Finite Difference Method Applied for the Beams on Elastic Foundation Theory. Metoda konečných diferencí použitá pro nosníky na pružném podkladu teorie

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1 Fne Dfference Meho Apple for he Beams on Elasc Founaon Theory Meoa onečných ferencí použá pro nosníy na pružném polau eore Mare IKODÝM, Karel FYDÝŠEK Absrac: Ths paper s focuse on he heory of beams on elasc (Wnler's) founaon. For soluon of hese problems of mechancs, he Fne Dfference Meho can be apple. The basc nformaon abou fne fferences an her applcaon are eplane. Absra: Článe je zaměřen na eor nosníů na pružném (Wnleroě) polau. Pro řešení ěcho úloh mechany, může bý použa meoa onečných ferencí. Zálaní nformace o onečných ferencích a jejch aplacích jsou ysěleny. Keywors: elasc founaon, beams, Cenral Dfference Meho, numercal mehos Klíčoá sloa: pružný pola, nosníy, meoa cenrálních ferencí, numercé meoy. Inroucon o he Theory of Beams on elasc Founaon Fg. Elemen of a Beam on Elasc Founaon. M.Sc. Mare IKODÝM, Ph.D., Deparmen of Mahemacs an Descrpe Geomery, VŠB - Techncal Unersy of Osraa, 7. lsopau /7, Osraa, Czech epublc, Phone , E-mal mare.noym@sb.cz. Assoc. Prof. M.Sc. Karel FYDÝŠEK, Ph.D., IG-PAED IGIP, Deparmen of Mechancs of Maerals, Faculy of Mechancal Engneerng, VŠB - Techncal Unersy of Osraa, 7. lsopau /7, Osraa, Czech epublc, Phone , E-mal arel.fryryse@sb.cz.

2 The basc analyss of benng of beams on an elasc founaon s eelope on he assumpon ha he srans are small. In hs cone, an elasc founaon s efne as a suppor whch s connuously or sconnuously srbue along he lengh of he beam. The reacon force /m / srbue n a founaon s recly proporonal o he eflecon () /m/ of a beam. The sragh beams on elasc founaons, see Fg., can be escrbe by ornary lnear fferenal euaon: where: GA m GA h (.) E /Pa/ s moulus of elascy of he beam, J y A /m / s he major prncpal secon momen of area A /m / of he beam cross-secon, // s shear eflecon consan of he beam, G /Pa/ s shear moulus of he beam, // s normal force, /m / s srbue loa (nensy of force), m // s srbue couple (nensy of momen), / eg / s coeffcen of hermal epanson of he beam, h /m/ s eph of he beam an / eg/ s ransersal emperaure ncreasng n he beam. Euaon (.) s ere for he suaons when npu parameers E, J,,, G, A, an h are consan. For more nformaon abou he eraon of e. (.), see references [] o []. From he Wnler's heory [], [] s een ha: bk, (.) /Pa/ where funcons: s sffness of he founaon an K K /m / s moulus of he founaon whch can be epresse as funcons of arable /m/ (.e. longunal changes n he founaon) an b /m/ s wh of he beam. Hence, from e. (.) an (.) follows: GA A m GA h.(.) In he mos suaons, he nfluences of shearng force, emperaure an nensy of momen can be neglece (or he beam s no epose o hem). Hence, from e. (.) follows:. (.). Fne Dfferences Le us e beam no noes "" eually space (wh sep /m/ ) along s lengh, see Fg.. Deflecon cure of a beam s appromae by polygon cure, see Fg..

3 Fg. Sole Beam s De no noes. Fg. Appromaon of he Deflecons by Polygon Cure an Appromaon of Frs Derae. Fne fferences can be efne as an appromaon of eraes. Hence, for he alue of frs erae, hree ypes of fferences can be efne accorng o Fg. : Bacwar fference a he pon : () ) an(. (.)

4 Forwar fference a he pon : () ) an(. (.) Cenral fference a he pon : ) an( () () (). (.) In some references (for eample []) are symbols "-", "+" noe as "-½" an "+½". Cenral fferences are more accurae, herefore hey wll be apple n he followng e. Smlarly, he hgher eraes (a he pon ) can be appromae by he cenral eraes as: (), (.) (). (.) (). (.) (). (.7) () 0. (.8). Cenral Dfference Meho Accorng he Cenral Dfference Meho, he fferenal euaons (.) can be appromae a he general pon as: () (), (.) where an are sffness of he founaon an srbue loa a he pon. From e. (.), (.) an (.) follows:, (.). (.) Euaon (.) can be wren for all noes n,,, 0, (.e. se of n+ lnear euaons followng from he screzaon of e. (.)). Ths se of euaons ogeher wh four screze bounary conons lea o he soluon of sysem of n+ lnear euaons.

5 Hence, alues of a each noe (.e. alues of n+ eflecons) can be recee, see also reference [] an []. oe: f sep 0 hen numercal soluon conerge o eac soluon.. Conclusons Ths fle shows eraons an way of applcaon of Cenral Dfference Meho as a numercal meho suable for he soluon of he beams on elasc founaon. For more nformaon abou applcaons of Cenral Dfference Meho, see [], [] an []. Anoher way of he soluons an applcaon of Cenral Dfference Meho n he branch of srucures on elasc founaon s presene n []. Acnowlegemens Ths wor was suppore by he Mnsry of Inusry an Trae of he Czech epublc as he par of projec MPO FT-TA/08. eferences [] Fryrýše, K.: osníy a rámy na pružném polau (Beams an Frames on Elasc Founaon ), monograph, Faculy of Mechancal Engneerng, VŠB - Techncal Unersy of Osraa, ISB , Osraa, Czech epublc, 00, pp.. [] Fryrýše, K., Jančo,.: osníy a rámy na pružném polau (Beams an Frames on Elasc Founaon ), monograph,všb - Techncal Unersy of Osraa, ISB , Osraa, Czech epublc, 008, pp.. [] Fryrýše, K., oým, M.: osníy a rámy na pružném polau (Beams an Frames on Elasc Founaon ), monograph, VŠB - Techncal Unersy of Osraa, ISB , Osraa, Czech epublc, 00. [] Jones, G.: Analyss of Beams on Elasc Founaons Usng Fne Dfference Theory, ISB , Thomas Telfor Publshng, Lonon, UK, 997, pp.. [] oým, M., Fryrýše, K.: Fne Dfference Meho Apple for he Beams on Elasc Founaon Eamples, In CD-OM: Boo of Proceengs MECHAICAL STUCTUES AD FOUDATIO EGIEEIG 00, ISB , VŠB - Techncal Unersy of Osraa, Faculy of Mechancal Engneerng, Deparmen of Mechancs of Maerals, Osraa, 00. eewer: prof. Ing. Hors GODEK, DrSc., VŠB - Techncal Unersy of Osraa, Czech epublc