Plate Bending Analysis by using a Modified Plate Theory

Transcription

1 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp , 2006 Plte Bending Anlysis y using Modified Plte Theory Y. Suetke 1 Astrct: Since Reissner nd Mindlin proposed their clssicl thick plte theories, mny uthors hve presented refined theories including trnsverse sher deformtion. Most of those plte theories hve tended to use higher order power series for displcements nd stresses long the thickness in order to chieve the higher ccurcy. However, they hve not crefully noticed lterl lod effect. In this pper, we py ttention to constitution of the lterl lods: ody force nd upper nd lower surfce trctions. Especilly we formulte modified theory for plte ending, in which the effect of ody force is distinguished from tht of surfce trctions. The present plte theory includes not only trnsverse sher deformtion ut lso trnsverse norml stress effect. In this pper, our ttention is focused on ending moment ehvior of pltes. keyword: Thick plte theory, Trnsverse sher deformtion, Lod effect, Trnsverse norml stress, Body force. 1 Introduction Since Reissner (1945) nd Mindlin (1951) proposed their clssicl thick plte theories, mny uthors hve presented refined theories including trnsverse sher deformtion. As well-known the Reissner s theory (1945) is n ssumed-stress theory nd the Mindlin s theory (1951) n ssumed-displcement one. In the Reissner s theory, prolic distriution of trnsverse sher stresses is ssumed nd we cn stisfy the sher-free condition on the upper nd lower surfces of pltes. In ddition trnsverse norml stress is lso incorported in the theory. A cuic distriutionof the trnsverse norml stress is ssumed, in which only n upper surfce trction is considered. Discussion of strins of pltes is supplemented in Reissner (1947). In the Mindlin s theory, liner distriutions of in-plne displcements nd constnt deflection re ssumed nd, 1 Ashikg Institute of Technology, Ohmecho, Ashikg, Tochigi, , Jpn. therefore, the trnsverse sher stresses distriute constntly long the thickness of pltes. This pprently contrdicts the sher-free condition on the plte surfces. In order to compenste the contrdiction, Mindlin introduced correction prmeter for the trnsverse sher stresses. In this theory, the trnsverse norml stress is neglected nd difference etween the upper nd lower surfce trctions is dopted s lterl lod. It is well-known tht the ove clssicl theories coincide when the trnsverse norml stress is neglected nd the sher correction prmeter is 5/6. Rtionl determintion of the correction prmeter, however, is not presented. In some cses we cnnot otin ccurte solutions when the prmeter is 5/6. The solutions depend on not only the sher correction prmeter ut lso constitutionof the lterl lods. Mny high order theories hve een presented in order to otin more ccurte solutions of the plte ending. Levinson (1980) nd Reddy (1984) ssume cuic distriution of in-plne displcements for thick pltes. Reissner (1975) nd Rehfield (1982 nd 1984) consider deflection chnge long the thickness of pltes. Lo, Christensen, nd Wu (1977, 1977, nd 1978) formulte high-order theory of pltes which includes oth in-plne nd out-of-plne modes of deformtion introducing 11 unknown prmeters. The ove plte theories re ssumed-displcement ones. Alterntive theories, which re ssumed-stress theories, re lso importnt. Amrtsumyn (1975) presents n ssumed-stress high-order plte theory, in which oth the trnsverse sher stresses nd the trnsverse norml stress is incorported. Voyidjis nd Bluch (1981) consider the trnsverse norml strin in ddition to the trnsverse stresses. Reissner (1983) lso formulte n ssumedstress high-order plte theory. Hirshim nd Murmtsu (1980), nd Hirshim nd Negishi ( st nd 2 nd ) estlish generlized highorder plte theory tht includes the ove theories s prticulr ones. Hirshim nd Negishi ( st ) lso discuss the ccurcy of the plte theories in detil. Krenk

2 104 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp , 2006 (1981) employs Legendre polynomils for representing the distriution of displcements nd stresses long the thickness. Green nd Nghdi (1972) lso present multidirector pproch for pltes nd shells, which surpsses the clssicl theories. Recently, mny sophisticted numericl pproches re pplied to the nlysis of the plte ending, for exmple, Long nd Atluri (2002), nd Qin, Btr, nd Chen (2003). In generl, the higher the order of the theories is, the higher the ccurcy of those is. However, the conciseness of the theories hs een lost. On the other hnd, the importnce of the lterl lod effect is not noticed in the ove high-order plte theories. The lterl lod of pltes consists of the upper nd lower surfce trctions nd the ody force. In the ove theories the ody force is not considered nd its effect on plte ending is not seprted from tht of the surfce trctions. In this pper, we py ttention to the constitution of the lterl lods s Suetke nd Tomod (2004) nd Suetke (2005), in which modified ending theory of thick pltes is formulted. In the modified theory employed here, the ody force is seprted from the surfce trctions. The present plte theory includes not only trnsverse sher deformtion ut lso trnsverse norml stress effect. The present modified theory cn e simple mens of comprison for numericl nlyses. In this pper, our ttention is focused on ending moment ehvior of pltes nd we mke sure tht the present modified theory gives us excellent results, even though it is s simple s the clssicl theory. 2 Modified Theory In this pper modified ending theory of pltes is presented y using the Levinson-Reddy type displcement field [Levinson (1980) nd Reddy (1984)]. We py ttention to the constitution of the lterl lods through considertion of the trnsverse norml stress. Consequently we cn tret with the surfce trctions nd the ody force seprtely. 2.1 Displcement-strin reltion Levinson (1980) nd Reddy (1984) ssume the 3 rd -order displcement field in order to represent distortion of norml to the mid-surfce. The displcement field ssumed here is given y U = ψ x z 4 3t 2 ϕ x z 3 ; ϕ x w x ψ x V = ψ y z 4 3t 2 ϕ y z 3 ; ϕ y w y ψ y W = w, (1) where x nd y re in-plne coordintes, z is coordinte norml to the mid-surfce of pltes, t is thickness of pltes, ψ x nd ψ y re deflection ngles, nd w is deflection of pltes. In Eq. (1), if we neglect the 3 rd - order terms, we hve the Mindlin type displcement field [Mindlin (1951)]. From the displcement field (1), we otin the following strin distriution: i) in-plne strins ε x = ψ x x z 4 3t 2 ϕ x x z3 ε y = ψ y y z 4 3t 2 ϕ y y z3 γ xy = ( ψ x y + ψ y x )z 4 3t 2 ( ϕ x y + ϕ y x )z3 ii) trnsverse sher strins, (2) γ xz = ϕ x (1 4 t 2 z2 ), γ yz = ϕ y (1 4 t 2 z2 ). (3) Note tht Eq.(3) stisfies the sher-free condition on the upper nd lower surfces of pltes. 2.2 Constitutive eqution Since we tret with isotropic elstic pltes here, we employ the Hooke s lw s constitutive eqution. Eliminting the trnsverse norml strin ε z from the 3-D Hooke s lw, we otin the following reltion: σ x = σ y = E (ε 1 ν 2 x +νε y )+ 1 ν ν σ z E (νε 1 ν 2 x +ε y )+ ν 1 ν σ z τ xy = Gγ xy ; G nd E 2(1+ν), (4) τ xz = Gγ xz, τ yz = Gγ yz, (5) where E is Young s modulus nd ν is Poisson s rtio.

3 Plte Bending Anlysis 105 Sustitution of Eqs. (2) nd (3) into Eqs. (4) nd (5) gives us σ x = E ( ψ x 1 ν 2 x +ν ψ y y )z + 4 ( ϕ 3t 2 x x +ν ϕ y y )z3 } + 1 ν ν σ z σ y = E (ν ψ x 1 ν 2 x + ψ y y )z + 4 (ν ϕ 3t 2 x x + ϕ y y )z3 } + ν 1 ν σ z τ xy = G( ψ x y + ψ y x )z + 4 3t 2 ( ϕ x y + ϕ y x )z3 } nd, (6) τ xz = Gϕ x (1 4 t 2 z2 ), τ yz = Gϕ y (1 4 t 2 z2 ). (7) Eqution (7) is the sme s the sher distriution in the Reissner s theory [Reissner (1945)]. In the Mindlin s theory [Mindlin (1951)], the 2 nd -nd3 rd -order terms re neglected nd the effect of the trnsverse norml stress is not incorported. 2.3 Equilirium condition Liner equilirium conditions for 3-D odies re given y σ x x + τ xy y + τ xz z +X = 0 τ xy x + σ y y + τ yz z +Y = 0 τ xz x + τ yz y + σ z z +Z = 0, (8) where X, Y,ndZ re ody forces nd, in this pper, we consider only Z, tht is, we set X = Y = 0. By using the 3 rd equilirium condition in Eqs. (8), we cn determine the distriution of σ z. Before doing tht, we should py ttention to the constitution of the lterl lods. The lterl lod of plte consists of the ody force Z = p 0 (x,y) / t nd the upper nd lower surfce trctions, p 1 (x,y) nd p 2 (x,y), s shown in Fig.1. Therefore we hve the following trction oundry conditions: σ z (x,y, t 2 )= p 1, σ z (x,y, t 2 )=p 2. (9) p 1 (x,y) Z=p 0 (x,y)/t O z p 2 (x,y) Figure 1 : Lterl Lod Constitution of Plte ing it with respect to z, in view of Eq. (10), we hve σ z = Gt 3 ( 2 w ψ x x ψ y y )(1 3 t z + 4 t 3 z3 ) + p 0 2 (1 2 t z)+p 2. (10) When we pply the trction oundry condition (9) to Eq.(10), we otin ψ x x + ψ y y = 2 w + 3 2Gt p ; p p 0 + p 1 + p 2. (11) Consequently we otin the distriution of σ z s σ z = p 2 (1 3 t z + 4 t 3 z3 )+ p 0 2 (1 2 t z)+p 2. (12) Integrting the 1 st nd 2 nd of Eq. (8) with respect to z, we otin the ordinry moment equilirium equtions. 2.4 Governing eqution Integrtion of Eqs.(6) nd (7) with respect to z,inviewof Eq. (12), gives us the following moment nd sher force expressions: M x = D 4 5 ( ψ x x +ν ψ y y )+1 5 ( 2 w x 2 +ν 2 w y 2 )} + νt2 60(1 ν) p 0 +6(p 1 + p 2 )}, (13) M y = D 4 5 (ν ψ x x + ψ y y )+1 5 (ν 2 w x w y 2 )} + νt2 60(1 ν) p 0 +6(p 1 + p 2 )}, (14) Sustituting Eq. (7) into the 3rd of Eq. (8) nd integrtx

4 106 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp , 2006 M xy = D 2 (1 ν)4 5 ( ψ x y + ψ y x )+2 5 nd 2 w }, (15) x y Q x = 2 3 Gt( w x ψ x), Q y = 2 3 Gt( w y ψ y), (16) 4 ψ y = 1 D y [p 0 + p 1 + p 2 t 2 ] + 60(1 ν) 2 (3 +ν)p 0 +3(1 +2ν)(p 1 + p 2 )}. 3 Fourier Anlysis where D is the ending rigidity of pltes: D Et 3/ 12(1 ν 2 For numericl clcultions we employ here the Fourier ). Sustituting Eqs. (13) to (16) into the series nlysis. In this section we explin the Fourier ordinry moment equilirium equtions, we hve nlyses of pltes nd 3-D odies. Two plte nlyses 4 ψ x D 5 ( 2 x ν 2 ψ x 2 y ν 2 } re presented here; one is sed on the present theory ψ y 2 x y )+1 5 x ( 2 w) nd the other on the clssicl one. We cn lso employ Gt( w x ψ x) νt2 p0 60(1 ν) x +6( p 1 x + p } lterntive pproches for the nlyses. 2 x ) 3.1 Plte nlysis sed on the present theory = 0, (17) nd 4 ψ y D 5 ( 2 y ν 2 ψ y 2 x ν 2 } ψ x 2 x y )+1 5 y ( 2 w) The deflection nd the deflection ngles re expressed y Gt( w y ψ y) νt2 p0 60(1 ν) y +6( p 1 y + p } the following trigonometric doule series here: 2 y ) w = W mn sin mπx = 0. (18) sin nπy n m ψ x = Φ mn cos mπx Equtions (11), (17), nd (18) re the present governing sin nπy. (23) n m equtions for the plte ending. We cn rewrite Eqs. (17) nd (18), in view of Eq.(11), s ψ y = Ψ mn sin mπx cos nπy n m (22) Elstic pltes dopted here s numericl exmples re simply supported rectngulr pltes sujected to lterl lods. The coordintes of the plte model is shown in Fig.2. ( 2 10 t 2 )( ψ x y ψ y )=0, (19) x 4 w = 1 D [p 0 + p 1 + p 2 t2 12 ν 6(1 ν) 2 10 p }] 5 (2 ν)(p 1 + p 2 ).(20) If we ssume tht ψ x / y = ψy / x, Eq. (19) cn e stisfied priori. In tht cse, we cn derive the governing equtions for ψ x nd ψ y from Eqs. (11) nd (17) to (19): 4 ψ x = 1 D x [p 0 + p 1 + p 2 t 2 ] + 60(1 ν) 2 (3 +ν)p 0 +3(1 +2ν)(p 1 + p 2 )}, (21) Note tht Eqs. (23) stisfy the oundry condition of simply supported pltes. In ddition, we expnd the lod functions into the following Fourier doule series: p i = n P (i) mn = 4 mn sin mπx nπy sin ; P (i) m Z Z 0 0 mid-surfce O z x=0 p i (x,y)sin mπx z=t/2 y=0 C x= y sin nπy dxdy. (24) x y= z=-t/2 Figure 2 : Rectngulr Plte nd Coordintes

5 Plte Bending Anlysis 107 Sustituting Eqs. (23) into Eqs. (20) to (22), we cn esily determine the coefficients W mn, Φ mn,ndψ mn s W mn = 1 [ λ 4 mn D P (0) mn +P (1) mn + λ2 mn t2 6(1 ν) Φ mn = 1 λ 4 mnd mπ λ2 mnt 2 60(1 ν) Ψ mn = 1 λ 4 mnd nπ λ2 mnt 2 60(1 ν) 12 ν 10 P(0) mn + 3 (2 ν)(p(1) 5 mn) [ P (0) mn +P (1) mn }], (25) (3 +ν)p (0) mn +3(1 +2ν)(P (1) mn)} ], [ P (0) mn +P (1) mn (26) (3 +ν)p (0) mn +3(1 +2ν)(P (1) mn)} ], where λ 2 mn (mπ/) 2 +(nπ / ) Plte nlysis sed on the clssicl theory (27) At this stge it is significnt to review the clssicl plte theories. The governing equtions of the sttic Mindlin s theory [Mindlin (1951)] re s follows: 4 w = 1 D 1 t 2 6(1 ν)κ 2 }p, (28) 4 ψ x = 1 p D x, 4 ψ y = 1 p D y, (29) where κ is the sher correction prmeter. In Eq. (28), if we set κ = 5/3(2 ν), the sttic Mindlin s theory coincides with the Reissner s one. If we use the trigonometric series (23) gin in the clssicl ending nlysis of pltes, the coefficients W mn, Φ mn, nd Ψ mn cn e determined s W mn = P(0) mn +P (1) mn λ 4 mnd 1 + λ2 mnt 2 }, (30) 6(1 ν)κ Φ mn = 1 λ 4 mnd mπ (P(0) mn +P (1) mn), (31) Ψ mn = 1 λ 4 mnd nπ (P(0) mn +P (1) mn). (32) As well-known, we cn clculte the ending nd twisting moments of pltes y using the following expressions insted of Eqs. (13) to (15): M x = D( ψ x x +ν ψ y ), (33) y M y = D(ν ψ x x + ψ y ), y (34) M xy = D 2 (1 ν)( ψ x y + ψ y ). x (35) D nlysis In order to evlute the ccurcy of the present plte theory, we perform 3-D elstic nlysis of the plte model. We employ the Fourier series gin for the 3-D nlysis. We explin the pproch riefly in this susection. Geometricl oundry conditions of the model s 3-D ody is given y V (0,y,z)=V(,y,z)=W(0,y,z)=W(,y,z)=0, U(x,0,z)=U(x,,z)=W(x,0,z)=W(x,,z)=0 (36) which corresponds to the conditions for simply supported pltes. To stisfy these conditions, we employ the following trigonometric series U = u mn (z)cos mπx sin nπy n m V = n m v mn (z)sin mπx W = n m w mn (z)sin mπx cos nπy sin nπy. (37) Trction oundry conditions employed here re σ z (x,y, t 2 )= p 1(x,y) σ z (x,y, t 2 )=p 2(x,y), τ xz (x,y,± t 2 )=τ yz(x,y,± t )=0. (38) 2 In ddition, the ody forces of the model re represented y X = Y = 0, Z = 1 t p 0(x,y). (39)

6 108 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp , 2006 Sustitution of Eq. (37) into the Nvier s eqution, which is governing eqution for 3-D elstic prolems, yields n ordinry differentil eqution system with respect to the unknown functions u mn (z), v mn (z),ndw mn (z). When we solve the differentil eqution system under Eqs. (38) nd (39), we cn determine those three unknown functions. 4 Numericl Models As numericl exmples we dopt simple plte ending prolems of squre plte. The plte model is simply supported long the ll edges. The width-thickness rtio µ t/ is chnged within the rnge of µ 0.5. Poisson s rtio is ν = 0.3. In the Fourier nlysis, we tke = terms in the doule series. A constitution of lterl lods dopted here is shown in Fig.3. The plte model is sujected to constnt ody force nd prtilly distriuted constnt lod on the upper surfce. The Fourier coefficients of the lod functions re given y P (0) mn = P (1) mn = 4p 1 π 2 mn 16p 0 π 2 (2 j 1)(2k 1) cos mπx 0 (m nd n: odd) 0 (m orn: even) cos nπy 0 cos nπ(y 0 + ) O z x 0,y 0 cos mπ(x } 0 + ) Z=p 0 (x,y)/t Z=p 0 /t O C, *, *, (40) }. (41) p 1 x, y Figure 3 : Figure 3: Lterl Lod Constitution Two different constitutions of lterl lods re dopted in the present numericl clcultions. One is symmetric distriution on the upper surfce; the other non-symmetric distriution. Numericl properties of the lods re s follows: i) symmetric distriution / / ˆχ 0 = 0.8, ˆχ 1 = 1.25, x 0 = y0 = 0.3, / = / = 0.4, (42) ii) non-symmetric distriution / / ˆχ 0 = 0.6, ˆχ 1 = 40, x 0 = y0 = 0.2, / = / = 0.1, (43) where the non-dimensionl lod prmeter ˆχ i is defined y ˆχ i = ˆp i 3 µd = 12(1 ν2 ) Eµ 4 ˆp i ; µ = t. (44) 5 Numericl Results In this pper, our ttention is focused on ending moment ehvior of pltes. In the numericl clcultion, we estimte the error of the present plte nlysis. Results of the 3-D elstic nlysis re employed s the stndrd vlues. Clssicl nlysis sed on the Mindlin s theory is lso performed in order to confirm the ccurcy of the present nlysis. Since we dopt the Levinson-Reddy type displcement field, Eq.(1), it is not so esy to predict locl ehviors for the lrge thickness of pltes. This is further issue to e improved. 5.1 Symmetric surfce trction In this susection, we discuss the cse of the symmetric surfce trction. As mentioned efore, the plte model is sujected to not only constnt ody force ut lso prtilly distriuted constnt lod on the upper surfce. In this cse, the surfce trction is symmetric. Numericl results re shown in Fig.4, in which the errors of the present nd the clssicl plte nlyses, ε, re plotted ginst the width-thickness rtio of the model, µ. In Fig.4, closed circles indicte the results of the present nlysis nd closed tringles tht of the clssicl nlysis, respectively. It follows from Fig.4 tht the present modified plte theory pproximtes the 3-D nlysis with high ccurcy. Especilly, we should note tht the error of the present

7 Plte Bending Anlysis 109 š ««š ««š š Figure 4 : Error of Moment t Center Point (Symmetric Model) Figure 5 : Error of Moment t Center Point (Nonsymmetric Model) nlysis remins quite smll even in the thick plte region ner µ = 0.5. On the other hnd, the error of the clssicl nlysis increses rpidly with the increse of the width-thickness rtio. The excellent pproximtion of the deflection ehvior hs lredy een confirmed [Suetke 2005]. The present investigtion shows the efficiency of the modified plte theory lso in the moment nlysis. 5.2 Non-symmetric surfce trction The results of the non-symmetric surfce trction model re presented in this susection. The lod dopted here consists of constnt ody force nd nonsymmetriclly distriuted constnt lod on the upper surfce. Numericl results of the moment error t the center point of the model re shown in Fig.5, which is depicted in the sme mnner s Fig.4. It cn e seen lso from Fig.5 tht the present modified plte theory gives us excellent results. In this cse, however, the clssicl theory lso mintins prcticlly sufficient ccurcy. 6 Concluding Remrks The following conclusions my e drwn from the present investigtion: 1) A modified plte ending theory is proposed, in which the effect of lterl lods is crefully considered. 2) The new theory gives us excellent pproximtions for moments even in thick plte region, while the clssicl one mintins prcticlly sufficient ccurcy within modertely thick plte region. 3) The constitution of the lterl lods plys key role in the plte ending nlyses of thick pltes. In prticulr, when the constitution of lods is not simple, we should use the modified theory insted of the clssicl one. References Amrtsumyn, S. A. (1975): Theory of Anisotropic Elstic Pltes (trnslted y Norio Kmiy), Morikit, Tokyo. (in Jpnese) Hirshim, K.; Murmtsu, M. (1980): The Effect of Trnsverse Components on the Bending of Pltes, Proc JSCE, Vol. 304, pp (in Jpnese) Hirshim, K.; Negishi, Y. (1983): Some Considertions on Accurcies of Typicl Two-Dimensionl Plte Theories including the Effects of Trnsverse Components, Proc JSCE, Vol. 330, pp (in Jpnese) Hirshim, K.; Negishi, Y. (1983): Study on Dynmic Chrcteristics (Free Virtion nd Dispersion Reltion) of Severl Plte Theories, Proc JSCE, Vol. 333, pp (in Jpnese) Krenk, S. (1981): Theory for Elstic Pltes Vi Orthogonl Polynomils, J Appl Mech, Vol. 48, pp Levinson, M. (1980): An Accurte, Simple Theory of the Sttic nd Dynmics of Elstic pltes, Mechnics Reserch Communictions, Vol. 7, pp Lo, K. H.; Christensen, R. M.; Wu, E. M. (1977): A High Order Theory of Plte Deformtion I. Homogeneous Pltes, J Appl Mech, Vol. 44, pp

8 110 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp , 2006 Lo, K. H.; Christensen, R. M.; Wu, E. M. (1977): A High Order Theory of Plte Deformtion II. Lminted Pltes, J Appl Mech, Vol. 44,pp Lo, K. H.; Christensen, R. M.; Wu, E. M. (1978): Stress Solution Determintion for High Order Plte Theory, Int J Solids Struct, Vol. 14, pp Long, S.; Atluri, S. N. (2002): A Meshless Locl Petrov-Glerkin Method for Solving the Bending Prolem of Thin Plte, CMES: Computer Modeling in Engineering & Sciences, Vol. 3, No.1, pp Mindlin,R.D.(1951): Influence of Rottory Inerti nd Sher on Flexurl Motions of Isotropic, Elsitc Pltes, J Appl Mech, Vol. 18, pp Nghdi, P.M. (1972): The Theory of Shells,inHnduch der Physik VI/2, C. Truesdell, ed., Springer-Verlg, Berlin, pp Qin,L.F.;Btr,R.C.;Chen,L.M.(2003): Elstosttic Deformtions of Thick Plte y using Higher- Order Sher nd Norml Deformle Plte Theory nd two Meshless Locl Petrov-Glerkin (MLPG) Methods, CMES: Computer Modeling in Engineering & Sciences, Vol. 4, No.1, pp Reddy, J. N. (1984): A Simple High-Order Theory for Lminted Composite Pltes, J Appl Mech, Vol. 45, pp Rehfield,L.W.;Murthy,P.L.N.(1982): Towrd New Engineering Theory of Bending: Fundmentls, AIAA J, Vol. 20, No. 5, pp Rehfield,L.W.;Vlisetty,R.R.(1984): A Simple, Refined Theory for Bending nd Stretching of Homogeneous Pltes, AIAA J, Vol. 22, No.1, pp Reissner, E. (1945): The Effect of Trnsverse Sher Deformtion on the Bending of Elstic Pltes, J Appl Mech, Vol. 12, pp. A69-A77. Reissner, E. (1947): On Bending of Elstic Pltes, Qurt Appl Mech, Vol. 5, pp Reissner, E. (1975): On Trnsverse Bending of Pltes including the Effects of Trnsverse Sher Deformtion, Int J Solids Struct, Vol. 11, pp Reissner, E. (1983): A Twelfth Order Theory of Trnsverse Bending of Trnsversely Isotropic Pltes, ZAMM, Vol. 63, pp Suetke, Y.; Tomod, T. (2004): A Considertion of Trnsverse Lord Effect in Thick Plte Anlyses, JApplied Mechnics (JSCE), Vol. 7, pp (in Jpnese) Suetke, Y. (2005): Influence of Lterl Lod Effect on Bending Anlyses of Thick Pltes, J Applied Mechnics (JSCE), Vol. 8, pp (in Jpnese) Voyidjis, G. Z.,; Bluch, M. N. (1981): Refined Theory for Flexurl Motions of Isotropic Elstic Pltes, J Sound Virtion, Vol. 76, pp