Application of an explicit dynamic method for discontinuous analysis by using rigid bodies spring model

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1 Appliatio of a epliit dai ethod for diotiuou aali uig rigid odie prig odel *Tadao Yagi ad Norio Takeuhi, Graduate Shool of Egieerig ad Deig, Hoei Uiverit, Toko, Japa tadao.agi.9u@tu.hoei.a.jp takeuhi@hoei.a.jp ABSTRACT Thi paper propoe a epliit dai odel to olve equetiall large deforatio ehavior; thi odel i epaded to ilude the rigid odie prig odel (RBS that i uitale for aali of progreive failure uh a ople ladlide. Firt, we defie RBS aed o the hrid priiple of virtual work, ad the, we derie a ew RBS approah for the otat ehai ad fritio harateriti. Sie the approah i iilar to the ditit eleet ethod, it i poile to eplai the ehavior of the failure ehai after the foratio. Fiall, to ofir the appliailit of the tailit aali of ladlide, we evaluate the dai ehavior i ultiple lok odel.. INTRODUCTION I the aali of frature prole, the ue of a direte eleet odel RBS i effetive. (Kawai 977 RBS wa developed a a uerial odel for geeralizig the liit aali i platiit, i whih a truture to e aalzed i idealized a a aelage of rigid odie oeted oral ad tagetial prig. O the other had, uig the priiple of hrid-tpe virtual work, the author have applied a epliit ethod to eah eleet i their previou tud. (Yagi 05 I thi paper, we illutrate the forulizatio of RBS that ha ee epaded to ilude the ditit eleet ethod (DE. Further, we derie a ew RBS approah for the otat ehai ad fritio harateriti. Sie the approah i iilar to the ditit eleet ethod, it i poile to eplai the ehavior of the failure ehai after the foratio. I the preet work, it i iteded to ofir the appliailit of the tailit aali of ladlide evaluatig the dai ehavior i ultiple lok odel. Graduate Studet Profeor

2 . BRIEF FORUATIONS We itrodue a uidiar oditio ito the fraework of the variatioal equatio with agrage ultiplier λ, uh that the hrid-tpe virtual work equatio a e deried a follow, withi the udoai ad iteretio oudar N: e σ : grad( u dv N λ ( u~ (a f udv u~ ( ds f α udv t u ds 0 ( Thi equatio iplie that the agrage ultiplier λ i the urfae fore o the ( a ( oudar a i the udoai ad, a how i Fig.. Fig. Sudoai ad it oo oudar Hee, the urfae fore i defied a follow: λ k δ ( a a where δ a deote the relative diplaeet at the oudar a, ad k deote the pealt futio. The equatio of otio, diretized with repet to pae utitutig the aoveetioed relatiohip i equatio (, i otaied a follow: U KU P (3 The aove equatio of otio i epreed i the gloal oordiate te a follow: d d d K ε K d K d d Pd K D ε P (4 where d deote rigid diplaeet ad rigid rotatio i the udoai, ad ε deote otat trai i the udoai. Auig a rigid diplaeet field ad a a atri that otai ol diagoal eleet, utitutio of the aove etioed relatiohip ito equatio (4 ield the followig: d P K d (5 d

3 Here, a how i Fig., we epad eleet ( ad the adjoiig eleet. I thi ae, the itegratio o the oudar edge, with a fou o eleet (, i relevat ol to eleet ( (4. Therefore, the other eleet are ot relevat i the iultaeou equatio. (4 ( (3 ( Fig. Eleet ( ad adjoiig eleet Fro the aove relatiohip, the tre eleet i otaied uig the urfae fore at the eleet oudar, ad a e epreed a follow: ( d P e N de ( t (6 d d Thu, the equatio of otio a e oputed for eah eleet. The eleet aeleratio a e otaied a the reultat of the otat fore, i a aer iilar to the DE approah, a follow: ~ P (7 U However, i DE, the otat ehai i aed o poit otat, wherea i RBS, it i aed o urfae otat. Therefore, it i eear to haraterize the ature of the otat. We propoe edge-to-edge otat, a how i Fig Blok urfae Fig. 3 Cotat ehai

4 Whe i a otat tate, the aout of peetratio i the oral ad hear diretio, δ ad δ, repetivel, are defied a follow: ( ( δ (8 ( ( δ (9 Thu, we deterie the otat fore per uit legth, applied i equatio (, a follow: δ δ k k λ λ 0 0 (0 The lip alog the oudar edge a e odeled with the ohr oulo failure riterio. 0 ta f ( where i the iteral fritio agle, ad i the joit oheio. If the oral tre atig o the oudar i teile, the two fae of the joit a eparate. 3. NUERICA EXAPE A a uerial eaple, to ofir the appliailit of the tailit aali of ladlide, we evaluated the dai ehavior i ultiple lok odel. Tale how the aaltial oditio ad aterial propertie, ad Fig. 4 how the odel of a lope i a gravitatioal field. Fig. 4 RBS odel of lope Slidig lok

5 Tale Blok propertie ad alulatio paraeter Paraeter Value Youg odulu (Pa 00 Poio ratio 0. Deit (kg/ 3 45 Tie ireet ( Fritio agle 30 Coheio of joit (kn/ 0.0 Teile tregth of joit (kn/ 0.0 Teio rak Fig. 5 Reult of RBS (lope with a eitig rak Slidig off Fig. 6 Reult of RBS (Crak grew, ad lok of the lope were lidig off

6 A per the reult of evaluatio, ladlide oveet ad ollape o aout of the teio rak were oerved o the upper part of the lok, a how i Fig. 5; uequetl, the lok tarted to ove ad lide dow, a how i Fig. 6. Tale Calulatio fator of afet Shear Stre Noral Stre Fator of Safet (kn/ (kn/ Preet ethod (RBS iit equiliriu aali (FEENIUS The fator of afet otaied fro preet ethod (RBS ad the liit equiliriu ethod are lited i Tale. Fro the preet ethod, it i 0.7, wherea fro the liit equiliriu aali, it i I the liit equiliriu aali, the oral ad hear fore atig o the lip urfae a e otaied ol whe the liit equiliriu oditio are atified, while i the preet ethod, a the kieati of the lok te i oidered, the atual iteratio fore etwee lok a e otaied, idiatig that thi ethod i ore tale. 4. CONCUSIONS Thi paper propoed a epliit dai odel to olve equetiall large deforatio ehavior; thi odel i epaded to ilude RBS that i uitale for aali of progreive failure, uh a ladlide. Firt, we defied the epliit ethod aed o the hrid priiple of virtual work, ad developed a ew RBS approah for otat ehai ad fritio harateriti. oreover, we ofired the appliailit of the tailit aali of ladlide oervig the iulated ehavior of ultiple lok odel. REFERENCES Kawai, T. (977, New eleet odel i direte trutural aali, J. So. Nav. Arhit. J., 4, Yagi, T. ad Takeuhi, N. (05, A epliit dai ethod of rigid odie-prig odel, It. J. Coput. eth., Vol.(4, -5.