THE GENERAL SOLUTION FOR A SLAB SUBJECTED TO CENTRE AND EDGE LOADS AND RESTING ON A KERR FOUNDATION

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1 THE GENERL SOLUTON FOR SL SUJECTED TO CENTRE ND EDGE LODS ND RESTNG ON KERR FOUNDTON Frn Vn Cr, Mrc S, ndré Jinki Rcid Ocor, ; Rid g, n hi r h gnr nyic oion o concr jcd o inrior nd dg od i rnd. n h mhmic driion o h cod orm nyic oion conidr h mn hin ring on Krr ondion. Krr oion r gnr dirni qion ring h qiirim o on grd chrcrid y mn o hr oi rmr. Th c o rnr hr dormion nd h inrocking cion o nd grd r conidrd in ccing h od indcd r in concr. o cohrn ondry condiion iho nncry conrin nd c o oing oi c i rnd. Th rnd oion roid h mn o drmin ny r or dcion ny dinc rom n inrior or dg od. Kyord: Concr mn, hin mod, Krr ondion mod, cod orm oion, odindcd r nrodcion Th dirni qion o h ric dcion,y o hin o in D jcd o dirid ric od q,y nd ondion inrc rr,y i Timohnko, 95 D, y q, y, y Th y o o hi qion i d on h mion o ondion rcion ncion o h dcion o h. Th im rrnion o conino ic ondion h n roidd y Winkr 867 ho md h h conid o coy cd, indndn inr ring. Sch ondion i qin o dn iqid. Th rion n h rr nd h dcion i hn, y k, y hr k Nmm³ i cd h ondion or grd mod. Wrgrd knon qion or r in concr 96, 98 r d on Winkr mion. cn n rom Winkr qion h or hi ondion mod h dicmn r conn or niorm rr nd ro oid h odd r. For mo mri hi hior i no ord in rcic. i hror gnry dmid h Wrgrd oion orim h o h dcion nd h nding r in h. Hor, c o hir iiy o com dg r, h rcin o Wrgrd oion h nr diminihd. P dirc corrondnc o: Mrc S, Min Krmnn 9, NL7LL Dnr, Th Nhrnd, hon , cimi , mi mrc@c.com

2 n n m o ind mor ccr rrnion, Prnk 95 md hr inrcion n h ring mn nd rood rion n rr nd dcion chrcrid y ric mod k nd hr mod G Nmm in h horion dircion, y k, y G, y. Vn Cr 999 dod h gnr oion or rcngr on Prnk ondion jcd o cnr nd dg od. Hor hi oion do no rcy iy h dg condiion. Uing h o ondry condiion i o k ino ccon h imiing c o h dg, h rd h momn in h nd hr orc in h nd in h grd nih h dg. Th corrc oion in c rqir hr ondry condiion: nihing o momn nd hr orc in h, coniniy o hr in h grd. h n chrcr o Prnk oion dirni qion o h orh ordr mk i imoi o r n odd nmr o ondry condiion. No h hi diicy do no ri ih Wrgrd oion hr h grd ony rc in h ric dircion nd hnc do no rqir ny hr coniniy qion. n h ry 6, Krr 96 rnd grd mod chrcrid y rmr, o ric modi c nd k, nd hr mod G. Hi rion n rr nd dcion ri k G, y Ł c ł c, y k, y G, y Thi mod r in dirni qion o h ih ordr. Th oion or imid rqir hr on dg or i o dg ondry condiion nd hnc h ormniond diicy cod orcom. Hor, Krr 965 nd ohr Knii, 985; Krr nd E Sii, 99 rood ony ricr nyic oion o h mod. Nrh or rcic gnr oion i incdd in h knon SL ini mn rogrm..rrnc. W rn in hi r, o or knodg or h ir im, h gnr nyic oion or ring on Krr ondion. To oion r rnd. Th ir oion i gin in r nd ii h dmnd or o inini n,.g. inrior condiion. n r, hi oion i ndd o o o ini n,.g. ih dg or join. icion o h cod orm oion r rnd in r. Th ndi gi h mhmic roion o Krr dirni qion.

3 Pr. Krr ondion mod.. Th gnr oion o Krr ondion mod. Conidr n inini o in D jcd o od q,y igr. D c Lod S Sring yr G Shr yr k Sring yr Figr Th ondion mod coni o o ring yr, ih conn c nd k, nd n inrmdi hr yr ih conn G. Th qiirim qion o h i GD 6 k D G k q G q kc Ł c ł k k Ł c ł k kc L Dk, Gk g, kc, Gc g nd q conn. Thn qion imii ino 6 6 g g q k No h or c h r ring yr nih nd h qiirim qion rnorm ino Prnk qion g q k Frhr no h or G g, h hr yr nih nd Prnk qion rnorm ino Wrgrd qion

4 q k Th oion o i 5 q q co in co y in dd do Forir ingr hich r niormy dirid od q or rcng., nd 6 q k g co in co y in g dd g. Roion o Krr qiirim qion. For h inrd rdr, h did domn o h roion i gin in h ndi. n ordr o rorm h ingrion o 6, or m ih rc o, cori h dnominor ih rc o h m ri. 6 g g 6 g g 7 g Th coriion i n o ii h ooing ingr oion 8 co in d k k k k [ ] hn > nd 9 co in d k k k k [ ] hn <.

5 5. Soion o h dcion ingr. Th oion o h dcion ingr dnd on h ign o h dicriminor di 7 hr ł Ł g ł Ł g g g Whn di >, h oion o h dcion ingr i d y g k q o in co 6 hr or > in co in co 5 in co in co or < 6 7 in co

6 6 in co 8 in co in co ih 9 g g 7 7 Whn di <, h oion o h dcion ingr i d y g k q o in co g hr or > 5 6 or <

7 ih d d d g d g co q g co ł Ł q g co ł Ł q 5 6 q rcco. Rion or momn nd hr orc. nding momn, momn o orion nd hr orc in h r gin or ny,y y h cic rion 7,, dy d d d D M dy d d d D M y m m 8 ddy d D M M y y,, m 9,, dy d d d dy d D T dy d d d d d D T y Prnk rion gi h hr orc in h grd c D d d G T grd, c D dy d G T grd y,

8 8.Eqiirim qion... Lodd. Pr. Th ini on Krr ondion. Conidr ih o dg, r ih h y dircion, ocd dinc L nd L rom h i o h od nd ring on n nimid grd. Thi iion rqir 8 ondry condiion ring nding momn, momn o orion nd hr orc oh dg o h nd hr orc in h grd hr yr. Hor, i cn hon Timohnko, 95 h h dg o h h o ondry condiion or momn o orion nd hr orc cn rdcd o h ooing niq condiion d d d T, D m d d dy Hnc h nmr o ondry condiion i rdcd o i. Tho ondry condiion rqir 6 ricr oion o h homogno qion drid rom qiirim qion 6 6 g g Th 6 ricr oion cn grod oo: hn h dicriminor di i oii co y in { r C hr co C D in E F }d q C k 6 g nd hn h dicriminor di i ngi co y in 5 { r C C D E F }d hr 6 C q k g

9 9 Th ri,,, nd h h m mning or. Th ondry conn F, r drmind y h ondry qion. For mor crn ri h dcion ingr o h inini 7 co y in in d nd h dcion ingr, oion o h homogno qion, 8 r co y in in d. Unodd grd id odd r dg. Th qiirim qion i ddcd rom Krr qiirim qion, hr q no od nd D no 9 k G Thi qion h o oion co y in g g 5 o,gr C M N d Th dcion m nih h inini. Hnc h k h oion ih h ngi onn h oii id o h i nd h oion ih h oii onn h ngi id o h i... Unodd id odd join. Th qiirim qion o n nodd i idny h homogno qion. Th oion r gin y or 5. Hr o h dcion m nih h inini. Hnc h oion ih ngi onn i id h oii id o h i nd h oion ih oii onn i id h ngi id o h i. W no h oion y o,.. ondry qion... ondry qion or ih r dg. Momn nd hr orc in h m ro h dg. For L nd L

10 , in in d d co y in d 5 M D m, in in d d d co y in d d 5 T D m Th hr orc in h hr yr h dg i ihr ro grd inrrd h dg, ihr h hr orc h id o h dg i q o h hr orc h ohr id coniniy o h grd.... Shr orc ro in h hr yr. For L nd L co y in ł 5 T G d, grd d D d Ł c Th 6 ondry conn F r drmind y h ym o qion 5, 5 nd Shr orc inid h dg q o hr orc oid h dg. W h oion o qiirim qion 5 o n nodd grd. in in For L nd L d D co y in 5 G d Ł c in in d ł G d d co y in o, gr d Hor, qion 5 inrodc o mnry conn, M nd N. Hnc nd o qion mor o o h ym o 8 ondry conn. Fooing dirn rir Eom, Pron nd Hjmd, 998, conidr h h dcion oid h dg dcion o h grd i rcion d o h dcion inid h dg dcion o h y in 55 d co in in d co y in o, gr d hr d. Th 8 ondry conn F, M, N r drmind y h ym o qion 5, 5, 5 nd 55.

11 Th rio n h dcion i h conqnc o om hr rnr hrogh h hr yr. On Winkr ondion rionhi n h dcion od imoi: h dcion oid h dg i y diniion ro... ondry condiion or ih join nodd id odd. Th dirnc n join nd r dg i h oiiiy o od rnr h join. i gnry dmid h noihnding h rnc o od rnr mdim, do or m, h momn in h rmin ro h join. Hnc, ondry qion 5 i no modiid. Frhr, ric qiirim rqir h h hr orc r q in h nd in h hr yr oh id o h join. in rio rgrh, o h h dcion o h nodd i rcion o h dcion o h odd. ondry qion, r h h dcion o h hr yr h nodd id i rcion o h dcion h odd id, h rio ing h m h n h dcion: dcion nd hr yr dcion r roorion o ch ohr. Th ondry qion r, or L nd L Momn r ro h join, in in d d co y in d 56 M D m co y in d, o, d o d 57 M D m Shr orc in h r q h join d d co y in d d 58 D m in in d D co y in d o, d m d o, d d Shr orc in h hr yr r q h join. d D d Ł c 59 G co y in ł in in d G d d Ł D c co y in o, d ł Th dcion oid h join r rcion o h dcion inid

12 y 6 d co in in in d co y in o, d Th conriion o h hr yr in h o dcion oid h join i rcion o h conriion o h hr yr inid h join D co y in 6 dg Ł c ł in in d hr d. G Ł D c co y ł in o, d Th ondry conn r drmind y h ym o qion 56, 57, 58, 59, 6 nd 6... ondry condiion h join ih od rnr. Whn h od rnr mdim r hr rnr.g. coninoy rinorcd concr, h rc dcion r q nd h ym o qion 56 o 6 cn imiid. ndd hn hr rnr occr, h hr orc in oh nd hr yr m ncriy q o h hr orc in n inini Vn Cr, S, 998. Hnc, ony hr ondry qion r rqird ch join: h nodd djcn nighor cn ignord. Momn r ro h dg o h odd d co y in d 6 D m in in d Shr orc h dg r q ih h hr orc in n inini d d co y in d d 6 D m in in d D co y in d d m d d in d Eqion 6 cn imiid co y in d d 6 D m d d d Shr orc in h hr yr r q ih h hr orc in h c o n inini in

13 d D co y in 65 G d Ł c in in d ł G d d Ł D c co y in ind ł Eqion 65 cn imiid d D co y in 66 G d Ł c in d ł Th 6 ondry conn r drmind y h ym o qion 6, 6 nd 66.. S imid y on dg. on i rinciy inrd in r or nry h dg or join, h innc o h ohr dg, ocd dinc gr nogh, cn in gnr ngcd. Hnc, in hi c, ony h o h ondry condiion, nmy ho h concrnd dg, h o kn ino ccon.

14 Pr. icion. Comr rogrm. Sr comr rogrm h n dod rd o h jc o hi r. rinci rogrm hnd h c rnd in...,...,. nd. o r. n ddiion, im rogrm h n rin o rorm h nmric ingrion o h dcion or ny nyic rnormion: rogrm or n inini jcd o niorm circr od. Thi rogrm o riying h idiy o h rinci rogrm in h c o o inini n. Frhr rogrm h n dod hnding h ondry c or m imid y n dg. Th imi o h r or nd o h r or m hn nrging on o h id o h rcngr od. Hnc, ho rogrm cn d o riy h rogrm in h c o o ini n... Emnry rogrm or n inini. Th od i conidrd niormy dirid or circr r. Hnc h oion cn rd on ingr in or coordin ih 67 r r r in h hyohi o Krr, y rnormion o qion 6 in or coordin, 68 q J mr J m dm 6 k gm m gm in h hyohi o Prnk, y ing in qion 68 i dind in o Pr, 69 q k J mr J m gm m dm in h hyohi o Wrgrd y ing g in qion 69 g i dind in o Pr, 7 q k J mr J m dm m Conidr od o N, rdi o 56, mm, rr o Nmm² nd o hickn mm, mod o Nmm², Poion rio o,. Th dcion comd y h ingr 68, 69 nd 7 r rnd in. Th rogrm cn oind y h corronding hor

15 5 T : Dcion ndrnh circr od k g c Wrgrd Prnk Krr To ri o grd chrcriic r conidrd: k., g, c. ih di > nd k., g.9, c ih di <. No h or c im h r in Krr hyohi nd o h r in Prnk hyohi; or c im nd g. im g h r nd o ho in Wrgrd hyohi. Thi i in grmn ih h concion o o r : h qiirim qion... Vriicion o h rinci rogrm.... Vriicion o n inini ing qion 7. Th dcion comd y qion or or qr od r comrd ih ho comd y 7 or circr od. D o h rinci o d S Vnn, hy m nry q or od o m igh nd rr. o h nding r r comrd in h i o h od. Th r r rnd in. T : Comrion n circr nd qr od. k g c dinc rom i , Th r riy oh rogrm.... Vriicion o h oion or n inini ing h rogrm or n inini m. Th od on h m h idh o mm. Th od on h h ngh o mm nd idh o mm conidrd o inini. Th rr i Nmm² in oh c. Poion rio o h i o ro c h oion or h m i indndn o Poion rio o n r. Th dcion nd h nding r r comd in h i o h od. Th r r rnd in.

16 6 T : Comrion n inini m nd inini k g c m m Th r riy oh rogrm.... Vriicion o h oion or ini ing h rogrm or ini m. Th m od in h rio rgrh r ocd h dg o m nd. Dcion nd r r comd h dg. Th r r rnd in. T : Comrion n ini m nd ini. ondry condiion d k g c m Th r riy oh rogrm or c o ondry condiion... nyi o h dod mod... Mchnic hior o h mod. Uing h rinci rogrm or h ondry condiion o. com dcion nd nding r in h i o h od or k., c nd o g rying rom.6 o.. n doing o cciy oion, oion nd gin oion i.. h com r cord y h dirn nyic oion. Th r r rnd in 5.

17 7 T 5: R in or rying o g. nini Joind : dg od d.5 g y y.6 di >.7 di >.8 di >.9 di <. di >. di >. di > Th ccion o h r hrogh h dirn g do no ho ny hi or diconiniy, hogh h iid nyic rion nd dir conidry... Gomric hior o h mod. Uing h dirn rogrm, com h dcion h dg o ini or od moing ord h dg. Th r r gin in 6 ih r dg nd ro hr orc in h hr yr, 7 ih r dg nd hr rnr in h hr yr, 8 ih join nd roorion dcion nd 9 ih join nd hr rnr in nd in hryr. Th comion r gin rormd ing oh ri o d. Th ngh r h dinc o h i o h od o h dg. T 6: Dcion or od moing ord r dg no hr rnr in hr yr: ondry condiion 5, 5, 5. k g c T 7: Dcion or od moing ord r dg hr rnr in hr yr: ondry condiion 5, 5, 5, 55. d k g c

18 8 T 8: Dcion or od moing ord joind dg roorion dcion: ondry condiion 56, 57, 58, 59, 6, 6. d k g c T 9: Dcion or od moing ord joind dg hr rnr in nd hr yr: ondry condiion 6, 6, 66. k g c on od normy h cd, h dcion incr rgry hn h od mo ord h dg. o h o h dcion or h ooing: h o h dcion o 6 r h m h o 7 nd 8 or δ : no hr rnr in h hr c. h o h dcion o 9 r h m h o 8 or δ : hr rnr, in nd in hr yr, in h o c. h o h dcion o 7 r highr hn h o 8 or idnic o δ > : hr rnr occr ony in h hr yr in h c o 7, hi i occr oh in hr yr nd in in h c o 8. Thi o o concd h h dirn o ondry condiion r conin ih ch ohr.. nnc o h dirn oi rmr. Uing h rinci rogrm riy ny h innc o h dirn oi rmr on h dcion in h i o h od. W conidr qr od o N, id o mm, rr o Nmm² nd o hickn mm, mod o Nmm², Poion rio o,. n, ny h innc o Krr ring conn c or gin o k. Nmm³ nd g.5 in h c o n inini.

19 9 T. Dcion in n inini or rying c c W no h h dcion dcr ih incring o c. Th h dcion r highr hn ho comd ih Wrgrd mod c. Frhr no h h dcion in com r ih dcring o c. n ny or q cnr dcion h ri innc o ir o k nd c g.5 in n inini. T. nnc in n inini o ir o k nd c k c W no h in h c on n inini, h o k nd c comn ch ohr in mnnr o king. Thi i h on od h cd. n ny or q cnr dcion h ri innc o ir o g nd c k. in n inini. T. nnc in n inini o ir o g nd c g c W no h h innc i mor ookn hn in. nod in, h dcion in com r ih dcring o c. Hor h innc o h dcr o h c i modrd y h incr o h g. n ny or h ir o k nd c o, h dcion or od ocd h join o ini ih hr rnr. Th dcion r comd ong h join or incring dinc rom h i o h od.

20 T. nnc in ini ih hr rnr o ir o k nd c k c n conr o h r in, h o k nd c do no comn ch ohr. Hnc h o h rmr cn dicrimind y mn o h dcion mrd h join. n ny gin or h ir o k nd c o, h dcion or od ocd h join o ini ih hr rnr. Hor hr h dcion r comd orhogon o h join or incring dinc rom h i o h od. T. nnc in ini o ir o k nd c k c Hr o h o h rmr cn dicrimind y mn o h dcion mrd orhogon o h join.. Drminion o h rmr k, G nd c... Drminion y dirc oi. Th drminion o k hrogh ring i knon inc Wrgrd im. Th coni o im icion o Winkr mion. cy h ndrd rqir h h rormd ih o 7 mm dimr. Th o k i gin y 7 k q D hr q i h id rr nd D h ord dcion. Th drminion o k nd G oghr cn rormd y dcrid y Prnk 95 him. Th i ry imir o Wrgrd ring ; hor h ric od i id ccnricy. Prnk r h qiirim o h ric orc od nd grd rcion nd o h momn. Thi r in ooing rion

21 7 R L K Q k K 7 R L K Q k K hr Q i h id od i h ccnriciy o h od k i Wrgrd grd mod R i h dcion on h odd id o h L i h dcion on h nodd id o h i h rdi o h K, K, K r h modiid ncion o h cond kind o ordr ro, on nd o Gk G i Prnk hr mod. Diid 7 y 7 in ordr o oin n qion in, hn drmin k y 7. W do no kno o ny o drmin h o c in i. Yo nd Hn 98 gi n miric rion n k nd G. 7 G N mm 5k N mm.. Drminion y dcion mrmn on m or. Thr i mny ckccion rogrm or h drminion o E Yong mod o h nd k d on dcion in mrd on m or on Winkr ondion. To r ry rcny ihd Khnoich,, F,. rcny dod ohr comr rogrm rn ry im nd cod orm gorihm or h comion o r nd dcion S,. Th PVERS rogrm ii n omic ckccion rogrm or h drminion o Prnk hr mod. im iri rocdr or h drminion o k nd G E i md o knon; drid rom ronnc mrmn d on dcion in mrd on m. Th rogrm ii cod orm orm o h dcion ndr n iod od. For gin o G, on com h o k corronding o h ord dcion ndr h od. Thn, ih oh, on com h dcion h ohr dinc o h dcion in nd h dirnc ih h mrd dcion. h dirnc r igniicn, h roc i rd ih n o G. n m i gin in 5.

22 T 5. Em o iri ck ccion o k nd G. G k Dinc SD Mrd dcion W do no kno ny ck ccion rogrm or h drminion o c. Th r o ho h ny ir o o k nd c cn iid o chrcri h dcion in o n inini. Th o k corronding ih o c q inini i hn o rrni o h grd mod. Hnc, rdy cod h cd, in h c o n inini cn o h rom y Prnk mod. W gg h ooing rocdr: drmin in ir h o k nd G rom dcion in ord in h midd o mion o n inini on Prnk grd. Wih h oind o G, com, no ing Krr mod, ri o ir o k nd c iying h m dcion in. Choo o o h ri, h ir o h gi h i ih h dcion in mrd r ih h join or orhogon o h join. Finding Thi r rn h nyic oion o Krr gnr dirni qion ring h qiirim o on grd chrcrid y mn o hr oi rmr. Thi no oion, oind hrogh ond horic nyi, i ciniic orrd in h concrnd id nd cn gi mor ccr ndrnding o h rom kn ndr conidrion. Wiho minimiing h rmndo oiiii o Fini Emn oion, nyic oion r ry h ony mn or ror riicion. Frhrmor, comr rogrm d on cod orm nyic oion r on r nd mor r rindy hn ho d on FEM. Th r o h nmric icion h h rormd hi g o or rrch, m o indic h Krr mod i rh no n inrinic chrcriic o h grd rhr rmr oing ring h oi inrcion n dg or join. Th ir ring yr o Krr mod hod rhr inkd ih h hn ih h grd. Krr mod cod hn rnmd S ih ri oom chrcriic on Prnk grd. ndd in h c o n inini inrior iion h o Krr mod c cn iho hrm kn inini, h hickn h c o h ring yr hn nih nd h dcion h oom o h ring yr com q ih h dcion on o o h ring yr nd on o o h : h yr i ord in h ciy i dr riing h imc.imh E. ci h i c c Th rom i od ming Prnk grd. r dg, h oion or h nodd grd id h gin i d on Prnk mod. i ony h

23 id o r dg or join h nd Krr oion. W hink hror h Krr mod i rmr gorning h hr orc diriion n nd grd n dg or join. Hnc h diicy, i no h imoiiiy, o h dirc drminion o h o h mod y on h grd nd h nciy, roy, o doing i y dcion mrmn rormd on r dg or join.

24 Rrnc Eom,.S., Pron,.D., nd Hjmd, K.D. 998: ini mn dy o Lod Trnr n Dod Pmn S. CROW Forh nrnion Workho on h Dign nd Eion o Concr Pmn. çco, Porg. F, T.F., Tn, K.H., L, S. : CodForm nd SmiCodForm gorihm or ck ccion o Concr Pmn Prmr. Non Drci Ting o Pmn nd ck Ccion o Modi. Third om Procding STM Conrnc. Tyji nd Lknn Edior. S,. Krr, D. 96: Eic nd icoic ondion mod. Jorn o id Mchnic, Vo., Nr.. Krr, D. 965: dy o n ondion mod. c Mchnic, Vo. Krr, D. 976: On h driion o od ondry rom in rcr mchnic. n. Jorn Soid nd Srcr, Vo. Krr, D. 98: On h orm domn o ic ondion mod. ngnir rchi 5 SringrVrg 98. Krr, D nd ESii 99: Grn ncion or coninoy ord. Zichri ür ngnd Mhmik nd Phyik, Vo.. Khnoich L. : SL. Khnoich L, McPk, T.J., Tyi, S.D., : LTPP Rigid Pmn FWD Dcion nyi nd ck ccion rocdr. Non Drci Ting o Pmn nd ck Ccion o Modi. Third om Procding STM Conrnc. Tyji nd Lknn Edior. S,. Knii, M.C. 985: nyi o on Krr ondion mod. Jorn o Enginring Mchnic, SCE, Vo., No.. S, M., Thn,.,, Vn Cr, F., nd Vrk, J.P. : Sndrd nd rcommndd rcic or FWD d ion nd roring rngh mid o NTO or nqiry. PVERS comr rogrm; Procding ir Eron FWD r gro ming, D Uniriy o Tchnoogy,. Timohnko, S. 95: Théori d Pq d Coq. Liriri Poychniq Ch. érngr. Pri, Lièg. Vn Cr, F. 99: Rigoro nyic oion o concr mid o inrior nd dg od. Proc.5 h n. Con. on Concr Pmn Dign. Prd Uniriy. ndin 99. Vn Cr, F. 999: Th imid rcngr on Prnk ondion. CROW. ici 6. Ed,999. Th Nhrnd.

25 5 Vn Cr, F., S, M. 998 CROW Forh nrnion Workho on h Dign nd Eion o Concr Pmn. çco, Porg. Won, G.N. 966: Thory o Fncion. Cmridg Uniriy Pr, 966. Wrgrd, H.M. 96: Sr in Concr Pmn comd y horic nyi. Procding Highy Rrch ord, Pr, 96. Wrgrd, H.M. 98: N Form or Sr in Concr Pmn o irid. Procding SCE, Vo, 98. Winkr, E. 867: Di Lhr on dr Eiciä nd Figki. Prg, Dominic, 867. Yo, Z.K., Hn, L.Y. 98: Erimn nyi on ring r in concr mn. Jorn o E Chin Highy, n, 98.

26 6 PPENDX F mhmic roion o Krr dirni qion. Krr mod or n inini. Conidr n inini o in D jcd o od q,y igr. Lod D q S c Sring yr G Shr yr k Sring yr Figr Th ondion mod coni o o ring yr, ih conn c nd k, nd n inrmdi hr yr ih conn G. Si h dcion ino o r, y, y, y hr i h conrcion o h r ring yr nd h dcion o h r o h ondion. Th conc rr ndr h nd h hr yr r gin y nd. Hnc, y Winkr mion,, y c c, y k Th qiirim qion o h i D q hr D Eh µ nd d d d d dy d dy. Vric qiirim in h hr yr cn ddcd rom igr. dt dt y 5 ddy ddy ddy dyd d dy 6 dt d dt dy y

27 7 T dy T y dt y d d dy T y d T dt dy Figr T nd T y r h hr orc r ni ngh in h hr yr cing on n orhogon o h nd y dircion. Prnk 95 m h h hr orc in h hr yr r roorion ih h riion o h dcion igr 7 T d G d T y d G dy T d T dt d Figr hnc 8 nd dt d dt y d d G dy d dy 9 G oh qiirim qion r in h ym D c q c k G

28 8 Eimining nd yid h qiirim qion o h on Krr ondion GD 6 k D G k q G q kc Ł c ł k k Ł c ł k kc L Dk, Gk g, kc, Gc g nd q conn. Eqion imii ino 6 6 g g q k No h or c, hich rqir, h r ring yr nih nd h qiirim qion rnorm ino Prnk qion g q k Frhr no h or G g, h hr yr nih nd Prnk qion rnorm ino Wrgrd qion 5 q k Krr rion n rr nd dcion nrodcion cn oind rom qion 6 q D From qion k Ł c ł k GD kc 6 7 [ q D ] Hnc GD 6 8 k G k c c Eqion 8 n o com h ric r on h grd. Dirniing yid 6 9 D Rcing 6 in 7 r in Krr rion Krr, 98 k G k G Ł c ł c. Roion o Krr qiirim qion. L G k q q co in co y in dd do Forir ingr hich r niormy dirid od q or rcng., nd

29 9 q, co in co y in dd Eqion i iid or, k [ g g ] Hnc h oion o h qiirim qion cn rin q k g co in co y in g dd g. Trnormion o h ih ordr dnominor o h ingrnd. n ordr o rorm ingrion, or m ih rc o, rdc y coriion h ordr o h dnominor, DEN, o h ingrnd. Th dnominor cn d ih cic qion in. Th roo cn r or com. W ir rnorm h dnominor L y g. Thn 6 5 DEN y y hr 6 Ł g ł 7 7 Ł g ł 9 g 7 g Th nr r or com o h roo o 5 dnd on h ign o h dicriminor di 7.. Nr o h roo. i i gr yid di 8 g 7 7 g g 8 Th ign o 8 dnd o h dicriminor, di, nd h roo o h orh ordr rion in g n h qr rck. di nyi o h dicriminor di.

30 > 8 hn 8 <, di < nd di > hr h o g. 8 hn 8, di nd 9 di g g Ł ł Hnc i 8 hn di >, n g. g hn di. < 8 hn 8 > nd h ign o di dnd on h o g ih rc o h roo g nd g o 8. g g On iy ho h < g < g. Hnc di > hn g < g or g > g nd di < hn g < g < g. Th nr o h roo, ncion o h o nd g, i rnd in. T : Nr o h roo o h dnominor o h ingrnd. g di Nr o h roo > 8 di > On r, o com 8 g <> di > On r, o com 8 g di Thr r, hr q < 8 g < g di > On r, o com < 8 g g di Thr r, o q < 8 g < g < g di < Thr r < 8 g g di Thr r, o q < 8 g > g di > On r, o com 5. Fcoriion o h dnominor o h dcion ingr. W cori h dnominor ih rc o h ri. 6 6 g g g g g

31 Th ym i iid or d d d g 5 d 6. Roion o h dcion ingr hn h dicriminor i oii. 6.. V o h roo. Whn h dicriminor i oii, h oion o h cic qion i oind y h cic gric mhod. Th roo r 6 g 7 g 8 g hr 7 7 W no h i r nd h nd r com. Rcing h i y hir, on oin 9 6 i i 6 i i g 6..Trnormion o h dcion ingr. Th dcion ingr cn no rin oo in co in co 6 y k q dd i g Th ingr cn od y Won, 966, r..5 k k k k in co d

32 hn > nd y co in d k k k k [ ] hn <. Hor qion nd rqir h h r r o κ i oii. Hnc ir h o ho h h r r o, nd r ngi. W r no o ro hi nyicy nd h h o ho i nmricy or h rg rng o nd g or hich di >. W k, o h conidr h o o di. Th o r gin in nd h o h r r o nd r gin in. T : V o g\ T : V o nd g\ W my concd h h r r o h o h roo r ngi. Th cn rnorm h dcion ingr in 5 q k 6 g [ ] co in co y in hr 6 g 7 i g i dd

33 8 i g 6.. Roion o h dcion ingr. Th ingr ih rc o cn i ino hr r 9 co in d 5 co in d 5 co in i d 6.. Roion o ingr Th oion i oind y icion o rion nd. For > 5 For < 5 hr 5 g 6.5. Roion o ngr Fir h o h com nmr nd. W ri 55 i g 56 i g On ddc h o nd 57 i 58 i hr 59 Hnc, or >

34 6 6 6 i i i i L i i i i L ying i co i in 6 in co in co nd or < 6 in co in co 6.6. Roion o h ngr n h m y or on oin or > 65 in co in co nd or < 66 in co in co 6.7. Gnr oion o h dcion ingr. Uing h oion or, nd, oin h com oion o h dcion ingr.

35 5 67 q k g [ ] co y in d 6 o hd ingrd h dcion ingr ih rc o, h oion od h rin 68 q k g [ ] co in 6 o d hr, nd in, nd r rciy rcd y, y nd. 7. Roion o h dcion ingr hn h dicriminor i ngi. 7.. V o h roo. Whn h dicriminor i ngi, h ocd rigonomric mhod. Conidr qion 5. L y coθ. Th dnominor rnorm in 69 DEN co q co q L 7 coq Th oion o h rigonomric qion r θ rcco, θ θ π, θ θ π, Th roo o h origin qion r hn co q co q Ł co q Ł g ł ł g g 7.. Roion o h dcion ingr. Sinc h roo r r, ingr nd cn id, iho rhr nyi, i h roo r ngi. gin ho i nmricy coring h rqird rng o nd g. Th imiing or g, dnding on h o, r hon in.

36 6 T : Limiing or g g g min g g m Th o h roo r comd, in 5, or. T 5: V o h roo G Th roo r ngi, hnc my ri h dcion ingr oo 7 q k g co in co y in g dd hr 75 d d 76 d nd i i. g d Th oion o h ingr r gin y or > nd y or <. Wih hn > nd

37 hn <. Th oion o h dcion ingr r q co y in 8 [ g ]d k g or o q co in 8 [ g ]d k g o hr, nd in, nd r rciy rcd y, y nd.

A study Of Salt-Finger Convection In a Nonlinear Magneto-Fluid Overlying a Porous Layer Affected By Rotation

A study Of Salt-Finger Convection In a Nonlinear Magneto-Fluid Overlying a Porous Layer Affected By Rotation Innion on o Mchnic & Mchonic Engining IMME-IEN Vo: No: A O -ing Concion In Nonin Mgno-i Oing oo Ac Roion M..A-hi Ac hi o in -ing concion in o o nonin gno-i oing oo c oion. o in h i i gon Ni-o qion n in