Analysis of cable membrane structures using the Dynamic Relaxation Method

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1 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 Poo, Pouga, June - Juy 4 A. Cuna,. Caeano, P. beo, G. üe (eds.) ISSN: -9; ISBN: Anayss of cabe membane sucues usng e Dynamc eaxaon eod. Hüne, J. áca, P. Fajman Dep. of ecancs, Facuy of Cv ng., Czec ec. Unvesy n Pague, Ing. oš Hüne, Pague, Czec ep. Dep. of ecancs, Facuy of Cv ng., Czec ec. Unvesy n Pague, Pof. Jří áca, CSc., Pague, Czec ep. Dep. of ecancs, Facuy of Cv ng., Czec ec. Unvesy n Pague, Doc. Pe Fajman, CSc., Pague, Czec ep. ema: mos.une@fsv.cvu.cz, maca@ fsv.cvu.cz, fajman@ fsv.cvu.cz ABSAC: s pape compaes e effecveness of dffeen scemes of dynamc eaxaon meod (D) fo e anayss of cabe and membane sucues. D s an eave pocess a s used o fnd sac equbum. D s no used fo e dynamc anayss of sucues; a dynamc souon s used fo a fcous damped sucue o aceve a sac souon. e saby of e meod depends on e fcous vaabes (.e. mass a dampng) and e sep. e effec of mass dsbuon aong e sucue s aso suded n e pape. g dffeen scemes D w be used n s pape. Scemes A and B ae based on e eoy of vscous dampng. Scemes C, D and ae based on e eoy of nec dampng (KD) w a pea n e mdde of e e sep and scemes F, G and H ae based on e eoy of KD w paaboc appoxaon. A cabe s appoxaed as a enson ba, a caenay (sevea enson bas) and a pefecy fexbe eemen. Fo membane sucues a angua eemen s consdeed. e cosen meods ae apped o sx consucons. e cabe sucues ae anayzed n xampes o, e membane sucues ae anayzed n xampes 4 o 6. e esus py a a s possbe o deemne e bes sceme. In s conex, may be noced a e meods based on nec dampng appea moe sabe and fase. Fo ba eemen, caenay and cabe eemens e esus confm a s benefca o dvde e same amoun of mass no a nodes of e sucue popoonay o e sffes node of e soved sucue (scemes C and F). Fo membane eemen s pefeed o use e nec dampng meod w e appoxaon of e nec enegy pea n e mdde of e e sep Δ. KY WODS: Dynamc eaxaon, Cabe sucues, embane sucues, Knec dampng. INODUCION e oad anayss of cabe-membane sucues s a geomeca nonnea pobem. Fo numeca modeng of cabe sucues can be used deazaon of e sucue no e eemens and nodes. e suface of membane sucue s dscezed no a sysem of jons and ange membane eemens. e edges of e ange fom e connecon beween e jons and ey ae caed ns. e jons can be dvded no wo goups suppoed o unsuppoed ones. e equbum of poson unsuppoed nodes oaded by e noda oads can be seaced eavey. Lage dspacemens of e sucue and sma defomaon of eemens ae consdeed. Sevea meods exs o sove ese sucues. e dynamc eaxaon meod (D) w be examned n s pape. e saby of e meod depends on e fcous vaabes (.e. mass a dampng) and e sep. s pape compaes e effecveness of dffeen scemes of D fo e anayss of cabe and membane sucues. e effec of mass dsbuon aong e sucue s aso suded n e pape. s pape deveops papes [] and [].. enson ba e ba connecs e endpons and caes ony posve noma foce. e nena foce (noma foce) n one ba eemen can be cacuaed accodng o e we-nown quaon (). A = ( s ), () s wee: s e Young s moduus of eascy, A s e coss-secona aea, s e dsance beween wo end jons n e cod decon (cuen eng), s s e un-eongaed eng of eemen (sac eng). If e foce s negave, en s equa o zeo. e deadweg of su as been assumed o be concenaed equay a s wo end jons. e used ba eemen can be seen n Fgue. LNS A cabe can be appoxaed as a enson ba, a caenay (sevea enson bas) and a pefecy fexbe eemen (wee bendng momens o zeo). Homogeneous maea w a consan coss-secon ougou s eng s assumed n a cases. Fo membane sucues a angua eemen s consdeed. Fgue. Ba eemen. 99

2 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4. Caenay e basc assumpon of s eoy s a e beavou of a cabe can be appoxaed by a few bas. ese bas ae neconneced by jons and susan ony posve noma foce. e beavou of ndvdua bas s descbed n Cape.. As found n [], fve bas ae we enoug o descbe coecy e caacescs of e cabe.. Cabe eemen e basc assumpon of e anayss of a fexbe easc cabe s a e cabe s egaded o be pefecy fexbe and s devod of any fexua gdy. Load on a cabe, wc mus ncude a eas sef-weg, s dsbued unfomy aong e cuve of e cabe wc s assumed o be a paaboa. e deaed anayss can be found n [4] and [5]. I s necessay fo e a of e sudy o use an nena foce, wc s aways posve and e poance of wc s sown n Fgue..4 embane eemen Fo a membane sucues e naua sffness eemen can be used fo cacuaon of nena foces. e ogna fomuaon of e naua sffness eemen s ceded o Agys [6] bu e fomuaon ee foows e wo of Banes [7] and oppng [8]. Fo e fomuaon of e naua sffness eemen a angua eemen s consdeed. s eemen as ony n-pane sffness so e eemen fomuaon s w espec o dspacemens n e oca coodnae decons. Usng equaons of equbum, s possbe o conve e suface sess wn e eemen no foces aong e sdes of e ange. Genea appcaon of s eemen s descbed e.g. n [8]. In s case e deazaon of a ypca eemen s as sown n Fgue wee e oca coodnae sysem s conveneny cosen suc a way a e axs concdes w e fs sde. e sesses n e eemen w espec o x and y decons, w σ z equa o zeo, ae e sandad pane sess fomuaon fo an soopc maea [9]. Fgue. Cabe eemen. Foce can be cacuaed eavey fom quaon (). g(,,, c, s A, ) = n c = c 4 ( a b ) ( a b ) c 8 b = n c s a () wee: s e Young s moduus of eascy, A s e coss-secona aea, s e dsance beween wo end jons n e cod decon (cuen eng), s s e un-eongaed eng of eemen (sac eng). s e ozona dsance beween e wo end jons, c s e veca sepaaon beween jon j and jon (can be negave), s e esuan of e veca unfom oad q acng vecay e ene eng of paaboc cuved cabe, we = qs. Fo easons of cay, quaon () noduces wo moe subsuons: a = 4c 4 4c, () b = 4c 4 4c. (4) In Fgue s: Fgue. embane eemen.,, s e eng of e edge,, s e ncnaon of e edge o e oca x axs, θ θ s e ncnaon of e edge o e oca x axs. e na foces, and of sdes, and ae defned fo membane eemen as: / = Ad and σ x σ = y τ xy wee: / ( ν ) ( ν ) ν ( ν ) ( ν ) ν / ( ν) ac ac c c ac ac a b ab ab ab σ x b σ y b τ xy c b Δ c Δ Δ b (5) (6) 9

3 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 A d ν s e aea of e membane eemen, s e cness of e membane, s e Young s moduus of eascy, s e Posson s ao, s e eng of e edge of e unoaded eemen, s e eng of e edge of e unoaded eemen, s e eng of e edge of e unoaded eemen, Δ, Δ, Δ s e eongaon of e edge, and Fuemoe, ey ae used subsuons (fo =,, ): and c a b = cos θ, (7) = sn θ, (8) = snθ cosθ, (9) = b. () c bc e dec sffness S of sde ( =,, ) s defned fo membane eemen as: S Ad =. () ( ) DYNAIC LAXAION e dynamc eaxaon meod (D) s an eave pocess a s used fo e sac anayss of sucues. D s no used fo e dynamc anayss of sucues; a dynamc souon s used fo a fcous damped sucue o aceve a sac souon. e eoy of s meod was fs descbed by Day []. Dung sevea yeas, e D ave been poved pogessvey. e nec dampng ecnque was suggesed by Cunda []. oppng [8] and Lews [] aso conbued o e nec dampng meod. Pacca exampes of e appcaon can be seen n [,,,,].. Pncpe e basc unnowns ae noda veoces, wc ae cacuaed fom noda dspacemens. e dscezaon fom ene w e sep Δ w be pefomed. Dung e sep Δ a nea cange of veocy s assumed. e acceeaon dung e sep Δ s us consdeed o be consan. By subsung e above assumpons e veocy fo jon n decon m ( x, y and z ) can be expessed n a new e pon ( Δ /) us: v ( Δ / ) = v ( Δ / ) / Δ C / Δ C / / / Δ C /,() wee: s e esdua foce a e noda pon, n e decon m and a e e, s e fcous mass a e noda pon and n e decon m, C s e fcous dampng faco fo e noda pon and n e decon m, ( / ) v Δ s e veocy a e noda pon n e decon m and a e e. e cuen coodnaes of e noda pon a e e nsan ( Δ ) may en be expessed as foows: x = Δ v. () ( Δ ) ( Δ / ) x Say, equaons may be wen fo e y and z coodnae decons. Fom e baance (beween exena and nena foces) n e node, we may cacuae e esdua foces ( m = x, y, o z ) fo e coespondng node a e e. z x y = P z = P x = P y x x j y y j z z j, (4) wee: s e ndex of e n (eemen o edge) eneng e noda pon. j s e second endpon on e n. P x P y P z x, s e exena oad a e noda pon n e decon x, s e exena oad a e noda pon n e decon y, s e exena oad a e noda pon n e decon z, s e esuan of e veca unfom oad fo eac n, y, z ae e cuen coodnaes of e noda pon, s e dsance beween wo end jons fo eac n. e nena foce fo eac n can be cacuaed fom quaon () fo e cabe eemen; fom quaon () fo e ba eemen; and fom quaons (5) fo e edge of membane eemen. If e sum of e foces n e n s ess an zeo so mus be se equa o zeo. In ode o sa cacuaons, e veocy a e e pon Δ / mus be cacuaed. Usng e na condons fo e e = wee v =, we oban: wee v ( Δ / ) Δ = ae esdua foces a e e =., (5) 9

4 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 A eac e nsan, s aso possbe o cacuae e nec enegy U ougou e sucue: U n ( Δ / ) n n p m ( Δ / ) ( v ) =, (6) wee n s e numbe of jons and p s e numbe of densons (D o D).. Scemes g dffeen scemes D w be used n s pape. Scemes A and B ae based on e eoy of vscous dampng []. Scemes C, D and ae based on e eoy of nec dampng (KD) w a pea n e mdde of e e sep [8] and scemes F, G and H ae based on e eoy of KD w paaboc appoxaon []... Sceme A Dscezed mass s cosen n s sceme e same fo eac node and a decons fom foows quaon (7): Δ = ( max S ), (7) wee S s e ages dec sffness of e - jon n e m decon. e sffness S of eac n (eneng no jon ) s fo e eemen (cabe o ba) epesened by wo componens G namey, e geomec sffness S and e easc sffness S. A G S = S S = s. (8) Sffness S D membane eemen soud be deemned fo eac edge of e equaon (). Hence, e foowng quaon (7) appes o e sffness of e node S : S = S, m, (9) wee S, m s e sffness S - fom quaon (8), especvey fom quaon () - dsbued n e m decon. Vscous dampng coeffcen C fo e woe sucue s cacuaed usng e coeffcen of cca dampng []: e eave agom conveges fases wen usng e ccay damped mode. In an undamped mode, e sucue w oscae aound s poson of equbum, and e vscous dampng coeffcen, nown as cca dampng, may be found fom quaon (): 4π C = S =, () NΔ wee N denoes e numbe of eaons equed o compee one cyce of oscaon. I may now be seen a n ode o oban e vaue of e vscous dampng coeffcen, an addona compue un s necessay, w C se o zeo... Sceme B In s sceme, e fcous vaues and C ae cacuaed fo eac jon sepaaey. e dscezed mass fo eac jon s cacuaed fom quaon (): wee Δ =, () S ( S, S S ) S = max,. () e vscous dampng coeffcen C fo e noda pon s cacuaed as foows: C = S = 8 () Δ.. Sceme C s sceme s based on e eoy of KD w a pea n e mdde of e e sep [8]. Wen e ecnque of nec dampng s empoyed, e vscous dampng coeffcen s aen as zeo. e sysem s boug o es by foowng a pocess soppng e eaons, weneve a pea n e nec enegy of e ene sysem s deeced, and en esang e compuaon fom e cuen confguaon, bu w zeo na veocy []. e coodnaes ae se o assumed o ave occued. en x ( Δ / ) x y z ( Δ / ) x wen e pea s Δ ( Δ / ) = x vx. (4) Say, equaons may be wen fo e y and z coodnae decons. e mass fo woe sucue s cacuaed fom quaon (7)...4 Sceme D s sceme s sa o e Sceme C, bu e dscezed mass fo eac jon s cacuaed sepaaey fom quaon ()...5 Sceme s sceme s sa o Sceme D bu masses ecacuaed afe eac esa of e nec enegy. ae..6 Scemes F, G and H ese scemes ae sa o Sceme C (especvey D and ) bu ee s used e eoy of KD w paaboc appoxaon []. e ace of nec enegy nea e pea can be appoxaed by a paaboc cuve. e coodnaes ae se o ( βδ) wen e pea s assumed o ave occued. en x wee x = x βδ v ( βδ ) ( Δ / ) x, (5) 9

5 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 K K K K K β =, (6) ( Δ / ) and wee K = U n s e nec enegy a e e ( Δ / ) ( Δ / ) pon ( Δ / ), K = U, K = U. n 4 XAPLS e cosen meods ae apped o sx sucues. e cabe sucues ae anayzed n xampes o, e membane sucues ae anayzed n xampes 4 o 6. e na geomey (no nena sess) s evden fom ndvdua fgues. e cacuaons wee emnaed wen e nec enegy of e sucue was ess an. -6 J and we e esdua foces of a e degees of feedom wee ess an e vaue (defned ndvduay fo eac exampe). e maxum numbe of eaons s cosen accodng o e eemens max. 5 eaons fo ba eemens max. 5 eaons fo cabe and membane eemens. e e sep s cosen Δ = s n a cacuaons. Sef-ceaed scps n ALAB (a) was used fo a cacuaons. e cacuaons wee caed ou on e compue ASUS pocesso AD -45 APU.65 GHz, memoy 4GB A. e esus of e cacuaons (e numbe of eaons and e CPU e) ae pesened n abes o xampe A suspended cabe ng sown n Fgue 4, wc as been dscussed n [], s anaysed ee o sow e accuacy and speed of e compuaons deveoped n s pape. n acs on e sucue. An unfom oad q =.55 - N/m acs on eac cabe. e coodnaes z of e unsuppoed nodes wee aways se o zeo. e vaue of a esdua foce s accuacy of e cacuaon s appoxaey ± cm. =. N. e abe. e numbe of eaons xampe. sceme/eemen ba caenay cabe vscous A 4-66 dampng B nec C dampng D Δ/ nec dampng βδ F G H abe. e of souon (CPU e n seconds) xampe. sceme/eemen ba caenay cabe vscous A.7-6. dampng B nec C dampng D Δ/ nec dampng βδ F G H xampe Hypa ne e exampe s aen fom Lews []: wo ses of sag ne cabes geneae a mode of a ypeboc paabood suface, sown n Fgue 5. Fgue 4. A pan vew of xampe. e sucue consss of 6 cabes conneced o 6 jons (-8 ae fee, A-H ae fxed) w an nne adus of 5 m and oue adus of 75 m. e sucues ave 8 ada cabes and 8 angena cabes. A cabes ave e same coss-secona aea A = m and e same Young's moduus = 7 GPa. e sac eng s of a ada cabes s 4 m, and a of a ng cabes s m. Any exena node oads Fgue 5. opoogy of Hypa ne. e sucue as 6 degees of feedom. e oad P z of.57 N s apped a a nena nodes, excep fo nodes 7,, and. e coss-secona aea of e cabes s.785 mm and Young s moduus s 4.8 N/mm. e pe-enson foce n a cabes s. N. e pevous anayss and expeena measuemens of e same sucue ae epoed n [4]. e vaue of a esdua foce s =. N. e accuacy of e cacuaon s appoxaey ± mm. 9

Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 abe. e numbe of eaons xampe.

sceme/eemen ba caenay cabe vscous A..4 5. dampng B 6.9-46.7 nec C.5 4. 55.8 dampng D.4-5.8 Δ/.. 88.4 nec dampng βδ F.5.7 55. G.5-66.9 H..9 86.

m, cabe 5 4.5 m, cabe 6.8 m. e oad s apped a jons 8 and 9 wee e foces exeed by e canvas ac. ese foces amoun o: P 8 z = N and P 9 = 6 N. e vaue of a esdua foce s =. N. e accuacy of e cacuaon s appoxaey ± 5 cm.

e canvas was epesened by foces exeed by e sucue s own weg and pesess n e mode. e bacgound fo e ceaon of e mode was e geodec suvey of e exsng sucue made n Novembe, see Fgue 7.

6 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 abe. e numbe of eaons xampe. sceme/eemen ba caenay cabe vscous A dampng B nec C dampng D 97-9 Δ/ nec dampng βδ F G 9-65 H abe 4. e of souon (CPU e n seconds) xampe. sceme/eemen ba caenay cabe vscous A dampng B nec C dampng D Δ/ nec dampng βδ F G H e paamees of cabes ae as foows: cabes 7: = 4 GPa, A = mm, q =. N/m cabes 8 : = 4 GPa, A = 8 mm, q =.7 N/m. e foowng cabe engs (seeced cabes) wee consdeed: cabe. m, cabe.68 m, cabe 4.4 m, cabe 4. m, cabe m, cabe 6.8 m. e oad s apped a jons 8 and 9 wee e foces exeed by e canvas ac. ese foces amoun o: P 8 z = N and P 9 = 6 N. e vaue of a esdua foce s =. N. e accuacy of e cacuaon s appoxaey ± 5 cm. 4. xampe am sop Fuemoe, one ea sucue was esed, see Fgue 6. Fgue 7. e pespecve vew of Baandov am sop. Fgue 6. Baandov am sop. I s a cabe-membane sucue w a cena symmey, ony one fou of e sucue was subjeced o modeng. e canvas was epesened by foces exeed by e sucue s own weg and pesess n e mode. e bacgound fo e ceaon of e mode was e geodec suvey of e exsng sucue made n Novembe, see Fgue 7. e measuemen of foces n cabes as been dscussed n [5]. e ancong of e cabes no anco bocs was modeed as a fxed suppo. e pon wee e cabes wee ancoed no anco bocs was modeed as a sdng jon aowng e jon s moon ony n e decon of e x and y axes. e sucua sceme of e sucue consdeed s evden fom Fgue 8. Fgue 8. A pan vew of a am sop. abe 5. e numbe of eaons xampe. sceme/eemen ba caenay cabe vscous A dampng B 5-9 nec C dampng D 5 - Δ/ 9 5 nec dampng βδ F G - 5 H

7 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 abe 6. e of souon (CPU e n seconds) xampe. sceme/eemen ba caenay cabe vscous A dampng B nec C dampng D Δ/ nec dampng βδ 4.4 xampe 4 F G H I s a membane sucue w 6 nodes (of wc ae unsuppoed) neconneced w 4 membane eemens. s sucue s sown n Fgue 9. e oad P z = N acs on bo unsuppoed jons. e paamees of membanes ae aways = 5 Pa, d = mm and ν =.. e vaue of a esdua foce s =. N. e accuacy of e cacuaon s appoxaey ± mm. I s a membane sucue w 5 nodes (of wc ae unsuppoed) neconneced w 6 membane eemens. s sucue s sown n Fgue. e oad P =.75 N. e paamees of membanes ae aways = 5 Pa, d = mm and ν =.. e vaue of a esdua foce s =. N. e accuacy of e cacuaon s appoxaey ± mm. abe 8. e numbe of eaons and e of souon (CPU e n seconds) xampe 5. sceme numbe of eaons e of souon vscous A 4.8 dampng nec dampng Δ/ nec dampng βδ B C.5 D F 49.8 G 4. H xampe 6 I s a membane sucue w 65 nodes (of wc 57 ae unsuppoed) neconneced w 96 membane eemens. e opoogy and na geomey of s sucue s sown n Fgue. Fgue 9. opoogy and na geomey of xampe 4. abe 7. e numbe of eaons and e of souon (CPU e n seconds) xampe 4. sceme numbe of eaons e of souon vscous A 66. dampng B nec C 4.7 dampng D 4.7 Δ/ 4.8 nec dampng βδ 4.5 xampe 5 F 4. G 4. H 6.8 Fgue. opoogy and na geomey of xampe 6. e oad P z = N fo nena jons and P z = 5 N fo a exena jons. e paamees of membanes ae aways = 5 Pa, d = mm and ν =.. e vaue of a esdua foce s = N. e accuacy of e cacuaon s appoxaey ± cm. abe 9. e numbe of eaons and e of souon (CPU e n seconds) xampe 6. sceme numbe of eaons e of souon vscous A 67.9 dampng B nec C dampng D Δ/ 6.8 nec dampng βδ F G H 59. Fgue. opoogy and na geomey of xampe 5. 95

8 Poceedngs of e 9 Inenaona Confeence on Sucua Dynamcs, UODYN 4 5 FINAL CONS e ovea anng of meods soed by e numbe of eos (sum of a exampes), e oa numbe of eaons and e oa CPU e ae sown n abes o. e esus py a a s possbe o deemne ceay e bes sceme. In s conex, may be noced a e meods based on nec dampng appea moe sabe and fase, wc confms e concusons pesened n [,,]. Fo ba eemen, caenay and cabe eemens e esus confm a s benefca o dvde e same amoun of mass no a nodes of e sucue popoonay o e sffes node of e soved sucue (scemes C and F). Fo membane eemen s pefeed o use e nec dampng meod w e appoxaon of e nec enegy pea n e mdde of e e sep Δ. 6 CONCLUSIONS I may be concuded a e Sceme C based on nec dampng w a pea n e mdde of e e sep and e equa mass dvded no a nodes popoonay o e sffes node as poved e mos compeensve esus. abe. e summay of esus ba eemen. sceme numbe of e of eos an eaons souon vscous A dampng B nec C dampng D Δ/. 6 nec dampng βδ F 6.5 G H 9. 5 abe. e summay of esus caenay. sceme numbe of e of eos an eaons souon vscous A dampng B 8 nec C dampng D 8 Δ/ nec dampng βδ F G 8 H abe. e summay of esus cabe eemen. sceme numbe of e of eos an eaons souon vscous A dampng B nec C dampng D 7 9. Δ/ nec dampng βδ F G H abe. e summay of esus membane eemen. sceme numbe of e of eos an eaons souon vscous A dampng B nec C 7. dampng D Δ/ nec dampng βδ F G H ACKNOWLDGNS e esus pesened n s pape ae oupus of e eseac pojec P5//59 - Cabe - membane sucues anayses suppoed by Czec Scence Foundaon and pojec SGS4/9/OHK// - Advanced agoms fo numeca modeng n mecancs of sucues and maeas suppoed by e Czec ecnca Unvesy n Pague. FNCS []. Hüne, J. áca and P. Fajman, Anayss of Cabe Sucues usng e Dynamc eaxaon eod, n B.H.V. oppng, P. Ivány, (dos), Poceedngs of e Foueen Inenaona Confeence on Cv, Sucua and nvonmena ngneeng Compung, Cv- Comp Pess, Sngse, UK, Pape 45,. do:.4/ccp..45 []. Hüne, J. áca, embane sucues - dynamc eaxaon, Poceedngs of e 4 Confeence Nano & aco ecancs. Pague, ČVU,, pp75-8. ISBN []. Hüne, J. áca, Cabe sucues - numeca anayss. Poceedngs of e d confeence: N Nano & aco mecancs. Pague, ČVU,, pp ISBN [4] H. Deng,.F. Jang, A.S.K. Kwan, Sape fndng of ncompee cabesu assembes conanng sac and pesessed eemens, Compues and Sucues, (8), 5, -, pp , ISSN [5] J. Kadčá, Saa nosnýc an vsuýc sřec. Academa, Pague, 99, 6 p. ISBN (n Czec). [6] J.H. Agys, ecen Advances n ax eods of Sucua Anayss, Pogess n Aeonauca Scences. Pegamon Pess, London, 964, 4. ISSN [7].. Banes, Fom and sess engneeng of enson sucues, Sucua ngneeng evew, 994, 6 (-4), pp 75-. ISSN [8] B.H.V. oppng, P. Ivány, Compue Aded Desgn of Cabe embane Sucues. Saxe-Cobug Pubcaons, Kppen, Sngse, Scoand, 7, p, ISBN [9] O. Zenewcz,. ayo, e fne eemen meod voume e bass. Buewo-Henemann, Oxfod, ngand. 5p. ISBN [] A.S. Day, An noducon o dynamc eaxaon. e ngnee. Januay 965, pp8-, ISSN -89. [] P.A. Cunda, xpc fne-dffeence meods n geomecancs, Poceedngs.F. conf. numeca meods n geomecancs. Bacsbug, 976, pp-5. [] W.J. Lews, enson sucues: Fom and beavou, omas efod, London,, p, ISBN []. ezaee-pajand, S.. Saafaz, H. ezaee, ffcency of dynamc eaxaon meods n nonnea anayss of uss and fame sucues, Compues and Sucues., (-), s 95-, ISSN [4] W.J. Lews, e ffcency of Numeca eods fo e Anayss of Pesessed Nes and Pn-joned Fame Sucues, Compues and Sucues,, pp79-8, 989. [5] P. Fajman,. Poá, Dopínání an nosné onsuce zasávy Baandov. Konsuce,, 4, ISSN 8-84 (n Czec). 96

Several Intensive Steel Quenching Models for Rectangular and Spherical Samples

Several Intensive Steel Quenching Models for Rectangular and Spherical Samples Recen Advances n Fud Mecancs and Hea & Mass ansfe Sevea Inensve See Quencng Modes fo Recangua and Speca Sampes SANDA BLOMKALNA MARGARIA BUIKE ANDRIS BUIKIS Unvesy of Lava Facuy of Pyscs and Maemacs Insue