Bending and Vibrations of a Thick Plate with Consideration of Bimoments

Transcription

1 Journal of Appled Mathematcs and Physcs, 6, 4, Publshed Onlne August 6 n ScRes. Bendng and Vbratons of a Thck Plate wth Consderaton of Bmoments Мakhamatal K. Usarov, Davronbek М. Usarov, Gayratjon T. Ayubov Insttute of Sesmc Stablty of Structures of the Academy of Scences of the Republc of Uzbekstan, Tashkent, Uzbekstan Receved July 6; accepted 7 August 6; publshed August 6 Copyrght 6 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract The paper s dedcated to the development of the theory of orthotropc thck plates wth consderaton of nternal forces, moments and bmoments. The equatons of moton of a plate are descrbed by two systems of sx equatons. New equatons of moton of the plate and the boundary condtons relatve to dsplacements, forces, moments, and bmoments are gven. As an example, the problems of free and forced oscllatons of a thck plate are consdered under the effect of snusodal perodc load. The problem s solved by Fnte Dfference Method. Egenfrequences of the plate are determned, numerc maxmum values of dsplacements, forces and moments of the plate are obtaned dependng on the frequency of external force. It s shown that when the value of the frequency of external effect approaches the egenfrequency, there occurs an ncrease n dsplacement, force and moment values; that testfes a gradual transton of the moton of plate ponts nto the resonant mode. Keywords Plate, Orthotropc, Isotropc, Dsplacement, Stress, Moment, Bmoment, Bendng, Vbratons. Introducton Theory of plates and shells has a specal place n desgn of structural elements. Specfed theores of plates are bult by many authors. All exstng specfed theores of plates are developed on the bass of a number of smplfyng hypotheses. An overvew of the man statements and common methods of constructng an mproved theory of plates and shells can be found n the works of S. A. Ambartsumyan [], K. Z. Galmov [], Sh. K. Galmov [], Kh. M. Mushtar [4] and others. Statc problem of the bendng of a thck sotropc plate n threedmensonal theory of elastcty s consdered by B.F. Vlasov n [5], whch gves an exact analytcal soluton n How to cte ths paper: Usarov, M.K., Usarov, D.М. and Ayubov, G.T. (6) Bendng and Vbratons of a Thck Plate wth Consderaton of Bmoments. Journal of Appled Mathematcs and Physcs, 4,

2 trgonometrc seres. Monograph by E.N. Bada [6] s devoted to solvng the problem of bendng of orthotropc plates n trgonometrc seres. Numercal results of dsplacements and stresses are obtaned. The authors n [7]-[] deal wth dynamc problems of plates wth ansotropc propertes. Karamooz Ravar M.R., Forouzan M.R. [7] have studed the problems of plates oscllatons. Frequency equatons of orthotropc crcular rng plate were obtaned for general boundary condtons n oscllaton plane. In [8] the soluton of transton oscllatons of rectangular vscous-elastc orthotropc plate are gven for concrete stran models accordng to Flugge and Tmoshenko-Mndln s theores. The paper [9] s devoted to analytcal soluton of the problem of forced steady-state vbratons of orthotropc plate. By the method of superposton the problem s reduced to a quas-regular nfnte system of lnear equatons. In [] an analytcal method of soluton of spatal problem of bendng of orthotropc elastc plates subjected to external loads on upper and lower edges s developed. In [] a problem s consdered of a bendng of orthotropc rectangular plate layng on two-parameter elastc foundaton. Research n the feld of thck plates has shown that n the case of spatal deformaton of a plate along ts thckness there occurs the nonlnear laws of dsplacements dstrbuton and the hypothess of plane sectons s volated. In the cross-sectons of the plate except for the tensle and shear forces, bendng and torsonal moments, there appear the addtonal force factors, the so-called bmoments. The author of the artcle addresses the problem of bendng and vbratons of thck plates based on bmoment theory of plates []-[5], bult as a part of three-dmensonal theory of elastcty, usng the method of dsplacements decomposton n one of the spatal coordnates n Maclaurn nfnte seres. Ths paper gves a bref descrpton of the technque of constructng a theory of plates wth consderaton of bmoments generated due to dsplacements dstrbuton of cross-secton ponts by a non-lnear law. Here the equatons of bmoments are bult wth the equaton of three-dmensonal dynamc theory of elastcty, descrbed on face surfaces of the plate. The bmoments are ntroduced n stress dmensons and are characterzed by the ntensty of generated bmoments. We would use the desgnatons and determnant correlatons of forces, moments, bmoments and equatons of moton relatve to these force factors. Unlke bmoment theory n [4] and [5], here the bmoment equatons are bult wth the equaton of threedmensonal dynamc theory of elastcty, descrbed on face surfaces of the plate. Bmoments are ntroduced n stress dmensons, and they characterze the ntensty of generated bmoments. Determnant relatonshps of forces, moments, bmoments and equatons of moton relatve to these force factors are gven.. Statement of the Problem Consder an orthotropc thck plate of constant thckness H = h and dmensons ab, n plane. Introduce the desgnatons: E, E, E elastcty modul; G, G, G shear modul; ν, ν, ν Posson rato of plate materal. When buldng an equaton of moton the plate s consdered as a three-dmensonal body and all components of stress and stran tensors: j, ε j, (, j =, ) are taken nto consderaton. The components of dsplacement vector are the functons of three spatal coordnates and tme u( x, x, zt, ), u( x, x, zt, ), u( x, x, zt, ). The components of stran tensor ε are determned from Cauchy relaton as: j ε ε ε =, =, =, x x z ε = +, ε = +, ε = +, x x z x z x For orthotropc plate, the Hooke law, n a general case, s wrtten as: E E E, (.а) (.b) = ε + ε + ε (.a) = ε + ε + ε (.b) E E E, = ε + ε + ε (.c) E E E, = G ε, = G ε, = G ε (.d) 644

3 where E, E,, E are the elastc constants, determned through Posson rato and the modul of elastcty n the form [4] [5]. As an equaton of moton of a plate we would use three-dmensonal equatons of dynamc theory of elastcty: x x z + + = ρu (.а) x x z + + = ρu (.b) + + = ρu x x z where ρ s a densty of plate materal. Boundary condtons on lower and upper face surfaces of the plate z = h and z = h are: ( ) ( ) ( ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) (.c) = q, = q, = q, at z = h; (4.a) ( ) ( ) ( ) = q, = q, = q, at z = h. (4.b) Here q, q, q and q, q, q are dstrbuted external loads, appled to upper and lower face surfaces of the plate z = h and z = h along the drecton of ox, ox, oz coordnates axes.. Method of Soluton The methods of buldng the bmoment theory of plates are based on Cauchy relaton (), generalzed Hooke s law (), three-dmensonal equatons of the theory of elastcty (), boundary condtons on face surfaces (4). A proposed bmoment theory of plates s also descrbed by two non-connected problems, each of whch s formulated on the bass of sx two-dmensonal equatons of moton wth correspondng boundary condtons. The components of dsplacement vector are expanded nto Maclaurn nfnte seres n the form: ( k ), ( k) ( k) ( k) ( k) ( k) z z z z uk = B + B + B + B + + B, ( k =, ) h h h h (5.а) z z z z = u A A A A A h h h h (5.b) Here B A are unknown functons of two spatal coordnates and tme B = B x, x, t, A = A x, x, t. In a general case, these functons are determned accordng to the formulae: ( ) u u B h k A h ( k ) k =, (, ), = =! z! z z= z= ( k) ( k ) ( ) The dsplacements n stresses n upper z = h and lower ponts z = h n plate fbers we would desgnate ( ) ( + ) ( ) ( + ) as u, u, ( =, ) and j, j, ( =, ; j =, ). The frst problem of bmoment theory descrbes tenson-compresson and transverse reducton of the plate, and the second one the bendng and transverse shear of the plate. Determnant relatonshps and correspondng equatons of moton of the plate n the frst and second problems are brefly descrbed below. The frst problem s descrbed by the forces and bmoments wth sx generalzed functons ψ, ψ, u, u, rw,, whch are determned by relatonshps: h uk uk k ψ k k h h ( ) u =, = u d z, k =,, (6) u h u W=, r= uzz d h (7) h 645

4 Introduce the external loads for the frst problem ( + ) ( ) q q q q q + q,, q = q = q = (8) The expressons of longtudnal and tangental forces are wrtten as []-[5]: N = EH + E H + EW x x N = E H + E H + E W, x x (9.а) N = N = G H + H x x The ntenstes of the bmoments p, p from tangental stresses, have the expressons ( u ψ ) r ( u ψ ) r p = G +, p = G + H x H x The ntensty of the bmoment p from normal stress s wrtten n the form: W p = E + E + E x x H (9.b) (.а) (.b) The equatons of moton relatve to longtudnal and tangental forces and bmoments from tangental and normal stresses have the form []-[5]: N N N N + + q = ρh ψ, + + q = ρh ψ () x x x x p p p q + + = ρ r () x x H H Note, that the expressons of force factors (9), (), and hence, the equatons of moton of the system (), () s rgorously bult. Ths system conssts of three equatons relatve to sx unknown functons ψ, ψ, u, u, rw,. As could be seen, three equatons are mssed. If n expressons (9.а) the terms EW,EW are omtted, then we would obtan two equatons of moton of classc theory of plates n the form (), snce the equaton of moton () becomes solated and fal. The second problem of bmoment theory conssts of the equatons for bendng moments, torsonal moments, shear forces relatve to sx knematc functons ψ, ψ, u, u, rw,, determned by formulae: h uk uk k ψ k k h h ( ) u =, = uzz d, k=,, () + u Introduce the generalzed external loads for the second problem h u W =, r = udz h (4) h ( + ) ( ) q + q q + q q q =, =, = q q q (5) Bendng, torsonal moments and shear forces, whch are rgorously bult, have the form []-[5]: ( r W ) H M = E + E E x x H (6.а) 646

5 ( r W ) H M = E + E E x x H M H = G + x x (6.b) (6.c) r r Q = G u + H, Q = G u + H (6.d) x x The system of equatons of moton of the second problem conssts of two equatons relatve to bendng, torsonal moments and one equaton relatve to shear force and t s wrtten n the form []-[5]: M M H M M H (7) x x x x + Q = ρψ, + Q = ρψ Q x Q + = Hρ r q (8) x Note, that the expressons of forces and moments (6), hence, the equatons of moton of the system (7), (8) are rgorously bult. Smlar to the frst problem, here three equatons are mssed. The system of equatons of moton (7), (8) conssts of three equatons relatve to sx unknown functons ψ, ψ, u, u, rw,. If n expressons of forces and moments, nto the equatons of moton (7) and (8) conventonally ntroduce u = ψ, u = ψ and E = E =, and the shear modulus G, G substtute for kg z, kg z, (where kz s a shear coeffcent), then an equaton of moton of plates could be obtaned accordng to Tmoshenko s theory. To complete the systems (), () and (7) and (8) t s necessary to buld two more systems, wth three equatons n each. Wrte down three equatons of moton of the theory of elastcty () on face surfaces of the plate z = h and z = + h. Addng and subtractng the equatons of the theory of elastcty () on face surfaces of the plate z = h and z = h, and takng nto account the Hooke s law (), surface condtons (4) and desgnatons (6), (7) and (), (4), two ndependent systems wth three equatons n each could be obtaned. The frst of these systems descrbes the frst problem and has the form: (9) x x H x x H + + = ρu, + + = ρu q q + + = ρw x x H () Here the ntenstes of the bmoments,, under transverse reducton and tenson-compresson of the plate, generated due to,, are: j + j j =, ( =, ; j =, ) (),, are the ntenstes of the bmoments generated due to transverse stresses,, ( + ) ( ) k k k, k,, = + ( = ) = H z z H z z : The second system of equatons obtaned from the equatons of the theory of elastcty () s wrtten n the form: + + = ρu, + + = ρu () () x x H x x H 647

6 x x H q q + + = ρw (4) Here,, are the ntenstes of the bmoments under transverse bendng and shear for the second problem generated due to the stresses,, : j j j =, ( =, ; j =, ) (5) The ntenstes of the bmoments,,, generated due to the stresses,,, under transverse shear and bendng are wrtten n the form: ( ) ( ) ( ) ( ) + + k k k k = = = +, (, ), H z z H z z The ntenstes of the bmoments,,,,, are determned from Hooke s law () wth consderaton of the condtons on face surfaces z = h and z = h (4) as: E E = E + E + q, = E + E + q, = G + x x E x x E x x (7) E E = E + E + q, = E + E + q, = G + x x E x x E x x (8) Here E E E E = E E, E = E E, E = E E. E E E,, The expressons of the ntenstes of the bmoments are determned by the soluton of the system of lnear algebrac equatons relatve to coeffcents of Maclaurn seres B ( ) ( ), B, A+, ( =,,, ), whch are obtaned by the substtuton of the seres (5) nto the condtons on face surfaces at z = h and z = h (4) and desgnatons (6), (7). (6) = G ψ u 6 W G H E + E q + q H x E x x x = G ψ u 6 W G H E + E q + q H x E x x x (9.а) (9.b) r W W q W q = E 4 8 HE HE H H x x G x x G () E + E q x x The expressons of the ntenstes of the bmoments,, are determned by the soluton of the system of lnear algebrac equatons relatve to coeffcents of Maclaurn seres B ( ) ( ) +, B+, A, ( =,,, ), whch are obtaned by the substtuton of the seres (5) nto the condtons on the face surfaces at z = h and z = h (4) and desgnatons (), (4). ψ u W G = G 4 8 H E + E q + q, (.а) H H x E x x x ψ u W G = G 4 8 H E + E q + q, (.b) H H x E x x x 648

7 r W W q W q = 6E EH EH H x x G x x G E + E q. x x Wrte down the formulae to determne the dsplacements on the face surfaces of the plate z = h and z = + h: ( ) u = u u, u = u + u, =,, u = W W, u = W + W. () ( ) ( + ) ( ) ( + ) Formulae for stresses on the face surfaces of the plate z ( ) ( + ) = h and z = h have the form: ( ) j = j j, j = j + j, =, ; j =, (4) Maxmum values of dsplacements and stresses of the plate are reached on the face surfaces of the plate and are determned by the solutons of the frst and second problems by the formulae () and (4). Note, that the expressons of ntenstes of the bmoments (), (7), (8), (9), (), () and () are bult for the frst tme and are new n the theory of plates. Consder the boundary condtons of a dscussed problem for the thck plates. ) On the border of the plate the dsplacements are zero. On the edges of the plate x = const and x = const the condtons should be as follows: () ψ =, ψ, r =, u = ; u =, W = (5) ψ =, ψ =, r =, u = ; u =, W = (6) ) On the border x = const the plate s supported. The followng condtons should be satsfed: N =, N =, r =, =, =, W = (7) M =, M =, r =, =, = W = (8) ) On the border x = const the plate s free of supports. The followng condtons should be satsfed N =, N =, p =, =, =, = (9) M =, M =, Q =, =, = = (4) Boundary condtons on the border x = const are smlarly wrtten. When studyng the problem of transverse bendng and shear t s enough to consder only the second problem wth the equatons of moton (7), (8), (), (4) and boundary condtons (5)-(4). 4. Soluton of Tests Problem As an example, consder the forced harmonc vbratons of a cantlever rectangular plate fxed on both ends under the effect of harmonc perodc external load: ( ) ( ) ( + ) ( + ) π x πx q =, q =, q =, q =, q =, q = qsn sn sn ( ωt + β) (4) a b where q, ω, β s an ampltude, frequency and the mode of vbraton of an external load, respectvely. Note, that f ω =, we obtan the problem of statc bendng of the plate. Substtutng (4) nto (8) and (5) determne the load terms of the equaton of moton. For a plate fxed on both ends the boundary condtons are wrtten n the form (5) and (6). 5. Numerc Results Frst determne egenfrequences of the plate. After dvdng the varables by spatal coordnates and tme, the 649

8 a problem s solved by Fnte Dfference Method. The step n spatal coordnates s x = x =. In calculatons, for sotropc plates ν = ν = ν =. are gven as an ntal data. For square plates wth dmensons a = b = H the value of egenfrequency s p = Wth ncreasng dmensons of the plate up to a = b = 5H the value of egenfrequency s p =.96. For square plates wth dmensons a = b = 8H the value of egenfrequency s p =.98. Table shows the results obtaned for the dsplacements, moments and forces n fxed square plates ρh ω a = b = H under dfferent values of dmensonless frequency ω =. When the value of the frequency E of external effect ω approaches the egenfrequency p =.7469 the values of the dsplacements, forces and moments dramatcally ncrease; ths testfes of gradual transton of the moton of plate ponts nto resonant mode. As seen, an abrupt ncrease n the values of dsplacements, forces and moments could be observed. Table and Table show numerc values of dsplacements, moments and forces, calculated for the fxed square plates wth dmensons a = b = 5H and a = b = 8H, respectvely, for dfferent values of dmensonless frequency ω. Calculatons show that when the value of the frequency of external effect ω approaches egenfrequency, an ncrease n the values of dsplacements, forces and moments s observed; ths testfes of gradual transton of the moton of plate ponts nto resonant mode. Table. Dsplacements, forces and moments at a= b= H. ω ψ E re WE M Q Table. Dsplacements, forces and moments at a= b= 5H. ω ψ E re WE M Q Table. Dsplacements, forces and moments at a= b= 8H. ω ψ E re WE M Q

9 Calculatons show that the equatons of moton of a plate () may be substtuted by knematc condtons relatve to tangental stresses: u ψ W u ψ W G + = q, G + = q, (6) H H x H H x Knematc equatons serve to determne the generalzed dsplacements u, u. The equatons (6) are determned by the soluton of the system of lnear algebrac equatons relatve to coeffcents of the seres (5) B+, B+, A, ( =,,, ), whch are obtaned by the substtuton of the seres (5) ( ) ( ) nto the condtons on the face surfaces at z = h and z = h (4) and desgnatons (), (4). 6. Concluson Based on these studes, we would note that usng the method of expanson n a seres as part of three-dmensonal dynamc theory of elastcty, a two-dmensonal bmoment theory of orthotropc thck plates was developed and the equatons of moton of the plate relatve tot forces, moments and bmoments were bult. It s shown that the problem n the general case s reduced to the defnton of twelve unknown functons of two spatal coordnates and tme. New expressons to determne the forces, moments and bmoments of the plates were bult, as well as the methods for solvng the problems of free and forced vbratons of plates based on Fnte Dfference Method. References [] Ambartsumyan, S.A. (987) Theory of Ansotropc Plates. Nauka, Ch. Ed. Sc. Lt., Moscow, 6 p. [] Galmov, K.Z. (977) Theory of Shells wth Account of Transverse Shear. Ed. Kazan Unversty, Kazan, p. [] Galmov, Sh.K. (976) Specfed Theory of Calculaton of Orthotropc Rectangular Plate under Lateral Load. Investgatons n Theory of Plates and Shells, Sat. artcles, Kazan, Vol. XII, [4] Mushtar, Kh.M. (99) Nonlnear Theory of Shells. Nauka, Moscow, p. [5] Vlasov, B.F. (95) On a Case of Bendng of a Rectangular Thck Plate. Vestnk MGU. Mechancs, Mathematcs, Astronomy and Chemstry, No., 5-4. [6] Bada, E.N. (98) Some Spatal Problems of Elastcty. Lenngrad Unversty, Lenngrad, p. [7] Karamooz Ravar, M.R. and Forouzan, M.R. () Frequency Equatons for the In-Plane Vbraton of Orthotropc Crcular Annular Plate. Archve of Appled Mechancs, 8, 7-. [8] Soukup, J., Valeš, F., Volek, J. and Skočlas, J. () Transent Vbraton of thn Vscoelastc Orthotropc Plates. Acta Mechanca Snca, 7, [9] Papkov, S.О. () Steady-State Forced Vbratons of a Rectangular Orthotropc Plate. Journal of Mathematcal Scences, 9, [] Chang, H.-H. and Tarn, J.-Q. () Three-Dmensonal Elastcty Solutons for Rectangular Orthotropc Plates. Journal of Elastcty, 8, [] Zenkour, A.M., Allam, M.N.M., Shaker, M.O. and Radwan, A.F. () On the Smple and Mxed Frst-Order Theores for Plates Restng on Elastc Foundatons. Acta Mechanca,, -46. [] Usarov, M.K. (4) Calculaton of Orthotropc Plates Based on the Theory of Bmoments. Uzbek Journal Problems of Mechancs, Tashkent, No. -4, 7-4. [] Usarov, M.K. (4) Bmoment Theory of Bendng and Vbratons of Orthotropc Thck Plates. Vestnk NUU, No. /, 7-. [4] Usarov, M.K. (5) Bendng of Orthotropc Plates wth Consderaton of Bmoments. St. Petersburg, Cvl Engneerng Journal,, 8-9. [5] Usarov, M.K. (5) On Soluton of the Problem of Bendng of Orthotropc Plates on the Bass of Bmoment Theory. Open Journal of Appled Scences, 5,

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