Author(s) Guenfoud, Salah; Bosakov, Sergey V.; Laefer, Debra F.

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1 Provded he uhor(s) d Uvers College Dul Lrr orde wh pulsher poles. Plese e he pulshed verso whe vlle. Tle Dm lss of ple resg o els hlf-spe wh dsruve properes Auhor(s) Guefoud Slh; Boskov Serge V.; Lefer Der F. Pulo de Coferee dels Lk o ole verso Iem reord/more formo The 4 orld Cogress o: Adves Cvl Evromel d Merls Reserh (ACEM 4) BECO Bus Kore 4-8 Augus 4 hp://em..om/em4.hm hp://hdl.hdle.e/97/77 Dowloded 8-4-7T:6:5Z The UCD ommu hs mde hs rle opel vlle. Plese shre how hs ess eefs ou. Your sor mers! (@ud_o) Some rghs reserved. For more formo plese see he em reord lk ove.

2 4 orld Cogress o Adves Cvl Evromel & Merls Reserh Dm lss of ple resg o els hlf-spe wh dsruve properes *Slh Guefoud ) Serge V. Boskov ) Der F. Lefer ) ) Deprme of Mehl Egeerg LMANM Loror Uvers of Guelm Alger ) Sruurl Mehs deprme Belrus Nol Tehl Uvers Msk Belrus ) Shool of Arheure Ldspe d Cvl Egeerg Uvers College Dul Ireld ) o@hoo.fr ABSTRACT Ths work gves sem-ll pproh for he dm lss of ple resg o els hlf-spe wh dsruve properes. Suh lulos hve ee ssoed wh sgf mheml hlleges ofe ledg o urelle ompug proesses. Therefore he dm lss of ems d ples erg wh he surfes of els foudos hs o de o ee ompleel solved. To dve hs work he defleos of he ple re deermed he R mehod d he dsplemes of he surfe of els foudo re deermed sudg Gree's fuo. The ouplg of hese wo sudes s heved med mehod kow he heor of els s Zhemohk s mehod whh llows deermo of reve fores he o oe d hee he deermo of oher phsl mgudes. The oed soluos e ppled o sud he dm ero ewee sols d sruures d o ssess umerl ompuos hrough vrous umerl mehods progrms. Nurl frequees url shpes d he dm respose of ple due o eerl hrmo eo re deermed. Vldo wh kler prolem llusres he dsruve proper effes o he resuls of he dm lss. KEY ORDS: Gree s fuo; dsruve properes; ple; Ege-frequees; Ege-shpes; dm respose.. INTRODUCTION: The dm ehvour of ems d ples resg o els foudo s op of hgh eres hroughou he desg d osuo seors. However he dm lss of suh sruures s ssoed wh sgf mheml dffules d leds ofe o mprl ompuol proesses. For hese resos he prolem

3 of dm lss of ples resg o els foudos s o ompleel solved.. Thus here s eed o develop prese mehods of lss of hese sruures d o provde more effeve mehods h hose h urrel es. Ths work provdes sem-ll pproh o deerme he url frequees d url shpes of regulr ple resg o els foudo wh dsruves properes (Boussesq's pe) d s respose o eerl hrmo eo (Fg. ). The followg s spes re egleed: dmpg er of he els foudo d fro he o oe ewee he ple d he surfe of he els foudo. The pproh s sed o he med mehod kow s Zhemohk's mehod [] where he regulr ple s dvded o fe umer of del elemes d he eer of eh eleme s pled rgd lk hrough whh he o ewee he ple d he surfe of els foudo s omplshed. The pproh ssumes h o ewee he sruure d he surfe of els foudo s repled fe o rgd lks d h he mss of eh eleme s oered s ere. h v Eν Fg.. Ple resg o els hlf spe wh dsruve properes. PROBLEM ASPECT.. DESCRIPTION OF THE METHOD OF ANALYSIS I ref he proedure o sud free vros of regulr ple resg o surfe of els foudos of Boussesq pe [] s s follows. A regulr ple wh mss Μ d ldrl rgd D ress o he surfe of els foudo wh dsruve properes wh modulus of els E d Posso's ro ν. The essel prmeers Zhemohk's mehod for he sud of he dm of ple resg o els foudo re show Fg.. The erl fores re ppled ol o he ple se he mss of he els foudo s o ke o ou whle he effors of oeo re ppled o he ple d o he surfe of he els foudo fg..

4 h v v J() () Eν J () J k() C () k-() k() k( ) J () () J () Eν φ () U () φ () Fg.. Illusro of he mehod of lss The ol ssem of equos llowg he dml sud of he regulr ple resg o he surfe of els foudo wh dsruve properes s epressed equo () []: ( v ) ( ) J ( ) ϕ ( ) ϕ ( ) u( ) = = = = = [ ( ) J ( ) ] = Ι ϕ ( ) ; [ ( ) J ( ) ] = Ι ϕ ( ) ; [ ( ) J ( ) ] = Μu ( ) Δ P = ; = () : oeo's fore ppled o he ple d o he surfe of he els foudo; J : er's fore ppled ol o he ple; v k : Gree's fuo defg he dspleme of he surfe of he els foudo he po due o fore ppled he po of he sme surfe; ϕ ϕ : gle of roo of he ple relve o he es O d O he emeddg po; u : l dspleme of he ple he emeddg po; k : defleo of he ple po due o fore ppled he po of he ple; Δ P: fuo hrerg he defleo of he ple po due o eerl fore P ppled he po of he ple (for free vros Δ P = ); : rms of he elemes of he ple relve o he

5 es of oorde; Μ : ol mss of he ple; Ι Ι : er's momes of he ple relve o he es of he oordes. The reloshp for he free vros s show equo () s per []: ω ω ω ω ω k ( ) = ke ; ϕ ( ) = ϕe ; ϕ ( ) = ϕ e ; u( ) = ue ; Jk ( ) = Jke. () Tkg o ou () he ssem from () kes he followg form: ( v ) = [ J ] = [ J ] = [ J ] = = = Ι = Ι = Μu. J ϕ ; ϕ ; ϕ ϕ u = ; = ().. GREEN FUNCTION DEFINING THE VERTICAL DISPLACEMENT OF THE SURFACE OF HALF SPACE Se he els foudo osdered s he els hlf-spe wh dsruve properes (prple of Boussesq) he he epresso of v k kes he form of equo (4) s per []: v k ν = πe Ω dξdη ( ξ ) ( η) (4) Ω = d : legh d wdh of he eleme; d : oordes of he po o he surfe where he dspleme s deermed; ξ d η : oordes of he po where he fore s ppled fg. 4. η Fg. 4. Geomer of he loded eleme

6 Afer égro epresso (4) eomes: k F k E v π ν = (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = F k l l l l l l l l Fgure 5 llusres he dsplemes of he surfe of hlf spe wh dsruve properes gve (5) he o oe due o oered fore ppled he ple's ere. Fg. 5. Surfe's deformo of hlf spe due o oere fore ppled he ple's ere.. FUNCTION OF THE DEFLEIONS OF PLATE I equo (6) s he fuo defg he defleos of he regulr ple po due o he fore ppled po. Bsed o he Clesh's soluo [4] kes he followg form: ( ) ( ) ( ) = = k A. (6) Aordg o [4] d for he se preseed here he erms of he epresso (6) h ssf he oudr odos of he suded ple re s follows:

7 ( ) ( ) ( ) ( ) ( ) ( ) B A B A k 4 = (7) where: ( ) ; l l l 4 l 6 = D P π ( ) = ; ( ) = ; ( ) = ; ( ) = 4 ; ( ) ν = h E D : ple's ldrl rgd; ν E : modulus of els d Posso's ro of he merl of he ple; h : ple's hkess; ( ) ν β = ; d : oordes of he po o he ple where he dspleme s deermed; d : oordes of he po where he fore s ppled fg.. The oeffes A B A d B re deermed he R's mehod [5] (.e. osderg he deformo's eerg of he ple). Afer smplfo hese oeffes ke he followg epressos: ( ) = 6 dd Q D A β ; ( ) = dd Q D B 6 β ; ( ) = dd Q D A β ; ( ) = 4 dd Q D B β. Epressos of ( ) Q = ( ) Q = ( ) Q = ( ) Q 4 = d lso he fl epressos of hese oeffes re ver log d hus o preseed here. Fgure 6 llusres he dsplemes of he ple gve (7) due o verl lod dsrued uforml over he ere ple.

8 Fg. 6. Ple's deformo due o he uforml dsrued verl lod. 4. REMARKS Se he dsplemes of he surfe of hlf-spe re osdered equl o he ple defleo (.e. v ) so he er fore J e gve he followg epresso s per []: d ( ) ( ) d v ν J = M = M = = M ω v Mω F. (8) d d π E = ν Here v = F d F gve (5). π E =. EAMPLE OF APPLICATION The ple s dvded o 5 del elemes d osderg h he po of emedme odes wh he ere of mss (fg. 7) v Eν Fg. 7. Elemes' umerg of he dsreed ple Also kg o ou he dm lulo he followg geomerl d mehl properes of he hlf-spe d ple re spefed: 8 = = m; ν = / ; E = N / m ; h =. m; = =. m; E = N m ; 4 / ν =/. Irodug suh d o he ssem usg () d kg o ou he epressos of eh of s prmeers s well s mheml smplfos he followg mr ssem s oed: [ A] 5 = (9) ϕ ϕ u

The erms of he mr [ A ] re ver log epressos resulg from mheml rsformos d hus re o luded here.

ssem (9). h hs log d ompled equo he roos represe he Ege frequees vlues of he ple. The lulo of he roos of he equo s eeued umerll usg Mhem.

9 The erms of he mr [ A ] re ver log epressos resulg from mheml rsformos d hus re o luded here... DETERMINATION OF THE PLATE S EIGEN FREQUENCIES The deermo of he ple s Ege frequees resg o he surfe of hlfspe wh dsruve properes s doe he resoluo of he equo of he deerm of he mr [ A ] of he ssem (9). h hs log d ompled equo he roos represe he Ege frequees vlues of he ple. The lulo of he roos of he equo s eeued umerll usg Mhem. Fgure 8 represes lose up of oe of roos of he deerm represeg he ple s Ege frequees. De@ AD w Fg. 8. Grphl lose up of he roo of deerm represeg Ege freque The sperum of Ege frequees (H) s: ω =.958 ω =. 467 ω = DETERMINATION OF THE PLATE S NATURAL SHAPES Deermo of he url shpes of he squre ple resg o he surfe of hlf-spe wh dsruve properes requres deermo of he url shpe orrespodg o eh Ege freque. To heve hs he frs equo of he mr ssem (9) s eluded d he ssem s solved wh equos d ukows. Susequel he url shpes of he em re deermed he formul (8) rodued Zhemohk d Ss [] for gve dsreo: = ; = 5. The pplo of hese operos llows he deermo of he ple s url shpes for he frs hree followg forms: ω ω ω

properes oers s reo due o he dm eerl verl lods. The se preseed s of he reo of he ple used he verl hrmo lod ppled po p odg wh he ere of he eleme 9 (fg. 7). The eerl verl hrmo lod s vred ordg o eq.

10 Fg. 9. Nurl shpes of he ple orrespodg o he frs hree Ege frequees 4. RESPONSE OF THE PLATE RESTING ON THE SURFACE OF A HALF-SPACE ITH DISTRIBUTIVE PROPERTIES DUE TO THE ETERNAL VERTICAL HARMONIC ECITATIONS The fored vros of he ple resg o he surfe of hlf-spe wh dsruve properes oers s reo due o he dm eerl verl lods. The se preseed s of he reo of he ple used he verl hrmo lod ppled po p odg wh he ere of he eleme 9 (fg. 7). The eerl verl hrmo lod s vred ordg o eq. : ( f ) P p = P9 = P os π () where P : mplude of eo; f : freque of eo; : me of eo. The vlues: P = N f = 5 H re osdered. Og he respose requres he resoluo of he ssem (). I hs se he prmeer Δ p s deermed followg formul () rodued Zhemohk d Ss []: Δ p = ppp = p= 9 P 9 () p : ple defleos he po due o he eerl lod P p ppled o he ple po p. These defleos re due o he eerl lods Pp s deermed (7) The soluo of he ssem () for he fored osllos gves he ukows vlues vrg wh he me of eo. Ukows represe reve effors he o oe. I hs se he vlue s vrl s epressed eq. : ( ) = ( ) os( f ) ( = 5) π. () Fll he ple dsplemes durg he me of eo represeg s respose re deermed eq. : 5 ν v = F () Ωπ E = Fgure shows he verl dsplemes vrl v of he ere of he ple resg o he surfe of hlf-spe wh dsruve properes D due o he dm eerl lod P 9 eh mome = Δ. Here = s; Δ =. s. Furhermore he mmum dspleme s oed he po whh he eerl eo s ppled. s

11 =. s =. 4s =. 5s Fg.. Respose of he ple resg o els foudo wh dsruve properes due o hrmo lod ppled he ere of eleme 9 5. RESULTS COMPARISON To esure he rell of he resuls he sme squre ple resg o wo dffere pes of els foudos (Boussesq's model d kler's model) re osdered. The ples re eed hrmo lod epressed equo () d ppled o he ere of he eleme 9 (fg. 7). Fgure ompres he verl dspleme vrl of he ple v 9 he po where he dm lod s I s he foudo wh dsruve properes (Boussesq's model) ppled where ( ) d ( ) II he foudo wh sprg model (kler s model). Of oe s h he verl dsplemes of he ple for he hlf-spe wh dsruve properes s lws less h he sme pe of dspleme whe he dsruve properes of hlf-spe re egleed hus provg h he hlf-spe dsruve properes fudmell fluees he dm lss. v ( m).6.4. ( II ) ( I ) ( s) -.6 Fg.. Comprso of he dspleme vrl of po of he ple wo dffere models

12 6. CONCLUSION Usg sem-l pproh dm lss of ems d ples resg o he surfe of els hlf-spe wh dffere models ws heved o deerme he Ege frequees url shpe ple respose o eerl dm lods d oher phsl mgudes. Deermo of he reve fores he o oe represeg he ero pheome ewee he ple d hlf-spe surfe s eessr o fd he ohers phsl mgudes. For hs purpose s mperve o sud Gree's fuo defg he dsplemes of he o oe (.e. o prolems pheome). The luled resuls were ompred ssforl o he sme ple resg o he surfe of els foudo usg kler's model. Addoll he oed soluo s sem-ll d herefore e redl ompued o e more omple wh egeerg pplos. As suh hs work represes fudmel dve he solvg of more ompled dms prolems.the e sep s o sud he ples resg o els foudo wh erl properes. REFERENCES Zhemohk B.N. d Ss A.P. (96) Prl Mehods of he Clulo of Bems d Ples Resg o Els Foudo. Srod Pulshg Comp. Mosou (Russ Edo). Gl L.A. e ll (976) Developme of he heor of o prolems he USSR. Nk Pulshg Comp. Mosou (Russ Edo). Guefoud S. Boskov S.V. d Lefer D.F. (9) Dm lss of em resg o els hlf-spe wh erl properes Sol Dms d Erhquke Egeerg 9 (9) Tmosheko S. d oowsk-kreger S. (959) Theor of ples d shells. MGrw-Hll Book Comp INC. New York (Russ Edo). Guefoud S. Boskov S.V. d Lefer D.F. () A R's mehod sed soluo for he o prolem of deformle regulr ple o els qurer-spe Ierol Jourl of Solds d Sruures 47 () 8-89.

Integral Equations and their Relationship to Differential Equations with Initial Conditions

Integral Equations and their Relationship to Differential Equations with Initial Conditions Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp