Dynamic Design of Thick Orthotropic Cantilever. Plates with Consideration of Bimoments.

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1 World Journal of Mecancs, 06, 6, 4-56 ttp:// ISSN Onlne: ISSN Prnt: X Dynamc Desgn of Tc Ortotropc Cantlever Plates wt Consderaton of Bmoments Мaamatal K. Usarov Insttute of Sesmc Stablty of Structures of te Academy of Scences of te Republc of Uzbestan, Tasent, Uzbestan ow to cte ts paper: Usarov, М.K. (06) Dynamc Desgn of Tc Ortotropc Cantlever Plates wt Consderaton of Bmoments. World Journal of Mecancs, 6, ttp://dx.do.org/0.46/wjm Receved: September 6, 06 Accepted: October 4, 06 Publsed: October 7, 06 Copyrgt 06 by autor and Scentfc Researc Publsng Inc. Ts wor s lcensed under te Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). ttp://creatvecommons.org/lcenses/by/4.0/ Open Access Abstract Te paper s devoted to dynamc desgn of tc ortotropc cantlever plates by applyng te bmoment teory of plates, wc taes nto account te forces, moments and bmoments; and te teory taes nto account nonlnear law of dsplacements dstrbuton n cross secton of te plate. Te metods for constructng bmoment teory are based on ooe s Law, tree-dmensonal equatons of te teory of dynamc elastcty and te metod of dsplacements expanson nto Maclaurn seres. Te artcle gves te expressons to determne te forces, moments and bmoments. Bmoment teory of plates s descrbed by two unrelated two-dmensonal systems wt nne equatons n eac. On eac edge of te plate, dependng on te type of fastenng, nne boundary condtons are gven. As an example, te soluton of te problem of dynamc bendng of tc sotropc and ortotropc plate under te nfluence of transverse dynamc loads n te form of te eavsde functon s gven. Te equatons of moton of te plate are solved by numercal metod of fnte dfferences. Te numercal results are obtaned for sotropc and ortotropc plate. Te graps of canges of dsplacements and stresses of faces surfaces of te plate are presented. Maxmum values of tese dsplacements are found and analyzed. It s sown tat by Tmoseno teory numercal values of stresses are muc smaller compared to te ones obtaned by bmoment teory of plates. Maxmum numercal values of generalzed dsplacements, forces, moments, and bmoments are obtaned and presented n tabular form. Te analyss of numercal results s done and te conclusons are drawn. Keywords ooe s Law, Tc Plate, Dynamc Teory of Elastcty, Tree-Dmensonal Problem, Bmoment Teory. Introducton Te teory and te metods of tc plate desgn are developed as an appled part of DOI: 0.46/wjm October 7, 06

2 te Mecancs of rgd body. Exstng teores of tc plates consderng transverse sear of plates are based on a number of smplfyng ypoteses proposed by many researcers. Tere are numerous papers and monograps of Russan and foregn autors n ts drecton. Lterature revew on te teory and desgn of plates wtn te specfed teory s gven n []-[4]. Statc problems of bendng of tc sotropc plates wtn te tree-dmensonal teory of elastcty are consdered n [5] (B. F. Vlasov); t gves an exact analytcal soluton n trgonometrc seres. Te monograp by E. N. Bada [6] trows lgt upon te queston of bendng of ortotropc plates n trgonometrc seres. Numercal results of dsplacements and stresses are obtaned. In recent years, a number of studes ave been publsed on statc and dynamc analyss of structural elements n te feld of te teory of plates. Te autors [7] are nvolved n dynamc tass of ansotropc plate vbratons. Foregn autors Karamooz Ravar M. R. and Forouzan M. R. [8] ave consdered te problem of free oscllatons of a crcular rng ortotropc plate. Frequency equatons ave been bult n vbraton plane for general boundary condtons. Te wor of te autors n [9] s devoted to solvng te problem of transent oscllatons of a rectangular vscoelastc ortotropc plate on te bass of Flügge and Tmoseno-Mndln deformaton models. Te paper [0] solves te problem of steady forced oscllatons of ortotropc plate by superposton metod, wc s reduced to a quas-regular nfnte system of lnear equatons; ts analytcal soluton s bult. In [] on te bass of te metod of separaton of varables a tree-dmensonal problem of elastcty teory s solved. Te metod of desgn of rectangular ortotropc elastc plates subjected to external loads on te upper and lower faces s developed Papers [] [] are devoted to te constructon of te teory of plate by dsplacements expanson nto a seres on one of te spatal coordnates orented along te normal of te plate. Dsplacements n te plate plane can be expanded n te form of a cubc parabola, and normal dsplacements n te form of a quadratc parabola. In [], a problem of plate bendng s solved and a comparatve analyss wt te results of oter autors s carred out. In [], dynamc bendng of tc rectangular plate under te acton of lumped dynamc forces s consdered. Numercal results are obtaned. If to consder te law of nonlnearty of dsplacements dstrbuton n te crosssectons of te plate, ten n addton to tensle and sear forces, bendng and torsonal moments, tere appear te addtonal force factors, called te bmoments. In [4]-[7] te development and soluton of te problem of bendng and vbratons of tc plates s based on bmoment teory of plates bult wtn te tree-dmensonal teory of elastcty wtout smplfyng ypoteses, usng te metod of dsplacements expanson nto Maclaurn nfnte seres on one of te spatal coordnates. Ts paper s dedcated to dynamc analyss of tc plates on te bass of bmoment teory of plates. To tae nto account all force factors of te plate, ncludng te bmoments, one sould consder all te components of stress and stran tensors: j, ε j,, j =,. Te components of dsplacement vector are presented n te form of 4

3 a functon of tree spatal coordnates and tme u( x, x, zt, ), u( x, x, zt, ), u( x, x, zt, ). Te statements of dynamc problem for tc plates n tree-dmensonal formulaton and te metods of reducng t to a two-dmensonal bmoment teory are brefly descrbed. Determnant correlatons of forces, moments, and bmoments, as well as te equatons of moton of te plate, gven n [4]-[6] are produced relatve to tese force factors.. Statement of te Problem Consder an ortotropc tc plate of constant tcness = and plan dmensons ab., Introduce te denotatons: E, E, E-elastcty modulus; G, G, G -sear modulus; ν, ν, ν -Posson rato of materal of te plate. To descrbe te moton of te plate a Cartesan system of coordnates wt varables x, x and z s ntroduced. Te orgn s taen n te md-surface of te plate. OZ axe s drected down. Let te dstrbuted surface, normal and tangent loads be appled to two face surfaces of te plate z = and z =. Normal loads q +, q are appled along OZ axe. ( + ) ( ) Tangent loads q, q, ( =, ) are appled n OX, OX axes. Te plate s consdered as a tree-dmensonal body, ts materal obeyng te ooe s generalzed Law. Tree-dmensonal equatons of dynamc teory of elastcty are used as an equaton of moton of te plate. Boundary condtons of face surfaces of te plate z = and z = ave te form: ( + ) ( + ) ( + ) = q, = q, = q at z =, (.а). Metod of Soluton ( ) ( ) ( ) = q, = q, = q at z =. (.b) Te metods of constructon of bmoment teory of plates are based on ooe s generalzed Law, tree-dmensonal teory of elastcty, boundary condtons of face surfaces () and dsplacements expanson nto Maclaurn seres n te form: ( ) ( ) ( ) ( ) ( ) z z z z u = B0 + B + B + B + + B +, ( =, ), (.а) z z z z 0 u = A + A + A + A + + A + (.b) ( were B ), A-are unnown functons of two spatal coordnates B = B x, x, t, A = A x, x, t : u u B A ( ) =, (, ), = =! z! z z= 0 z= 0 Dsplacements of te ponts of face surfaces z = and z = + of te plate are ( ) ( + ) denoted by u, u, ( =, ), and stresses on face surfaces z = and z = + by,, and +, +, +.. 4

4 Note tat bmoment teory of plates s descrbed by two unrelated problems, eac of wc s formulated on te bass of nne two-dmensonal equatons wt approprate boundary condtons. Determnant equatons and equatons of moton of bmoment teory of plates are brefly descrbed. Te frst problem conssts of two equatons for longtudnal and tangental forces and four subsdary bult equatons for bmoments for te nne unnown nematc functons: ( + ) ( ) u + u ψ β (.а) u =, = ud, z = uzd, z =, ( + ) ( ) u u W= r= uzz = uz z, d, γ 4 d. (.b) Introduce load terms to te equaton of moton for te frst problem, q, =,, q -are determned by formulae: ( + ) + + q q q q q =, =,, q =. (4) Te forces N, N, N and bmoments T, T, T are determned by te expressons: ψ ψ ψ ψ N = E + E + E W, N = E + E + E W β β W 4r β β W 4r T = E + E + E, T = E + E + E (6) ψ ψ β β N = N = G +, T = T = G +. (7) Te ntenstes of transverse bmoments p, p and τ, τ from tangental stresses, ave te form: ( u ψ ) γ ( u β) r p = G +, τ = G +, ( =, ). (8) And te ntenstes of normal bmoments p and τ from normal stress are determned by te formula: ψ ψ W β β W 4r p = E + E + E, τ = E + E + E. (9) x x x x Te equaton of moton relatve to longtudnal and tangental forces, actng n te plane of te plate, as te form: N N N + + q = ρ ψ N + + q = ρ ψ. (5) (0.а) (0.b) 44

5 As could be seen, te systems of two Equaton (0) contans tree unnown functons ψ, ψ,w. To complete ts system two equatons of moton relatve to longtudnal and tangental bmoments are wrtten T T + 4p + q = ρ β (.а) T T + 4p + q = ρ β (.b) and two more equatons of moton relatve to te ntensty of transverse bmoments n te followng form: p p p q + + = ρ r () τ τ 6τ q + + = ργ. () Usng Maclaurn seres () and te correlatons (), boundary condtons () are presented n te form of te system of tree equatons W q u = + = G ( β ψ ), (, ) E u E u q W = r E x E x 0 E ( γ 7 ) (4). (5) Equatons of moton (0) - (5) comprse a combned system of dfferental equatons from nne equatons on unnown functons: ψ, ψ, β, β, u, u, r, γ, W. Note tat all formulae of force factors (5) - (9) and equatons of moton of te plate of te frst problem (0) - () are strctly bult. Approxmaton exsts n dervaton of te Equaton (4) and Equaton (5) only. Equaton (4) s bult wt te fourt order of accuracy, and Equaton (5) wt te sxt order of accuracy relatve to small para- meter of te plate δ =. ere a-s a small sze n plate plan. 0a Te second problem conssts n equatons for bendng moments, torsonal moments, sear forces and bmoments relatve to nne unnown nematc functons: ( + ) ( ) + u u W = r = u z = uz z, (6.а), d, d γ ( + ) ( ) u u ψ β 4 u =, = uzz d, = uzd, z =,. (6.b) Load terms of te second problem equaton q, ( =, ), q followng form: are determned n te + + q + q q q q =, ( =, ), q =. (7) Bendng and torsonal moments M, M, M and P, P, P are wrtten as follows 45

6 ( r W ) ψ ψ M = E + E E ( r W ) ψ ψ M = E + E E ( γ W ) β β P = E + E E ( γ W ) β β P = E + E E ψ ψ M = M = G +, β β P = P = G +. Expressons to defne sear forces ave te form:,, (8) (9) (0) r r Q = G u +, Q = G u +. () Te ntensty of transverse and normal bmoments p, p and p are determned by te expressons ( r W ) u 4ψ γ ψ ψ p = G +, ( =, ), p = E + E E. () Equatons of moton of te second problem are also descrbed by a system of sx equatons of moton of te plate. Te frst tree equatons of moton are wrtten for bendng and torsonal moments and one equaton-for sear forces: M M, (.а) + Q + q = ρψ M M, (.b) + Q + q = ρψ Q Q + + q = ρr. (4) Tree more equatons of moton of te plate would be wrtten for bmoments; two of tem for bendng and torsonal bmoments ave te form: P + P p + q = ρβ, (5.а) P + P p + q = ρβ. (5.b) Te sxt equaton of plate moton for te ntensty of transverse bmoments s wrtten as follows: 46

7 p p + + = ργ. (6) 4p q Usng Maclaurn seres () and relatonsps (6), boundary condtons () are presented n te form of te system of tree equatons, wrtten as: W q u = ( β 7 ψ ) +, ( =, ), (7) 0 0 G E u E u q W = ( γ r ) + +. (8) 4 0 E x E x 0 E Te system of dfferental equatons of moton () - (8) comprses a combned system of nne equatons relatve to nne unnown functons ψ, ψ, u, u, β, β, r, γ, W. It sould be noted tat all formulae of force factors (8) - () and equatons of moton of te plate for te second problem () - (6) are strctly bult. Approxmaton exsts n dervaton of Equaton (7) and Equaton (8) only. Equaton (8) s bult wt te fourt order of accuracy, and Equaton (7) wt te sxt order of accuracy relatve to small parameter of te plate δ. Te stresses on te upper and lower face surfaces z = and z = + are denoted by,, and +, +, +. Usng tese expressons, one would ntroduce te force factors,, and,,, defned by formula: ( + ) ( ), + j + j j j j = j =, ( =, ; j =, ). (9) Te values,, and,, are referred as bmoment ntenstes under tenson-compresson wt consderaton of transverse reducton and lateral bendng wt cross sear of te plate. * * * * Te ntenstes of te bmoments, and, are ntroduced by te dfferences and sums of dervatves n z-coordnate from normal stresses, : * *, = =, (0) z z=+ z z= z z=+ z z= * *, = + = +. () z z=+ z z= z z=+ z z= On te bass of ooe s Law and boundary condtons (.а) and (.b) te expressons for alf-dfference and alf-sum of te frst derved functons u, u and u are found n z-coordnate on face surfaces of te plate z = and z = + u u q W, (, ) z z=+ z = = (.а) z= G u u u u q E E z + z=+ z =, (.b) z= E u u q W + =, ( =, ) z z, (.а) G z=+ z= 47

8 u u u u q E E z z=+ z =. (.b) z= E Usng expressons (), () from ooe s Law one may determne te expressons * * * * for bmoment ntenstes,,,,, and,,,. Te ntenstes of bmoments,, are determned n te form: E u E u E = E E + E E + q, E E E = E E E u + E E E u E + q, E E E (4.а) (4.b) u u = G +. (4.c) Te ntenstes of bmoments,, are determned by te followng formula: E u E u E = E E + E E + q, (5.а) E E E E u E u E = E E + E E + q, (5.b) E E E Te ntenstes of bmoments u = G +. (5.c) * * * * u,,, ave te expressons: W W q q R, x G x G * = E E + E + E + E x x * = E E + E + E + E x x (6.а) W W q q R, (6.b) x G x G * W W q q R = E E + E + E + E, (7.а) G G * W W q q R = E E + E + E + E. (7.b) G G Unnown functons R and R n expressons (6) and (7) are determned from te system of algebrac equatons relatve to coeffcents of te seres (), obtaned from denotatons () and (6), and presented as ( 5 ) ( 4 ) R = 4 A A + = 40 W + 6r 5γ. (8) R = 4 A + 4 A + = R = 60 W + 4r γ. (9) ere are te formulae to determne te dsplacements on face surfaces of te plate z = and z = + : ( ) ( + ) ( ) ( + ) u = u u, u = u + u, =,, u = W W, u = W + W. (40) 48

9 Formulae for te stresses on face surfaces of te plate z = and z = + ave te form: ( ) ( + ) =, = +, =, ; =,. (4) j j j j j j j Note down te boundary condtons for a cantlever plate. Let te edge of te plate x = 0 be rgdly fxed. Remanng edges of te plate are free from supports. Te fxed edge of te plate as zero dsplacement and te boundary condtons on te edge x = 0 are: ψ = 0, ψ = 0, β = 0, β = 0, r = 0, γ = 0, u = 0, u = 0, W = 0. (4.а) ψ = 0, ψ = 0, β = 0, β = 0, r = 0, γ = 0, u = 0, u = 0, W = 0. (4.b) On te free edge of te plate x = b boundary condtons are: M = 0, M = 0, P = 0, P = 0, Q = 0, p = 0, = 0, = 0, = 0. (4.а) * N = 0, N = 0, T = 0, T = 0, = 0, = 0, p = 0, τ = 0, = 0. (4.b) * On two free opposte edges of te plate x = 0, x = a te followng condtons sould be fulflled: M = 0, M = 0, P = 0, P = 0, Q = 0, p = 0, = 0, = 0, = 0. (44.а) * N = 0, N = 0, T = 0, T = 0, = 0, = 0, p = 0, τ = 0, = 0. (44.b) * On two angular ponts of te plate, free from supports and external forces, x = 0, x = b and x = ax, = b te followng boundary condtons sould be fulflled: M = 0, M = 0, P = 0, P = 0, Q = 0, p = 0, = 0, = 0, = 0. (45.а) * N = 0, N = 0, T = 0, T = 0, = 0, = 0, p = 0, τ = 0, = 0. (45.b) * M = 0, M = 0, P = 0, P = 0, Q = 0, p = 0, = 0, = 0, = 0. (45.c) * N = 0, N = 0, T = 0, T = 0, = 0, = 0, p = 0, τ = 0, = 0. (45.d) * At ntal moment of tme t = 0 ntal condtons are taen as zero ones. Te advantage of bmoment teory, wen compared to exstng ones, s ts g accuracy and good applcablty to solvng practcal problems of evaluaton of stresses and dsplacements n ortotropc plates. 4. Soluton of Tests Problem Assume tat a plate s under te acton of external unformly dstrbuted surface normal load q on oz-axs n te form of eavsde functon appled to face surface of te plate z = : 0, at t 0; q = q0, at t > 0, were q 0 s a parameter of external force. Remanng components of external forces are zero. 49

10 Wle obtanng numercal results on dsplacements, a dmensonless functon s ntroduced: f Ef q =. Dmensonless stresses and ntenstes of bmoments are ntroduced accordng to te followng formulae: j j =, ( =, ; j =, ). q 0 Te problem s solved by te metod of fnte dfferences. A fnte-dfference approxmaton of dsplacements dervatves n spatal coordnates s gven ere. To approxmate te nternal ponts of dsplacements dervatves, te expressons of central dfference scemes are used. To approxmate te frst dervatves one would use te followng expressons wt respect to te central ponts f, j f, j f, j f, j f, j f, j = +, = +. x x Te second dsplacement dervatves are approxmated by te followng expressons: f f f + f f f f + f, j, j, j, j, j, j, j, j = +, = + x x Te second dervatve wt respect to tme, usng fnte-dfference equaton, s represented n te form: f f f + f +, j, j, j, j = τ 0 τ. ere ct E τ = -s a dmensonless tme, were c =. ρ 5. Numerc Results Calculatons are carred out for square plates wt dmensons n plan a = b =. Materal of te plate s taen as sotropc wt elastcty modulus E = E = E = E0, sear modulus G = G = G = E0 ( + ν ), Posson rato ν = ν = ν = ν = 0. and as ortotropc materal 5: wt elastcty modulus E = 4.6 E0, E =.6E0, sear 4 modulus G = 0.56 E0, G = 0.4 E0, G = 0.E0, ere E 0 = 0 MPa, Posson rato ν = 0.7, ν = 0., ν = Fgures - sow te dagrams of canges of dmensonless values of dsplacements ( + ) ( ) u, u, ( =, ) of te ponts on face surface z= + z, =, obtaned from te soluton of te frst and second problems of bmoment teory of plates by formulae (40). Te studes ave ndcated tat te form of te bend of generalzed dsplacements u +, u s antsymmetrc, and te form of te bend of generalzed dsplacements ( + ) ( ) u, u, =, s symmetrc. Maxmum dmensonless values of generalzed dsplacements ( + ), u u occur on te lmtng ponts of a free edge of te plate x = b 50

11 Fgure. Dagram of canges n dsplacements tme. u + -(a), u -(b) of te ponts on face surface of te plate z=+ z, = vs Fgure. Dagram of canges n dsplacements tme. u + -(a), u -(b) of te ponts on face surface of te plate z=+ z, = vs Fgure. Dagrams of canges n dsplacements tme. u + -(a), u -(b) of te ponts on face surface of te plate z=+ z, = vs 5

12 and ave te followng values: u + max =.67 (Fgure (а)) and u max =.77 (Fgure (b)). Maxmum dmensonless values of dsplacements u +, u of te ponts of face surface z= + z, = occur n te mddle of a free edge of te plate x = b, tey ave te followng values: u + max = (Fgure (а)) and u max = (Fgure (b)). Maxmum dmensonless values of dsplacements u +, u of te ponts on face surface z= + z, = occur n te mddle of a free edge of te plate x = b and ave te followng values: u + max = 6.96 (Fgure (а)) and u + max = 6.4 (Fgure (b)). Fgure 4 and Fgure 5 ndcate te dagrams of canges n dmensonless values of normal stresses of te ponts on face surface of te plate z= + z, =, obtaned by formulae (4) from te solutons of te frst and second problems of bmoment teory of plates. Maxmum dmensonless values occur n te mddle of a fxed edge of te plate x = b and ave te followng values + = (Fgure 4(а)), = (Fgure 4(b)). Fgure 4. Dagrams of canges n stresses + -(a), -(b) of te ponts on face surface of te plate z=+ z, = vs tme. Fgure 5. Dagram of canges n stresses + -(a), -(b) of te ponts on face surface of te plate z=+ z, = vs tme. 5

13 ( + ) ( ) Maxmum dmensonless values of stresses, of te ponts on face surface of te plate z= + z, = occur n te mddle of te fxed edge of te plate x = b. Maxmum dmensonless values ave te followng values: + = 8.7 (Fgure 5(а)) and = 86.4 (Fgure 5(b)). As could be seen, numercal values of dsplacements and stresses u, are substantally greater tan numercal values of dsplacements and stresses u, n te same observed ponts on face surface of te plate. Fgure 6 and Fgure 7 ndcate te dagrams of canges n dmensonless values of normal stresses on face surface of ortotropc plate z= + z, =, obtaned from te solutons of te frst and second problems of bmoment teory of plates by formulae (4). Maxmum dmensonless values occur n te mddle of a fxed edge of ortotropc plate x = b and ave te followng values + =.599 (Fgure 6(а)) and ( + ) ( ) = 4.78 (Fgure 6(b)). Maxmum dmensonless stresses, of te ponts on face surface z= + z, = of ortotropc plate occur n te mddle of a fxed edge Fgure 6. Dagrams of canges n stresses tme. + -(a), -(b) of te ponts on face surface of ortotropc plate z=+ z, = vs Fgure 7. Dagrams of canges n stresses tme. + -(a), -(b) of te ponts on face surface of ortotropc plate z=+ z, = vs 5

14 of te plate x = b. Maxmum dmensonless values are + = (Fgure 7(а)) and = (Fgure 7(b)). If to solve ts problem by Tmoseno teory, te maxmum stresses for sotropc plates equal to = 6.5, = 55.8, аnd for ortotropc plate equal to = 4.5, = As seen, numercal values of stresses, obtaned by Tmoseno teory are consderably less compared to bmoment teory of plates. Te laws of canges of generalzed dsplacements and force factors n tme for te frst and second problems are dentcal to te laws of dsplacement canges n tme, presented n Fgures -7. Furter consder only maxmum values of generalzed dsplacements, forces, moments, and bmoments obtaned from te soluton of te frst and second problems. Tables -4 sow maxmum values of nematc and force factors of te problems. Table and Table sow dmensonless numercal results of nematc functons calculaton for sotropc and ortotropc plate, obtaned from te soluton of te second problem. Table gves numercal results of calculaton of dmensonless longtudnal forces N N n =, n = and bmoments,, p, p, t = T, t = T. As could q0 q0 q0 q0 q0 q0 be seen, te values of forces and bmoments of te plate N, Т, p, p are commensurable, and te values of bmoments, are substantally greater tan te Table. Te values of nematc functons of te frst problem. Materal E u q E u q E q ψ E β q 0 0 E r q E W q 0 sotropc ± ortotropc ± Table. Te values of nematc functons of te second problem. Materal E u q 0 E u q Eψ q 0 0 E q 0 β E r q 0 E W q 0 sotropc ± ortotropc ± Table. Te values of longtudnal forces and bmoments of te frst problem. Materal q 0 q 0 0 t q n q p q p q sotropc ortotropc Table 4. Te values of moments, bmoments and sear forces of te second problem. Materal q 0 q 0 0 m q p q p q 0 0 Q q 0 sotropc ortotropc

15 values of remanng bmoments. Table 4 presents dmensonless numercal results of calculaton of bendng moments, M forces and bmoments,, m =, longtudnal bendng bmoments P p, p = and sear force Q q 0. Smlarly, numercal values of forces and bmoments are commensurable, and te values of bmoments, are many tmes greater tan te values of remanng forces and bmoments. a b A step n calculaton on dmensonless coordnates s taen as х =, х = Te stablty of teraton n dmensonless tme s provded by explct sceme wt τ = 0.0 step. Accordng to te analyss of results sown n Tables -4, te followng conclusons can be drawn: numercal values of nematc functons and force factors (Table and Table ), obtaned by solvng te frst problem, caracterze te tenson-compresson n longtudnal drecton, tang nto account te transverse reducton of te plate; numercal values of nematc functons and force factors (Table and Table 4), obtaned by solvng te second problem, caracterze te lateral bendng wt consderaton of transverse sear of te plate. Comparng te numercal results of te frst and second problems, t could be noted tat te numercal values of dsplacements and force factors n te second problem s muc greater tan te correspondng dsplacement values and force factors of te frst problem. 6. Conclusons Tecnque of constructng a bmoment teory of te plate, wc taes nto account te forces, moments and bmoments, developed by nonlnear law of dsplacements dstrbuton n cross-sectons of te plate s brefly presented ere. Exact expressons of nternal forces, moments and bmoments are gven, as well as te equatons of moton and boundary condtons for ortotropc tc plate. Bmoment teory of te plate s appled to solvng te dynamc problem of forced oscllatons of ortotropc tc plate. An example of forced oscllatons of cantlever plate under te nfluence of transverse dynamc loads n te form of te eavsde functon s consdered. Based on te metod of fnte dfferences, te metods for calculatng te dynamc cantlever plate are developed. Numercal results of dsplacements, forces, moments, bmoments and stresses for cantlever plate are obtaned and followed by analyss. Based on te analyss of numercal results, a concluson s drawn tat Tmoseno teory s not acceptable for te calculaton of dsplacements and stresses of te plate under dynamc effects. References [] Ambartsumyan, S.A. (987) Teory of Ansotropc Plates. Naua, Moscow, 60 p. [] Galmov, K.Z. (977) Teory of Sells wt Account of Transverse Sear. Kazan Unversty, Kazan, p. 55

16 [] Galmov, S.K. (976) Specfed Teory of Calculaton of Ortotropc Rectangular Plate under Lateral Load. Investgatons n Teory of Plates and Sells, SAT Artcles, Kazan, Vol. XII, [4] Mustar, K.M. (990) Nonlnear Teory of Sells. Naua, Moscow, p. [5] Vlasov, B.F. (95) On a Case of Bendng of a Rectangular Tc Plate Vestn MGU Mecancs. Matematcs, Astronomy and Cemstry, 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. (0) Frequency Equatons for te In-Plane Vbraton of Ortotropc Crcular Annular Plate. Arcve of Appled Mecancs, 8, 07-. ttp://dx.do.org/0.007/s [8] Cang,.-. and Tarn, J.-Q. (0) Tree-Dmensonal Elastcty Solutons for Rectangular Ortotropc Plates. Journal of Elastcty, 08, ttp://dx.do.org/0.007/s [9] Zenour, A.M., Allam, M.N.M., Saer, M.O. and Radwan, A.F. (0) On te Smple and Mxed Frst-Order Teores for Plates Restng on Elastc Foundatons. Acta Mecanca, 0, -46. ttp://dx.do.org/0.007/s [0] Amedov, A.B. (007) Comparatve Analyss of Specfed Teores of Bendng of Tc Plates. Uzbe Journal Problems of Mecancs, No., [] Amedov, A.B. (007) Dynamc Effect of Concentrated Forces n Tc Plates. Uzbe Journal Problems of Mecancs, No. 4, -6. [] Usarov, M.K. (04) Calculaton of Ortotropc Plates Based on te Teory of Bmoments. Uzbe Journal Problems of Mecancs, No. -4, 7-4. [] Usarov, M.K. (04) Bmoment Teory of Bendng and Vbratons of Ortotropc Tc Plates. Vestn NUU, No. /, 7-. [4] Usarov, M.K. (05) Bendng of Ortotropc Plates wt Consderaton of Bmoments. Cvl Engneerng Journal, No., [5] Usarov, M.K. (05) On Soluton of te Problem of Bendng of Ortotropc Plates on te Bass of Bmoment Teory. Open Journal of Appled Scences, 5, -9. ttp://dx.do.org/0.46/ojapps [6] Usarov M.K. (05) Bendng of Ortotropc Plates wt Consderaton of Bmoments. Cvl Engneerng Journal, No., [7] Usarov M.K. (05) On Soluton of te Problem of Bendng of Ortotropc Plates on te Bass of Bmoment Teory. Open Journal of Appled Scences, 5, -9. ttp://dx.do.org/0.46/ojapps

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