COMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS

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1 ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant Lectue Pofesso of echancs Insttute of ppled echancs Unvesty of skolc skolc-egyetemváos H-355 Hungay e-mal: mechlen@un-mskolc.hu mechecs@un-mskolc.hu bstact In ths pape the complementay enegy method s fomulated fo cuved two-laye composte beam wth ntelaye slp. It s assumed that each cuved laye sepaately follows the Eule-Benoull hypothess and the load-slp elaton fo the flexble shea connecton s a lnea elatonshp. n example llustates the applcaton of the developed fomulaton.. INTRODUCTION The consdeed cuved two-laye composte beam wth unfom coss secton s shown n Fg.. In the cylndcal coodnate system the cuved laye ( ) occupes the space doman B ( ) O z B z z 0 () whee s the coss secton of beam component B ( ) and the common bounday of and s denoted by B B B B z c 0 z 0.5 t () whee t s the thckness of the coss secton. The plane z 0 s the plane of symmety fo the whole cuved beam. The connecton between and s pefect n adal decton but n the dsplacements t may have jump n ccumfeental decton whch s called the ntelaye slp. The devaton of the govenng equatons s based on the next dsplacement feld [3] B u ue 0 ve we z w u u z B B B (3) d u v z B. (4) d Fom the defnton of the ntelaye slp t follows that (Fg. ) s c (5). B DOI: /musc.06.

2 Fgue. Two-laye composte beam and ts coss secton. ccodng to pape [] the expesson of the ntelaye shea foce s whee u u k Fgue. Illustaton of ntenal foces and couples. T kc t (6) s the slp modulus. Fomulae of ntenal foces and couples n tems of ae as follows (Fg. ) and E N d W E (7) 0 d R d d d E W E (8) d 0 (9) N d N N d dn S d f (0) d Q K K kc t () 3. In Eqs. (7-) N s the nomal foce s the bendng moment S s the shea foce Q s the moment of ntelaye shea foce about axs z f s the appled extenal foce n ccumfeental decton [] as shown n Fg. futhemoe E E d () E0

3 E0 d d d u R d (3) E E W u. Hee E s the modulus of elastcty of beam component B. The expesson of the stan enegy of the two-laye composte beam wth mpefect shea connecton n tems of W W s as follows [3] E0 d d U W EW EW R d d 0 d d E E K d. d d (4) Fom the system of equatons we obtan E0 d d W E E N R d d (5) d EW E d (5) E W E d d (5)3 N W H (6) E0 E N d E R E (7) d H E0 E N d E R E (8) d H H E0. E E R (9) The expesson of stan enegy n tems of N and Q can be obtaned fom Eq. (4) and Eqs. (6-9).

4 E0 U N Q N H R 0 E0 E E N N E R E E0 E E N N E R E E0 E E N E R E E0 E H E N Q d. E R E K (0). PRINCIPLE OF THE INIU OF STRIN ENERGY Equatons of equlbum fo the two-laye composte beam wth weak shea connecton can be wtten n the fom [3] d N df 0 N f d d () d d Q 0 Q 0. d d () Next we gve some typcal bounday condtons [] Fxed end: u 0 du d (3) Smply suppoted end n adal decton: N u 0. (4) Fee end: N 0 S (5) Intenal foces whch satsfy equatons of equlbum whch ae Eq. (0) Eqs. (- ) and the foce bounday condtons whch efe to N S and ae called statcally admssble ntenal foces. Next we fomulate the pncple of mnmum of complementay enegy fo homogeneous geometcal bounday condtons such as (3) (4)4. In ths case the pncple of mnmum of complementay enegy says that whee N N S S U N S Q U N S Q (6) Q Q ae the exact solutons of the consdeed bounday value poblem of statcal equlbum and N N S S Q Q ae only a statcally

5 admssble felds of the system of ntenal foces fo the consdeed equlbum poblem. Equalty n (6) s eached only f N N S S Q Q. (7) 3. DETERINTION OF STRESSES Fom Eqs. (3) (4) and the stan-dsplacement elatonshp and Hooke s law the followng fomula can be deved fo the ccumfeental nomal stess [3] W d E z B. d (8) ssumng that z 0 and statng fom the equlbum equaton z 0 z (9) we get fo the sheang stess b d b a. (30) 4. NUERICL EXPLE The cuved beam and ts load s shown n Fg. 3. The next data ae used: a 0.04 m b 0.0 m c 0.03 m t 0.03 m E E 8 0 f 000 N. The statcally admssble ntenal foces 3 and couples ae as follows Fgue 3. Smply suppoted cuved beam loaded by unfom adal load.

6 N f cos sn tan (3) q p mp sn p (3) q pmp p Q cos. (33) p These ntenal foces and couples satsfy the equatons of equlbum () () and the foced bounday condtons that s N 0 N 0 0. (34) In the pesent case we have U U m p q.... p The unknown constant m p... q s obtaned fom the mnmum condton of the complementay p enegy whch s expessed as U m p 0 p... q. (35) Fgue 4 shows the bendng moment dagams and fo two dffeent values 3 of the slp modulus k 50 N / m and k 50 N / m. The slp functon s obtaned as c s Q (36) K 3 and ts fgues fo k 50 N / m and k 50 N / m ae shown n Fg. 5. The nomal stess dstbuton n some coss sectons ae llustated n Fg. 6. The 3 postons of the coss sectons ae gven by Nm Nm 3 k 5 0 N / m Fgue 4. Bendng moment dagams. k 5 0 N / m

7 s m s m 3 k 50 N / m k 50 N / m Fgue 5. Plots of slp functons. m m 3 k 50 N / m / 4 3 / 4 / 0 0 Fgue 6. Plots of k. 50 N / m / 4 3 / 4 / 0 0 m m 3 k 50 N / m k 50 N / m 0 3 / 4 /4 / 0 In the same coss secton as n above the sheang stess dstbutons ae pesented n Fg. 7. The gaphs of the von ses stesses 3 ae shown n Fg. 8 fo Fgue 7. Gaphs of.

8 3 k 50 N / m m k 50 N / m 0 / 4 3 / 4 / 0 / 4 3 / 4 / some typcal coss secton as consdeed n Fgs. 6 and 7. ll numecal solutons ae obtaned wth 4. CONCLUSIONS q 5. The pape pesents the applcaton of the mnmum theoem of complementay enegy fo two-laye composte cuved beam wth weak shea connecton. By the combnaton of fomulated mnmum theoem wth the Rtz method the soluton of two-laye composte beam wth smply suppoted end condtons loaded by unfom adal load s gven. The obtaned numecal esults can be used as benchmak soluton to check the accuacy of solutons obtaned by dffeent appoxmate methods such as FE fnte dffeences etc. CKNOWLEDGEENTS Ths eseach was (patally) caed out n the famewok of the Cente of Excellence of Innovatve Engneeng Desgn and Technologes at the Unvesty of skolc and suppoted by the Natonal Reseach Development and Innovaton Offce NKFIH K570. REFERENCES Fgue 8. The plots of Von ses stesses. [] ECSEDI I. DLUHI K.: lnea model fo the statc and dynamc analyss of non-homogeneous cuved beams. ppled athematcal odellng 9() [] ECSEDI I. LENGYEL. J.: Cuved composte beam wth ntelaye slp loaded by adal load. Cuved and Layeed Stuctues () [3] ECSEDI I. LENGYEL. J.: Enegy methods fo cuved composte beams wth patal shea nteacton. Cuved and Layeed Stuctues ()