The solution of Euler-Bernoulli beams using variational derivative method

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1 Scetfc Research ad Essays Vol. 5(9), pp. 9-4, 4 May Avalable ole at ISSN Academc Jourals Full egth Research Paper The soluto of Euler-Beroull beams usg varatoal dervatve method Atlla Özütok * ad Arfe Ak Selcuk Uversty, Egeerg Archtectural Faculty, 43, Koya, Turkey. Accepted 7 Aprl, Fte dfferece methods are ofte used for aalyzg structures govered by complex dfferetal equatos. The fte dfferece method, well kow as a effcet umercal method, was formerly appled to the case of beam ad plate problems. The basc dsadvatages of ths method are the requremet of out-of-rego pots durg the soluto process ad the dffculty of mplemetg the boudary codtos alog rregular boudares. I ths study, the varatoal dervatve method was proposed to solve the fuctoal of the Euler Beroull beam obtaed by Gâteaux dfferetal method. The ma reaso for the preferece of ths method over the fte dfferece method was the elmato of the eed for the out-of-rego doma pots that complcate the applcato of the fte dfferece method. The momets ad deflectos of beams wth costat ad varyg cross-sectos ad varous support types were calculated order to demostrate the applcablty of the method. The performace of ths formulato s verfed by comparg the obtaed results wth the results of the umercal examples the lterature. Key words: Fte dfferece, varatoal dervatve, beams. INTRODUCTION We should bear md that all methods of structural aalyss are essetally cocered wth solvg the basc dfferetal equatos of equlbrum ad compatblty, although, some of the methods ths fact may be obscured. Aalytcal solutos are lmted to the cases where the load dstrbuto, secto propertes ad boudary codtos ca be descrbed by mathematcal expressos. However, the umercal aalyss methods are geerally more practcal for complex structures. Moreover, varous methods for fdg sutable approxmate solutos have bee uder cotuous developmet for decades. Amog these methods, Fte Dfferece Method (FDM), Raylegh- Rtz Method, Galerk Method, east-squares Method, Fte Elemet Methods (FEM) have domated the applcatos to problems egeerg. I FDM, a umercal soluto of the dfferetal equato for dsplacemet or stress resultat s obtaed for chose pots o the structure, referred to *Correspodg author: E-mal: aozutok@selcuk.edu.tr. Tel: Fax: as odes or smply as pot of dvso. Durg the study performed o the fte dfferece aalyss of the systems ad the comparso of the methods used wth fte elemet techques, we readly recogzed a very close relatoshp betwee fte dfferece ad fte elemet procedures. However, the FEM may appear to be a better choce tha FDM owg the easer defto of the boudary codtos. The FDM has bee comprehedsvely formulated may studes. We ca classfy the FDM to two groups accordg to the lterature amely, Covetoal Fte Dfferece Method (CFDM) ad Fte Dfferece Eergy Method (FDEM). The soluto by the CFDM s obtaed by applyg the fte dfferece expressos o the area equatos drectly. The dsadvatages of the method clude the dffcultes of mplemetg the boudary codtos o rregular boudares ad accurately represetg the rregular domas. The varatoal method s a powerful method that ca be used for both approxmate solutos ad formulato of problems (Reddy, 986). May researchers have developed the FDEM based o varato order to overcome the dffculty stated prevously. Houbolt used ths method 958 to perform

2 Sc. Res. Essays the statc aalyss of the beams ad plates. I 983, Barve ad Dey ad 99 Sgh ad Dey appled the FDEM to the statc, vbrato ad stablty tests of the plates. Arsla (4) studed o the fuctoal of the Euler-Beroull beam obtaed by usg the Gâteaux dfferetal approach to solve the fuctoal by the ad of the varatoal dervatve method based o FDM, ad cosequetly, the momets ad deflectos of the beams wth costat ad varyg cross-sectos ad varous support types were determed by cosderg the out-ofrego pots. Gelfad ad Fom (963) carred out the bucklg aalyss of the elastc bars usg the varatoal dervatve method. Gürsoy (4); Eratl ad Gürsoy (5) solved all the bucklg problems of elastc bars usg the methods of varatoal dervatve ad fte dfferece ad to determed the elemet matrx of the bar elemet usg varatoal dervatve method. As a result of ths study, the varatoal dervatve method was foud to be extremely smlar but more advatageous whe compared to FDM; due to ot beg eed of outof-rego pots that complcate the applcato of FDM. I ths work, we cosder the soluto for the Euler- Beroull beam s fuctoal obtaed by usg the Gâteaux dfferetal approach. The purpose of ths study s to solve the fuctoal by the ad of the varatoal dervatve method based o FDM. The momet ad deflecto values were selected as the ukows of the fuctoal, modfed by Aköz (985). Some sgfcat advatages of the Gâteaux dfferetal approach were explaed by Özütok (). The elemet matrx for Euler-Beroull beam s derved based o the varatoal dervatve. The performace of the matrx for bedg aalyss s verfed wth a good accuracy by comparg wth the umercal examples ad aalytcal solutos preseted the lterature. The fuctoal for Euler-Beroullı beam The equatos volvg the boudary codtos for Euler-Beroull beam whch ca be wrtte the operator form as: Q y f () Havg obtaed the feld equatos, oe eeds a method to obta the fuctoal. We beleve that the Gâteaux dfferetal method s sutable for ths am. Sce ths method was extesvely used ad explaed other studes (Aköz et al., 99), for the sake of smplcty, the basc steps ad deftos wll be summarzed brefly. The Gâteaux dervatve of a operator s defed as Q ( y + τ y ) d Q ( y ; y ) τ () τ Where, s a scalar. A ecessary ad suffcet codto for Q to be a potetal s (Ode ad Reddy, 976) dq( y; y), y * dq( y; y * ), y (3) Where, paretheses dcate the er products. If the operator Q s a potetal, the the fuctoal whch correspods to the feld equatos s gve by l I ( y) Q( sy), y ds (4) Where, s s a scalar quatty. The clear form of the fuctoal correspodg to the feld equatos are obtaed as the followg (Arsla, 4), M I ( y ) [ M, v ] [ q, v ], [ ˆ, ] M v M E I ε v, Tˆ ( M Mˆ ), v ( vˆ + v ), T σ σ (5) Where, the subscrpts ε ad σ dcate the geometrc ad dyamc boudary codtos, respectvely. As see Equato (5), the momet ad deflecto values selected as ukows of the fuctoal are tegrated smultaeously to the fuctoal ad are obtaed easly wthout eedg the boudary codtos to be kow. Varatoal dervatve f ( x, x,... x ), f For a fucto wth multple argumets, the dfferetal df ca be wrtte as df g( x, x,.. x) dx, the g ( x, x,.. x) dx s called the dervatve of f wth respect to,..., ad s gve by f x g ( x, x,.. x ) Notce that the dervatve of fucto ε x, for (6) f ( x,,.. x ) wth respect to x wll be aother fucto amed g ( x,,.. x ). Smlarly, f the frst varatoal dervatve as of a fuctoal [ ] b (7) I y ( x ) F ( x, y, ) dx a

3 Ozutok ad Ak M - M v v - M + Fgure. Nodal pots ad the elemets. s wrtte as b δ I y( x) g( x) δ y( x) dx [ ] a v + (8) The, the fuctoal dervatve of I wll be δ I g( x) (9) δ y( x) The fuctoal dervatve of a fuctoal I [ y( x )] s aother fucto g( x ). Image that we approxmate the fucto y( x) by a pece-wse lear curve that passes through a set of pots ( x, y ),,..., +, where x x, y y( x ), x a ad +. et y( x ) be subjected to essetal boudary codtos, so that are fxed. The, the y y( a) ad y y( b) + fuctoal I [ y( x )] ca be approxmated by a sum. y + y I ( y, y,... y ) F( x, y, ) The partal dervatve of argumets s, x () b I wth respect to oe of ts I F y + F y y F y k yk yk yk yk k y x + () Comparg ths wth the defto of the fuctoal dervatve, the followg equato s wrtte δi d I F ( x, y, ) ( F ( x, y, ) lm δ y y dx yk () The, the above expresso s called varatoal dervatve (Kag et al., 6), ad the frst codto for y(x) to make I [y(x)] fuctoal maxmum s I. Usg ths codto, the δ I δ y expresso wll be equal to zero at every pot. Varatoal dervatve method dffers from FDM by ot eedg to use the out-of-rego pots (Eratlı ad Gürsoy, 5). A formulato based o varatoal dervatve method that does ot requre ay out-of-rego pots s developed to calculate the bedg momets ad deflectos of the bars makg use of the fuctoal obtaed here. The, the elemet matrx ca be wrtte as M e I M e v e q v e EI (3) by usg ths equato [ ( )] I y x. Here, x + x ad e s elemet umber. If the followg relatoshps are replaced wth the dervatve expressos the fuctoal v v M M v, M + + (4) ad the dervatves are take wth respect to the momets ad deflectos defed as the ukows at each odal pot of th elemet of the bar system show Fgure ad are equalzed to zero, as follows δ I d I F ( x, y, ) ( F ( x, y, ) lm δ y y dx yk the, the elemet matrx s obtaed as λ M M + v q v + Where, λ EI. Numercal examples (5) (6) Several problems of beams wth varous boudary codtos have bee studed the past. I order to check the performace of varatoal dervatve method, varous problems are solved ad the results are compared wth those of smlar examples the lterature.

4 Sc. Res. Essays q Fgure. The smply supported beam to be aalyzed uder uformly dstrbuted trasverse loadg. Table. Comparso of the VD ad exact solutos. Values ( v exact 4 5 q α ) 384 EI Number of members CFD FDEM VD of (Arsla, 4) VD of ths study Exact Mx 3, v Fgure 3. The dstrbuto of momet ad deflecto alog the beam obtaed wth VD ad MFEM. Smply-supported beam wth uform loadg Frst of all, we decded to cosder the accuracy ad covergece of the varatoal dervatve elemet matrx o a smply supported beam subjected to a uform vertcal loadg of q kn/m as show Fgure. The materal propertes ad dmesos are E x 7 kn/m, m, h.5 m ad b. m. The maxmum deflectos obtaed from varous types of aalyses cludg CFD, FDEM ad VD of Arsla (4) gve the lterature ad the curret study are preseted Table to allow comparso wth the exact soluto. As t s see, the deflectos foud from the curret study coverge to the exact results. CFD: covetoal fte dfferece method; FDEM: fte dfferece eergy method, VD: Varatoal dervatve. I addto to the maxmum deflecto values, the dstrbuto of momet ad deflecto alog the beam axs from VD of ths study ad MFEM s plotted ad show Fgure 3. The effect of the umber of elemets

5 Ozutok ad Ak 3.4. M (KNm), v (m) Fgure 4. Deflecto ad momet covergece vs. umber of elemets. Fgure 5. A catlever beam wth uform dstrbuted loadg. q Table. Comparso of VD results wth the lterature ad exact results. m h b/h s Gul, 3 MFEM (v x ) Deflecto VD of ths study Exact used beam dscretzato o the deflecto ad momet are show Fgure 4, where the odmesoalzed maxmum deflecto v v EI q 4 ad momet M M q of a smply supported beam subjected to a uformly dstrbtued load s plotted. Catlever beam wth uform dstrbuted loadg: Costat cross-secto case Cosderg the problem of fdg the trasverse deflecto of a catlever beam uder a uform dstrbuted loadg by usg varatoal dervatve method Fgure 5; the deflecto ad momet of the free tp versus the umber of elemets are plotted as show Fgure 6. I the fgure, the deflecto ad momet coverge to the exact soluto results wth a error proportoal to the umber of elemets used. Catlever beam wth uform dstrbuted loadg: Varable cross-secto case I order to test the effect of heght varato o the deflecto ad momet, a catlever beam subjected to a uform dstrbuted loadg as show Fgure 7 was aalyzed for dfferet heghts. I Fgure 7, h b ad h s are the heght of beam cross-secto at the left ad rght had sdes of the beam, respectvely. The varato of the momet of erta alog the beam ca be defed for elemet by I 3 bh + 3 z (7) Where, m-, m h b /h s ad z s the legth of elemet. The maxmum deflectos for four cases wth dfferet heght ratos are show Table. I the table, results of the study by (Gul, 3) ad exact results are gve for comparso. DISCUSSION AND CONCUSION I ths study, the solutos of Euler Beroull beams of costat ad varyg cross-sectos uder uformly dstrbuted loadg are carred out usg varatoal dervatve method wthout the eed of out-of-rego pots

6 4 Sc. Res. Essays.3.5 V3 Mm Fgure 6. Deflecto ad momet covergeces wth respect to umber of elemets. q h b h s Fgure 7. A catlever beam wth varable cross-secto uder uformly dstrbuted loadg. of FDM. The elemet matrx of the beam s obtaed by applyg the fuctoal (Arsla, 4) to the varatoal dervatve method whch was derved for straght-axs bars usg Gâteaux dfferetal method. Besdes, observg the smlarty of the proposed elemet matrx to that of FDM, the obtaed matrx s superor beam problems sce t does ot requre ay out-of-rego pots. A computer code s developed Fortra to compute the proposed elemet matrx by cosderg the boudary codtos. Usg ths computer code, the ukow momets ad the dsplacemets at the odal pots are calculated ad foud to be agreemet wth the results of the prevous studes the lterature ad the exact results. I the etre example problems cosdered, the proposed approach coverges to the exact soluto as log as the umber of members s creased. Therefore, t s expected that the method wll provde a better approach to the case of hgher degree beam problems. REFERENCES Aköz AY (985). A ew fuctoal for bars ad ts applcatos, ( Turksh), IV. Natoal Mechacs Meetg, Turkey pp. -. Aköz AY, Omurtag MH, Doruolu AN (99). The mxed fte elemet formulato for three dmesoal bars. It. J. Solds Struct. Ege 8: Aköz AY, Özutok A (). A fuctoal for shells of arbtrary geometry ad a mxed fte elemet method for parabolc ad crcular cyldrcal shells, It. J. Numer. Meth. Ege 47: Arsla A (4). The soluto of Euler-Beroull beams by varatoal dervatve method, ( Turksh), M.Sc. Dssertato, Selcuk Uversty, Koya. Barve VD, Dey SS (983). Isoparametrc fte dfferece eergy method for plate bedg problems, Computer Struct. 7: Eratl N, Gürsoy A (5). Varatoal dervatve for bucklg loads of beams ad frames, It. Appl. Mech. 4(7): Gelfad IM, Fom SV (963). Calculus of varatos, Pretce-Hall, New Jersey. Gul M (3). Dyamcs aalyss of stffeed plate, ( Turksh), M.Sc. Dssertato,.T.Ü., stabul. Gürsoy A (4). The aalyss of the bucklg loads for beams ad frames wth varatoal dervatve ( Turksh), M.Sc. Dssertato,.T.Ü., stabul. Houbolt JC (958). A study of several aerothermoelastc problems of arcraft structures Hgh-Speed Flght, eema, Zurch Kag K, Weberger C, Ca W (6). A short essay o varatoal calculus, Staford Uversty. Ode JT, Reddy JN (976). Varatoal Method Theortcal Mechacs. Sprger, Berl. Reddy JN (986). A Itroducto to the Fte Elemet Method, McGraw-Hll, New York. Sgh JP, Dey SS (99). Varatoal fte dfferece method for free vbrato of sector plates, J. Soud Vb. 36: 9-4.