is taken as quasi-stati an the influene of inertia fores is, therefore, ignore. The wrinkles evelop progressively an the phenoenon is known as ynai progressive bukling. In the present stuy, ost of the aluiniu tubes suffer syetri rushing an soe of the il steel tubes suffer extensional rushing. Many rushe il steel tubes have a ixe type of rushing whih onsists of type I an type II oes an a general ixe ollapse oe is stuie. For low veloity ipat with large striking asses, the theoretial preitions for ynai progressive axial rushing of thin-walle square tubes gives a reasonable agreeent with the orresponing experiental results provie the effetive rushing istane is reognize an, for il steel, the influene of aterial strain rate sensitivity is retaine for the ynai behavior. However, for the ases of high veloity ipat with sall striking asses, the inertia effet is signifiant an the theoretial preition using a quasi-stati etho is unsuitable. Aknowlegeent The author wishes to thank to Dr. R. S. Birh, for his unfailing support, while Mrs. M. White, Mr. G. Swallow an Mr. J. Curran provie invaluable assistane throughout the projet. Referenes. Yang, C. C., Dynai Axial Crushing of Aluiniu Square Tubes, roeeings of the 8th National Conferene on Mehanial Engineering, Taiwan, R.O.C., Vol., pp-4,.. Yang, C. C., Dynai Axial Crushing of Square Mil Steel Tubes, The 5th Conferene on Theoretial an Applie Mehanis, Taiwan, R.O.C., pp697-7,.. Yang, C. C., Strain Analysis in Bukling Behavior of Square Tube Uner Axial Ipat, The 7th ROC Syposiu on Frature Siene, Taiwan, ppa-5- ~A-5-, R.O.C.,. 4. Yang, C. C., Strain Analysis in Bukling Behavior of Mil Steel Square Tubes uner Axial Ipat, The 6th Conferene on Theoretial an Applie Mehanis, Taiwan, R.O.C., H, pp-,. 5. Jones, N., Strutural Ipat, Cabrige University ress, Cabrige, U.K., Hayuk, R. J. an Wierzbiki, T., Extensional ollapse oes of strutural ebers, Coputers & Strutures, 8(), , Wierzbiki, T. an Abraowiz, W., On the rushing ehanis of thin-walle strutures, Journal of Applie Mehanis, 5, 77-74, Abraowiz, W. an Jones, N., Dynai axial rushing of square tubes, International Journal of Ipat Engineering, (), 79-8, Abraowiz, W. an Jones, N., Dynai progressive bukling of irular an square tubes, International Journal of Ipat Engineering, 4(4), 4-7,

6 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber. Wierzbiki, T., Crushing behavior of plate intersetions, Strutural Crashworthiness, Es.: Jones, N. an Wierzbiki, T., Chapter, 66-95, Butterworths, Lonon, 98.. Abraowiz, W., The effetive rushing istane in axially opresse thin-walle etal oluns, International Journal of Ipat Engineering, (), 9-7, 98.. Yang, C. C., Material roperties of Square Tube, The 9th Funaental Sientifi Conferene of Chinese Are Fores Aaey, Taiwan, R.O.C., Vol., ppb~,.. Yang, C. C. an Jones, N., Stati Copression of Mil Steel Square Tubes, roeeings of the 6th National Conferene on Mehanial Engineering, Taiwan, R.O.C., Vol., pp5-58, Yang, C. C., Stati Axial Crushing of Aluiniu Square Tubes, The 77th Anniversary Conferene of Chinese Military Aaey, The Chinese Military Aaey, Taiwan, R.O.C.,. 5. Callaine, C. R. an English, R. W., Strain rate an inertia effets in the ollapse of two types of energyabsorbing struture, International Journal of Mehanial Sienes, 6(/), 689-7, Ta, L. L. an Callaine, C. R., Inertia an strainrate effets in a siple plate struture uner ipat loaing, International Journal of Ipat Engineering,, 49-77, Su, X. Y., Yu, T. X. an Rei, S. R., Inertia-sensitive ipat energy-absorbing strutures part II: effet of strain rate, International Journal of Ipat Engineering, 6(4), , 995. speien No Table Experiental results of il steel square tubes with a laser Doppler veloieter syste. (n L L LDV V LDV V E E a µ (kg) () () (/s) (/s) (kj) (kj) (kn) (kn) (kn) ) Moe of eforation L General ixe L Syetri B Extensional (n): theoretial preition without strain rate effet by using Equations (4), (6) an (7). speien No Table Experiental results of aluiniu square tubes with a laser Doppler veloieter syste. L L LDV V LDV V E E µ a (kg) () () (/s) (/s) (kj) (kj) (kn) (kn) Moe of eforation H Syetri H General ixe H Syetri 5

7 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber Table. Experiental results for il steel square tubes. Spe. µ L L V E No. () () (/s) (kj) (kn) = 5.55kg n I n II C E C S C G C G C G C G =.87kg Y E Y G C G Y E C E C G C G S G C G C G =.95kg Y G Y E C G Y E C E Y E C E Y E Y E C G =.44kg Y E Y E S E C G Y G C G =.8kg Y E n I: nuber of type I ollapse eleent; n II: nuber of type II ollapse eleent S: Syetri; E: Extensional; G: General ixe 5 Table 4. Experiental results for aluiniu square tubes. Spe. No. µ L L V E () () (/s) (kj) (kn) = 5.55kg n I N S N S N S N G N S N S N S N S =.87kg G S N S N S G S N G G G G S G S N S =.95kg N G N S N S N S N S =.44kg N S N S G S N S N G =.8kg N E N S N G N G =.6kg N G N S H S N E =.kg N S N S N S N S n I: nuber of type I ollapse eleent; n II: nuber of type II ollapse eleent S: Syetri; E: Extensional; G: General ixe n II

8 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber (a) (b) Figure. Basi ollapse eleents. (a) Type I an (b) Type II 7 5 strain gauge h 46 Figure. Geoetry of a square tube speien (in ). Figure. (a) (b) () Dynai rushing oe (ynai progressive bukling) for a thin-walle square tube. (a) Syetri rushing oe (N99); (b) Extensional rushing oe (Y); () General ixe rushing oe (S4). 54

9 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber. / _.. = 5.55kg =.87kg =.95kg =.44kg =.8kg V (/se) Figure 4. Ratio of experiental result to theoretial preition versus ipat veloity for il steel tubes.. / _ V (/se) = 5.55kg =.87kg =.95kg =.44kg =.8kg =.6kg =.kg Figure 5. Ratio of experiental result to theoretial preition versus ipat veloity for aluiniu tubes. 55

10 THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber 56