Transactions of the VŠB Technical University of Ostrava, Mechanical Series. article No. 1907

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1 Transactons of the VŠB Techncal Unversty of Ostrava, Mechancal Seres No., 0, vol. LVIII artcle No. 907 Marek NIKODÝM *, Karel FYDÝŠEK ** FINITE DIFFEENCE METHOD USED FO THE BEAMS ON ELASTIC FOUNDATION PAT (THEOY) METODA KONEČNÝCH DIFEENCÍ POUŽITÁ PO NOSNÍKY NA PUŽNÉM PODKLADU ČÁST (TEOIE) Abstract Ths artcle s focused on the theory of straght and curved beams on elastc (Wnkler's) foundaton. For soluton of these problems of mechancs, the Fnte Dfference Method (.e. Central Dfference Method) can be appled. The basc nformaton about fnte dfferences and ther applcaton are explaned. Abstrakt Článek je zaměřen na teor přímých a křvých nosníků na pružném (Wnklerově) podkladu. Pro řešení těchto úloh mechanky, může být použta metoda konečných dferencí (tj. metoda centrálních dferencí). Základní nformace o konečných dferencích a jejch aplkacích jsou vysvětleny. INTODUCTION TO THE THEOY OF BEAMS ON ELASTIC FOUNDATION The basc analyss of bendng of beams on an elastc foundaton, see references [] to [], s developed on the assumpton that the strans are small. In ths context, an elastc foundaton s defned as a support whch s contnuously or dscontnuously dstrbuted along the length of the beam. The reacton Fg. Element of a straght beam on elastc foundaton. force q q x /Nm / dstrbuted n a foundaton s drectly proportonal to the deflecton v v(x) /m/ of a straght beam, see Fg., or proportonal to the radal dsplacement u u () /m/ of a curved beam (crcular arches), see Fg.. * M.Sc. Marek NIKODÝM Ph.D., VŠB - Techncal Unversty of Ostrava, Department of Mathematcs and Descrptve Geometry, 7. lstopadu /7, Ostrava, Czech epublc, Phone , E-mal marek.nkodym@vsb.cz ** Assocate Prof., MSc. Karel FYDÝŠEK Ph.D., ING-PAED IGIP, VŠB - Techncal Unversty of Ostrava, Faculty of Mechancal Engneerng, Department of Mechancs of Materals, 7. lstopadu, Ostrava, tel. (+0) 9 79, e-mal karel.frydrysek@vsb.cz 7

2 Fg. Example of a curved beam on elastc foundaton and ts element. Ths artcle s focused on the soluton of the straght and curved beams on elastc foundaton, see Fg. and, whch leads to the soluton of lnear dfferental equatons va Fnte Dfference Method (.e. Central Dfference Method). DIFFEENTIAL EQUATION FO STAIGHT BEAMS ON ELASTIC FOUNDATION The bendng of straght beams on elastc foundatons, see Fg., can be descrbed by ordnary lnear dfferental equaton: d v N d v d q q dm d q t d t t q () GA GA h where: E /Pa/ s modulus of elastcty of the beam, J /m / s the major prncpal second moment of area A /m / of the beam cross-secton, // s shear deflecton constant of the beam, G /Pa/ s shear modulus of the beam, N /N/ s normal force, q q /Nm / x s dstrbuted load (ntensty of force), m /N/ s dstrbuted couple (ntensty of moment), / deg / s coeffcent of thermal expanson of the beam, h /m/ s depth of the beam and t t / deg / s transversal temperature ncreasng n the beam. Equaton () s derved for the stuatons when nput parameters E, J, N,, G, A, and t h are constant. For more nformaton about the dervaton of eq. (), see references [] to [6]. From the Wnkler's theory, see references [] to [], t holds that: kv bkv () q t 8

3 /Pa/ where functons: k k x s stffness of the foundaton and K Kx /Nm / s modulus of the foundaton whch can be expressed as functons of varable x /m/ (.e. longtudnal changes n the foundaton) and b /m/ s wdth of the beam (see Fg. ). Hence, from eq. () and () follows: d v N d v d kv kv dm d q t d t t q () GA GA h In the most stuatons, the nfluences of shearng force, temperature and ntensty of moment can be neglected (or the beam s not exposed to them). Hence, from eq. () follows the smple form: d v N d v kv q. () DIFFEENTIAL EQUATION FO CUVED BEAMS ON ELASTIC FOUNDATION The bendng of curved beams on elastc foundatons, see Fg., can be descrbed by ordnary lnear dfferental equaton: d u d u dq u, () d d d where: /m/ s radus of the beam, /rad/ s angle varable and parameter // s gven by equaton: k. (6) q From the Wnkler's theory, see references [] to [], t s evdent that: ku bku (7) All others parameters mentoned n equatons () to (7) are explaned n former text. FINITE DIFFEENCES Let us consder an equdstant partton of the beam wth a step (nodes) "" along ts length, see Fg. (unloaded beam) and Fg. (loaded beam). /m/ and nodal ponts Fg. Solved straght beam s dvded nto nodes "". 9

4 Fg. Solved straght beam s dvded nto nodes "". Deflecton curve v = v(x) of a straght beam and ts dervatves are approxmated by polygon curves, see Fg.. Fg. Approxmaton of the deflectons by polygon curve and approxmaton of frst dervatve. Fnte dfferences can be defned as an approxmaton of dervatves. Hence, for the value of the frst dervatve, three types of dfferences can be defned accordng to Fg. : Backward dfference at the pont "": () dvx x v v v tan( ). (8) Forward dfference at the pont "": () dvx x v v v tan( ). (9) Central dfference at the pont "": () () v v () dv x x v v v tan( ). (0) In some references (for example [6]) are symbols "-", "+" noted as "-½" and "+½". Central dfferences (CD) are more accurate, therefore they wll be appled n the followng text. Smlarly, the hgher dervatves (at the pont "") can be approxmated by the central dfferences as: () d vx x v v v v, () () d vx x v v v v v, () 60

5 () d v x x v v 6v v v v, () () d vx x v v v v v v v, () 6 (6) d vx x v 6v v 0v v 6v v v. () 6 6 Smlarly, for a curved beams (.e. approxmatons for dervatves of functon u u () ), CD formulas can be derved by substtuton of varables v (for example u () du d u tan( ) () u () u CENTAL DIFFEENCE METHOD (CDM) FO STAIGHT BEAMS Accordng the Central Dfference Method (CDM), the dfferental equatons () for straght beams can be approxmated at the general pont "" (see eq. () and ()) as: N N k N q v v 6 v v v. (6) where k and q are the stffness of the foundaton and the dstrbuted loadng at the pont "", respectvely. Equaton (6) can be wrtten for all nodes 0,,,, n (.e. set of n+ lnear equatons followng from the dscretzaton of eq. ()), see Fg. 6. Ths set of equatons, together wth four dscretzed boundary condtons, lead to the soluton of system of n+ lnear equatons. There are always four fcttous nodes (-, - and n+, n+) outsde the ends of the beam, see Fg. 6. Hence, values of v at each node "" (.e. values of n+ deflectons) can be receved, see also reference [] and [7]. u etc.). Fg. 6 Example solved n reference [] (straght beam on elastc foundaton loaded by couple M). Note: Theoretcally, f step 0 (.e. n ) then the numercal soluton converge to exact soluton. Fg. 7 and 8 show the sparse matrces arsng from CDM for beam on elastc foundaton loaded by couple M, see Fg. 6. Ths example s solved n reference []. 6

6 Fg. 7 Example solved n reference [] (sparsty patterns of matrces n CDM, number of elements n = ). Fg. 8 Example solved n reference [] (sparsty patterns of matrces n CDM, number of elements n = 0). 6

7 6 CENTAL DIFFEENCE METHOD (CDM) FO CUVED BEAMS Accordng the CDM, the dfferental equatons () for curved beams can be approxmated at the general pont "" (see modfed eq. () and ()) as: u u u. (7) () u u q, u where k, k and q () dq are stffness of the foundaton, parameter and d frst dervatve of dstrbuted load at the pont "", respectvely. Equaton (7) can be wrtten for all nodes 0,,,, n (.e. set of n+ lnear equatons followng from the dscretzaton of eq. ()). Ths set of equatons, together wth fve dscretzed boundary condtons, lead to the soluton of system of n+7 lnear equatons. Hence, values of u at each node "" (.e. values of n+7 deflectons) can be receved, see also reference [] and [7]. Note: Theoretcally, f step 0 (.e. n ) then numercal soluton converge to exact soluton. CONCLUSION Ths artcle shows dervatons and way of applcaton of the Central Dfference Method (CDM) as a numercal method sutable for the soluton of the straght or curved beams on elastc foundaton. For more nformaton about applcatons of CDM, see [], [6], [7] and [8]. CDM seems to be a good alternatve to wdely spread Fnte Element Method. Another ways of the solutons and applcatons of structures on elastc foundaton are presented n [] to [9]. ACKNOWLEDGEMENT Ths work has been supported by the Czech-Slovak project 7AMBSK and Slovak-Czech project SK-CZ EFEENCES [] FYDÝŠEK, K.: Beams and Frames on Elastc Foundaton (Nosníky a rámy na pružném podkladu ), monograph, Faculty of Mechancal Engneerng, VŠB - Techncal Unversty of Ostrava, ISBN , Ostrava, Czech epublc, 006, pp.6. [] FYDÝŠEK, K., JANČO,. et al: Beams and Frames on Elastc Foundaton (Nosníky a rámy na pružném podkladu ), monograph,všb - Techncal Unversty of Ostrava, ISBN , Ostrava, Czech epublc, 008, pp.6. [] FYDÝŠEK, K., NIKODÝM, M. et al: Beams and Frames on Elastc Foundaton (Nosníky a rámy na pružném podkladu ), monograph, VŠB - Techncal Unversty of Ostrava, ISBN , Ostrava, Czech epublc, 00. [] HETÉNYI, M.: Beams on Elastc Foundaton, Ann Arbor, Unversty of Mchgan Studes, USA, 96. [] JANČO,.: Numercal Methods of Soluton of Beam on Elastc Foundaton, In: th Internatonal Conference MECHANICAL ENGINEEING 00, Faculty of Mechancal Engneerng, Slovak Unversty of Technology n Bratslava, ISBN , Bratslava, Slovaka, 00, pp. S-60 S-6. [6] JONES, G.: Analyss of Beams on Elastc Foundatons Usng Fnte Dfference Theory, ISBN , Thomas Telford Publshng, London, UK, 997, pp.6. [7] NIKODÝM, M., FYDÝŠEK, K.: Fnte Dfference Method Used for the Beams on Elastc Foundaton Part (Applcatons), Transactons of the VŠB Techncal Unversty of Ostrava, Mechancal Seres, vol. LVIII, 0 (n ths journal, n prnt). 6

8 [8] KAMIŃSKI, M.: A generalzed verson of the perturbaton-based stochastc fnte dfference method for elastc beams, Journal of Theoretcal and Appled Mechancs, 009, 7,, pp [9] TVDÁ, K.; DICKÝ, J.: Comparson of Optmzaton Methods. In: Proceedngs of the -th Internatonal Conference on New Trends n Statcs and Dynamcs of Buldngs, Bratslava, Slovaka, 00, pp. 6-6, ISBN