Plate Bending Analysis by Using a Modified Plate Theory

Plate Bending Analysis by using a Modiﬁed Plate Theory

Y. Suetake 1

Abstract: Since Reissner and Mindlin proposed their therefore, the transverse shear stresses distribute con- classical thick plate theories, many authors have pre- stantly along the thickness of plates. This apparently sented refined theories including transverse shear defor- contradicts the shear-free condition on the plate surfaces. mation. Most of those plate theories have tended to use In order to compensate the contradiction, Mindlin in- higher order power series for displacements and stresses troduced a correction parameter for the transverse shear along the thickness in order to achieve the higher accu- stresses. In this theory, the transverse normal stress is ne- racy. However, they have not carefully noticed lateral glected and difference between the upper and lower sur- load effect. In this paper, we pay attention to constitution face tractions is adopted as a lateral load. of the lateral loads: a body force and upper and lower sur- It is well-known that the above classical theories coincide face tractions. Especially we formulate a modified theory when the transverse normal stress is neglected and the for plate bending, in which the effect of a body force is shear correction parameter is 5/6. Rational determination distinguished from that of surface tractions. The present of the correction parameter, however, is not presented. plate theory includes not only transverse shear deforma- In some cases we cannot obtain accurate solutions when tion but also transverse normal stress effect. In this paper, the parameter is 5/6. The solutions depend on not only our attention is focused on bending moment behavior of the shear correction parameter but also constitutionof the plates. lateral loads. keyword: Thick plate theory, Transverse shear defor- Many high order theories have been presented in order mation, Load effect, Transverse normal stress, Body to obtain more accurate solutions of the plate bending. force. Levinson (1980) and Reddy (1984) assume a cubic distribution of in-plane displacements for thick plates. Reiss- ner (1975) and Rehfield (1982 and 1984) consider de- 1 Introduction flection change along the thickness of plates. Lo, Chris- Since Reissner (1945) and Mindlin (1951) proposed their tensen, and Wu (1977a, 1977b, and 1978) formulate a classical thick plate theories, many authors have pre- high-order theory of plates which includes both in-plane sented refined theories including transverse shear defor- and out-of-plane modes of deformation introducing 11 mation. As well-known the Reissner’s theory (1945) is unknown parameters. an assumed-stress theory and the Mindlin’s theory (1951) The above plate theories are assumed-displacement ones. an assumed-displacement one. Alternative theories, which are assumed-stress theories, In the Reissner’s theory, a parabolic distribution of trans- are also important. Ambartsumyan (1975) presents an verse shear stresses is assumed and we can satisfy the assumed-stress high-order plate theory, in which both the shear-free condition on the upper and lower surfaces of transverse shear stresses and the transverse normal stress plates. In addition a transverse normal stress is also in- is incorporated. Voyiadjis and Baluch (1981) consider corporated in the theory. A cubic distributionof the trans- the transverse normal strain in addition to the transverse verse normal stress is assumed, in which only an upper stresses. Reissner (1983) also formulate an assumed- surface traction is considered. Discussion of strains of stress high-order plate theory. plates is supplemented in Reissner (1947). Hirashima and Muramatsu (1980), and Hirashima and In the Mindlin’s theory, linear distributions of in-plane Negishi (1983 1st and 2nd) establish a generalized high- displacements and a constant deflection are assumed and, order plate theory that includes the above theories as particular ones. Hirashima and Negishi (1983 1st ) also dis- 1 Ashikaga Institute of Technology, 268-1 Ohmaecho, Ashikaga, cuss the accuracy of the plate theories in detail. Krenk Tochigi, 326-8558, Japan. 104 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp.103-110, 2006

(1981) employs Legendre polynomials for representing sumed here is given by the distribution of displacements and stresses along the ⎫ = −ψ − 4 ϕ 3 ϕ ≡ ∂w −ψ ⎪ thickness. Green and Naghdi (1972) also present a multi- U xz 2 xz ; x ∂x x ⎪ 3t ⎪ director approach for plates and shells, which surpasses ⎬ = −ψ − 4 ϕ 3 ϕ ≡ ∂w −ψ , the classical theories. Recently, many sophisticated nu- V yz 2 yz ; y y (1) 3t ∂y ⎪ merical approaches are applied to the analysis of the plate ⎪ ⎭⎪ bending, for example, Long and Atluri (2002), and Qian, W = w Batra, and Chen (2003). In general, the higher the order of the theories is, the where x and y are in-plane coordinates, z is a coordinate higher the accuracy of those is. However, the concise- normal to the mid-surface of plates, t is a thickness of ψ ψ ness of the theories has been lost. On the other hand, plates, x and y are deflection angles, and w is a de- rd the importance of the lateral load effect is not noticed in flection of plates. In Eq. (1), if we neglect the 3 - the above high-order plate theories. The lateral load of order terms, we have the Mindlin type displacement field plates consists of the upper and lower surface tractions [Mindlin (1951)]. From the displacement field (1), we and the body force. In the above theories the body force obtain the following strain distribution: is not considered and its effect on plate bending is not i) in-plane strains separated from that of the surface tractions. ⎫ ∂ψ ∂ϕ ε = − x − 4 x 3 ⎪ x ∂ z 2 ∂ z ⎪ In this paper, we pay attention to the constitution of the x 3t x ⎪ ⎬⎪ lateral loads as Suetake and Tomoda (2004) and Sue- ∂ψ ∂ϕ ε = − y z − 4 y z3 , (2) take (2005), in which a modified bending theory of thick y ∂y 3t2 ∂y ⎪ ⎪ plates is formulated. In the modified theory employed ⎪ ∂ψ ∂ψ ∂ϕ ∂ϕ ⎪ here, the body force is separated from the surface trac- γ = −( x + y ) − 4 ( x + y ) 3 ⎭ xy ∂y ∂x z 3t2 ∂y ∂x z tions. The present plate theory includes not only transverse shear deformation but also transverse normal stress ii) transverse shear strains effect. The present modified theory can be a simple means of comparison for numerical analyses. In this pa- 4 2 4 2 γxz = ϕx(1 − z ) , γyz = ϕy(1 − z ). (3) per, our attention is focused on bending moment behavior t2 t2 of plates and we make sure that the present modified the- Note that Eq.(3) satisfies the shear-free condition on the ory gives us excellent results, even though it is as simple upper and lower surfaces of plates. as the classical theory. 2.2 Constitutive equation

2 Modified Theory Since we treat with isotropic elastic plates here, we employ the Hooke’s law as a constitutive equation. Elim- In this paper a modified bending theory of plates is pre- inating the transverse normal strain εz from the 3-D sented by using the Levinson-Reddy type displacement Hooke’s law, we obtain the following relation: field [Levinson (1980) and Reddy (1984)]. We pay atten- ⎫ σ = E (ε +νε )+ ν σ ⎪ tion to the constitution of the lateral loads through con- x 1−ν2 x y 1−ν z ⎪ ⎪ sideration of the transverse normal stress. Consequently ⎬ σ = E (νε +ε )+ ν σ , we can treat with the surface tractions and the body force y 1−ν2 x y 1−ν z ⎪ (4) ⎪ separately. ⎪ τ = γ ≡ E ⎭ xy G xy ; G 2(1+ν)

2.1 Displacement-strain relation and

rd Levinson (1980) and Reddy (1984) assume the 3 -order τxz = Gγxz , τyz = Gγyz, (5) displacement ﬁeld in order to represent distortion of a normal to the mid-surface. The displacement ﬁeld as- where E is Young’s modulus and ν is Poisson’s ratio. Plate Bending Analysis 105

Substitution of Eqs. (2) and (3) into Eqs. (4) and (5) p(x,y) 1 gives us ⎫ ∂ψ ∂ψ σ = − E {( x +ν y ) ⎪ Z=p (x,y)/t x 1−ν2 ∂x ∂y z ⎪ 0 x ∂ϕ ∂ϕ ⎪ + 4 ( x +ν y ) 3} + ν σ ⎪ O 2 ∂ ∂ z −ν z ⎪ 3t x y 1 ⎪ ⎬⎪ ∂ψ ∂ψ σ = − E {(ν x + y )z , (6) y 1−ν2 ∂x ∂y ⎪ ∂ϕ ∂ϕ ν ⎪ + 4 (ν x + y )z3} + σ ⎪ p(x,y) t2 ∂x ∂y 1−ν z ⎪ z 2 3 ⎪ ⎪ ∂ψ ∂ψ ∂ϕ ∂ϕ ⎭⎪ Figure 1 : Lateral Load Constitution of Plate τ = − {( x + y ) + 4 ( x + y ) 3} xy G ∂y ∂x z 3t2 ∂y ∂x z and ing it with respect to z, in view of Eq. (10), we have 4 4 τ = Gϕ (1 − z2) , τ = Gϕ (1 − z2). (7) Gt ∂ψ ∂ψ 3 4 xz x t2 yz y t2 σ = (∇2w− x − y )(1 − z + z3) z 3 ∂x ∂y t t3 Equation (7) is the same as the shear distribution in the + p0 ( − 2 )+ . 1 z p2 (10) Reissner’s theory [Reissner (1945)]. In the Mindlin’s the- 2 t ory [Mindlin (1951)], the 2nd-and3rd-order terms are neglected and the effect of the transverse normal stress is When we apply the traction boundary condition (9) to not incorporated. Eq.(10), we obtain ∂ψ ∂ψ 3 x + y = ∇2w+ p ; p ≡ p + p + p . (11) 2.3 Equilibrium condition ∂x ∂y 2Gt 0 1 2 Linear equilibrium conditions for 3-D bodies are given σ by Consequently we obtain the distribution of z as ⎫ ∂σ ∂τ ∂τ p 3 4 3 p0 2 x + xy + xz + = ⎪ σ = − (1 − z + z )+ (1 − z)+p . (12) ∂ ∂ ∂ X 0 ⎪ z 3 2 x y z ⎪ 2 t t 2 t ⎬⎪ ∂τ ∂σ ∂τ xy + y + yz + = , (8) Integrating the 1st and 2nd of Eq. (8) with respect to z, ∂x ∂y ∂z Y 0 ⎪ ⎪ we obtain the ordinary moment equilibrium equations. ⎪ ∂τ ∂τ ∂σ ⎭⎪ xz + yz + z +Z = 0 ∂x ∂y ∂z 2.4 Governing equation where X, Y,andZ are body forces and, in this paper, we Integration of Eqs.(6) and (7) with respect to z,inviewof consider only Z, that is, we set X = Y = 0. Eq. (12), gives us the following moment and shear force expressions: By using the 3rd equilibrium condition in Eqs. (8), we σ can determine the distribution of z. Before doing that, 4 ∂ψ ∂ψ 1 ∂2w ∂2w M = −D{ ( x +ν y )+ ( +ν )} we should pay attention to the constitution of the lateral x 5 ∂x ∂y 5 ∂x2 ∂y2 loads. The lateral load of a plate consists of the body νt2 force Z = p (x,y) t and the upper and lower surface trac- + { + ( + )}, 0 ( −ν) p0 6 p1 p2 (13) ( , ) ( , ) 60 1 tions, p1 x y and p2 x y , as shown in Fig.1. Therefore we have the following traction boundary conditions: 4 ∂ψ ∂ψ 1 ∂2w ∂2w t t M = −D{ (ν x + y )+ (ν + )} σz(x,y,− )=−p , σz(x,y, )=p . (9) y ∂ ∂ ∂ 2 ∂ 2 2 1 2 2 5 x y 5 x y νt2 + {p +6(p + p )}, (14) Substituting Eq. (7) into the 3rd of Eq. (8) and integrat- 60(1 −ν) 0 1 2 106 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp.103-110, 2006

D 4 ∂ψ ∂ψ 2 ∂2w 1 ∂ = − ( −ν){ ( x + y )+ }, ∇4ψ = [p + p + p Mxy 1 ∂ ∂ ∂ ∂ (15) y ∂ 0 1 2 2 5 y x 5 x y D y t2 + ∇2 {(3 +ν)p +3(1 +2ν)(p + p )} . and 60(1 −ν) 0 1 2 2 ∂w 2 ∂w (22) Q = Gt( −ψ ) , Q = Gt( −ψ ), (16) x 3 ∂x x y 3 ∂y y 3 Fourier Analysis where D is the bending rigidity of plates: D ≡ For numerical calculations we employ here the Fourier Et3 12(1 −ν2). Substituting Eqs. (13) to (16) into the series analysis. In this section we explain the Fourier ordinary moment equilibrium equations, we have analyses of plates and 3-D bodies. Two plate analyses 2 2 2 are presented here; one is based on the present theory 4 ∂ ψx 1 −ν ∂ ψx 1 +ν ∂ ψy 1 ∂ D ( + + )+ (∇2w) and the other on the classical one. We can also employ 5 ∂x2 2 ∂y2 2 ∂x∂y 5 ∂x alternative approaches for the analyses. ∂ ν 2 ∂ ∂ ∂ 2 w t p0 p1 p2 + Gt( −ψx) − +6( + ) 3 ∂x 60(1 −ν) ∂x ∂x ∂x 3.1 Plate analysis based on the present theory = 0, (17) Elastic plates adopted here as numerical examples are simply supported rectangular plates subjected to lateral and loads. The coordinates of the plate model is shown in 4 ∂2ψ 1 −ν ∂2ψ 1 +ν ∂2ψ 1 ∂ Fig.2. D ( y + y + x )+ (∇2w) 5 ∂y2 2 ∂x2 2 ∂x∂y 5 ∂y The deﬂection and the deﬂection angles are expressed by ∂ ν 2 ∂ ∂ ∂ the following trigonometric double series here: 2 w t p0 p1 p2 + Gt( −ψy) − +6( + ) ⎫ 3 ∂y 60(1 −ν) ∂y ∂y ∂y = ∑∑ mπx nπy ⎪ w Wmn sin a sin b ⎪ = . n m ⎪ 0 (18) ⎪ ⎬⎪ ψ = ∑∑Φ mπx nπy x mn cos a sin b . (23) Equations (11), (17), and (18) are the present governing n m ⎪ ⎪ equations for the plate bending. We can rewrite Eqs. (17) ⎪ ψ = ∑∑Ψ mπx nπy ⎭⎪ and (18), in view of Eq.(11), as y mn sin a cos b n m

10 ∂ψ ∂ψy (∇2 − )( x − )=0, (19) Note that Eqs. (23) satisfy the boundary condition of t2 ∂y ∂x simply supported plates. In addition, we expand the load functions into the following Fourier double series: 1 ∇4 = [ + + ( ) mπx nπy w p0 p1 p2 = i D pi ∑∑Pmn sin sin ; n m a b t2 12 −ν 3 Z Z − ∇2 p + (2 −ν)(p + p ) (20). b a π π 6(1 −ν) 10 0 5 1 2 (i) = 4 ( , ) m x n y . Pmn pi x y sin sin dxdy (24) ab 0 0 a b If we assume that ∂ψx ∂y = ∂ψy ∂x, Eq. (19) can be sat- isﬁed a priori. In that case, we can derive the governing ψ ψ y=0 x=a equations for x and y from Eqs. (11) and (17) to (19): mid- surface x z=-t/2 ∂ O ∇4ψ = 1 [ + + C x ∂ p0 p1 p2 D x 2 z x=0 t y=b + ∇2 {(3 +ν)p +3(1 +2ν)(p + p )} , y 60(1 −ν) 0 1 2 z=t/2 (21) Figure 2 : Rectangular Plate and Coordinates Plate Bending Analysis 107

1 nπ ( ) ( ) ( ) Substituting Eqs. (23) into Eqs. (20) to (22), we can Ψ = · (P 0 +P 1 +P 2 ). (32) mn λ4 mn mn mn easily determine the coefficients Wmn, Φmn,andΨmn as mnD b As well-known, we can calculate the bending and twist- 1 (0) (1) (2) W = P +P +P ing moments of plates by using the following expressions mn λ4 D mn mn mn mn instead of Eqs. (13) to (15): λ2 t2 12 −ν ( ) 3 ( ) ( ) + mn P 0 + (2 −ν)(P 1 +P 2 ) , ∂ψ ∂ψ 6(1 −ν) 10 mn 5 mn mn = − ( x +ν y ), Mx D ∂ ∂ (33) (25) x y ∂ψ ∂ψ M = −D(ν x + y ), (34) y ∂x ∂y 1 mπ ( ) ( ) ( ) Φ = · P 0 +P 1 +P 2 mn λ4 mn mn mn D ∂ψx ∂ψy mnD a M = − (1 −ν)( + ). (35) xy 2 ∂y ∂x λ2 t2 ( ) ( ) ( ) − mn (3 +ν)P 0 +3(1 +2ν)(P 1 +P 2 ) , 60(1 −ν) mn mn mn 3.3 3-D analysis (26) In order to evaluate the accuracy of the present plate theory, we perform a 3-D elastic analysis of the plate model. 1 nπ (0) (1) (2) We employ the Fourier series again for the 3-D analysis. Ψ = · P +P +P mn λ4 mn mn mn mnD b We explain the approach briefly in this subsection. Geo- λ2 t2 ( ) ( ) ( ) metrical boundary conditions of the model as a 3-D body − mn (3 +ν)P 0 +3(1 +2ν)(P 1 +P 2 ) , 60(1 −ν) mn mn mn is given by ⎫ (27) V (0,y,z)=V(a,y,z)=W (0,y,z)=W(a,y,z)=0 ⎬ , λ2 ≡ ( π )2 +( π )2 ⎭ where mn m /a n b . U(x,0,z)=U(x,b,z)=W(x,0,z)=W(x,b,z)=0 3.2 Plate analysis based on the classical theory (36) At this stage it is significant to review the classical plate which corresponds to the conditions for simply supported theories. The governing equations of the static Mindlin’s plates. To satisfy these conditions, we employ the follow- theory [Mindlin (1951)] are as follows: ing trigonometric series ⎫ mπx nπy 2 U = ∑∑umn(z)cos sin ⎪ 4 1 t 2 a b ⎪ ∇ w = {1 − ∇ }p, (28) n m ⎪ D 6(1 −ν)κ ⎬⎪ π π V = ∑∑v (z)sin m x cos n y . ∂ ∂ mn a b (37) 4 1 p 4 1 p n m ⎪ ∇ ψx = , ∇ ψy = , (29) ⎪ D ∂x D ∂y ⎪ = ∑∑ ( ) mπx nπy ⎭⎪ W wmn z sin a sin b where κ is the shear correction parameter. In Eq. (28), if n m we Traction boundary conditions employed here are κ = / ( − ν) set 5 3 2 , the static Mindlin’s theory coincides t with the Reissner’s one. σ (x,y,− )=−p (x,y) z 2 1 If we use the trigonometric series (23) again in the classi- σ ( , , t )= ( , ) , z x y p2 x y cal bending analysis of plates, the coefficients Wmn, Φmn, 2 and Ψmn can be determined as t t τxz(x,y,± )=τyz(x,y,± )=0. (38) ( ) ( ) ( ) 2 2 P 0 +P 1 +P 2 λ2 t2 W = mn mn mn {1 + mn }, (30) In addition, the body forces of the model are represented mn λ4 D 6(1 −ν)κ mn by 1 mπ (0) (1) (2) Φ = · (P +P +P ), (31) = = , = 1 ( , ). mn λ4 mn mn mn X Y 0 Z p0 x y (39) mnD a t 108 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp.103-110, 2006

Substitution of Eq. (37) into the Navier’s equation, which non-symmetric distribution. Numerical properties of the is a governing equation for 3-D elastic problems, yields loads are as follows: an ordinary differential equation system with respect to i) symmetric distribution ( ) ( ) ( ) the unknown functions umn z , vmn z ,andwmn z .When we solve the differential equation system under Eqs. (38) χˆ = . , χˆ = . , = = . , 0 0 8 1 1 25 x0 a y0 b 0 3 and (39), we can determine those three unknown func- a∗/a = b∗ b = 0.4, (42) tions. ii) non-symmetric distribution 4 Numerical Models χˆ = . , χˆ = , = = . , As numerical examples we adopt simple plate bending 0 0 6 1 40 x0 a y0 b 0 2 problems of a square plate. The plate model is simply a∗/a = b∗ b = 0.1 , (43) supported along the all edges. The width-thickness ratio ≡ . ≤ ≤ . µ t/a is changed within the range of 0 001 µ 0 5. where the non-dimensional load parameter χˆ i is defined Poisson’s ratio is ν = 0.3. In the Fourier analysis, we take by 200 ×200 = 40000 terms in the double series. pˆ a3 12(1 −ν2) t A constitution of lateral loads adopted here is shown in χˆ = i = = . i 4 pî ; µ (44) Fig.3. The plate model is subjected to a constant body µD Eµ a force and a partially distributed constant load on the up- 5 Numerical Results per surface. The Fourier coefficients of the load functions are given In this paper, our attention is focused on bending moment by behavior of plates. In the numerical calculation, we esti- ⎧ mate the error of the present plate analysis. Results of the ⎪ 16p0 ⎨ π2( − )( − ) (m and n: odd) 3-D elastic analysis are employed as the standard values. (0) 2 j 1 2k 1 P = , (40) Classical analysis based on the Mindlin’s theory is also mn ⎩⎪ 0(morn:even) performed in order to confirm the accuracy of the present analysis. Since we adopt the Levinson-Reddy type displacement ( ) 4p mπx mπ(x +a∗) P 1 = 1 cos 0 −cos 0 field, Eq.(1), it is not so easy to predict local behaviors mn π2mn a a for the large thickness of plates. This is a further issue to nπy nπ(y +b∗) × cos 0 −cos 0 . (41) be improved. b b 5.1 Symmetric surface traction In this subsection, we discuss the case of the symmetric * a*,b surface traction. As mentioned before, the plate model x0,y0 p 1 is subjected to not only a constant body force but also a partially distributed constant load on the upper surface. Z=p (x,y)/tZ=p /t 0 0 x, y In this case, the surface traction is symmetric. O O C Numerical results are shown in Fig.4, in which the er- rors of the present and the classical plate analyses, ε,are a, b plotted against the width-thickness ratio of the model, µ. z In Fig.4, closed circles indicate the results of the present Figure 3 : Figure 3: Lateral Load Constitution analysis and closed triangles that of the classical analysis, respectively. Two different constitutions of lateral loads are adopted It follows from Fig.4 that the present modified plate the- in the present numerical calculations. One is a sym- ory approximates the 3-D analysis with high accuracy. metric distribution on the upper surface; the other a Especially, we should note that the error of the present Plate Bending Analysis 109

10 1

8 Present 0.8 Present Classical Classical ε 6 ε 0.6 ［%］ 4 ［%］ 0.4

2 0.2

0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 μ μ

Figure 4 : Error of Moment at Center Point (Symmetric Figure 5 : Error of Moment at Center Point (Non- Model) symmetric Model)

analysis remains quite small even in the thick plate re- a moderately thick plate region. gion near µ = 0.5. On the other hand, the error of the 3) The constitution of the lateral loads plays a key role in classical analysis increases rapidly with the increase of the plate bending analyses of thick plates. In particular, the width-thickness ratio. when the constitution of loads is not simple, we should The excellent approximation of the deflection behavior use the modified theory instead of the classical one. has already been confirmed [Suetake 2005]. The present investigation shows the efficiency of the modified plate theory also in the moment analysis. References Ambartsumyan, S. A. (1975): Theory of Anisotropic 5.2 Non-symmetric surface traction Elastic Plates (translated by Norio Kamiya), Morikita, The results of the non-symmetric surface traction model Tokyo. (in Japanese) are presented in this subsection. The load adopted Hirashima, K.; Muramatsu, M. (1980): The Effect of here consists of a constant body force and a non- Transverse Components on the Bending of Plates, Proc symmetrically distributed constant load on the upper sur- JSCE, Vol. 304, pp. 33-46. (in Japanese) face. Hirashima, K.; Negishi, Y. (1983): Some Considera- Numerical results of the moment error at the center point tions on Accuracies of Typical Two-Dimensional Plate of the model are shown in Fig.5, which is depicted in the Theories including the Effects of Transverse Compo- same manner as Fig.4. It can be seen also from Fig.5 nents, Proc JSCE, Vol. 330, pp. 1-14. (in Japanese) that the present modified plate theory gives us excellent Hirashima, K.; Negishi, Y. (1983): Study on Dynamic results. In this case, however, the classical theory also Characteristics (Free Vibration and Dispersion Relation) maintains practically sufficient accuracy. of Several Plate Theories, Proc JSCE, Vol. 333, pp. 21- 34. (in Japanese) 6 Concluding Remarks Krenk, S. (1981): Theory for Elastic Plates Via Orthog- The following conclusions may be drawn from the onal Polynomials, J Appl Mech, Vol. 48, pp. 900-904. present investigation: Levinson, M. (1980): An Accurate, Simple Theory of 1) A modified plate bending theory is proposed, in which the Static and Dynamics of Elastic plates, Mechanics Re- the effect of lateral loads is carefully considered. search Communications, Vol. 7, pp. 343-350. 2) The new theory gives us excellent approximations for Lo, K. H.; Christensen, R. M.; Wu, E. M. (1977): A moments even in thick plate region, while the classi- High Order Theory of Plate Deformation – I. Homoge- cal one maintains practically sufficient accuracy within neous Plates, J Appl Mech, Vol. 44, pp. 663-668. 110 Copyright c 2006 Tech Science Press CMES, vol.11, no.3, pp.103-110, 2006

Lo, K. H.; Christensen, R. M.; Wu, E. M. (1977): A Suetake, Y. (2005): Influence of Lateral Load Effect on High Order Theory of Plate Deformation – II. Laminated Bending Analyses of Thick Plates, J Applied Mechanics Plates, J Appl Mech, Vol. 44, pp. 669-676. (JSCE), Vol. 8, pp. 25-32. (in Japanese) Lo, K. H.; Christensen, R. M.; Wu, E. M. (1978): Voyiadjis, G. Z.,; Baluch, M. N. (1981): Refined The- Stress Solution Determination for High Order Plate The- ory for Flexural Motions of Isotropic Elastic Plates, J ory, Int J Solids Struct, Vol. 14, pp. 655-662. Sound Vibration, Vol. 76, pp. 57-64. Long,S.;Atluri,S.N.(2002): A Meshless Local Petrov-Galerkin Method for Solving the Bending Prob- lem of a Thin Plate, CMES: Computer Modeling in En- gineering & Sciences, Vol. 3, No.1, pp.53-63. Mindlin,R.D.(1951): Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elasitc Plates, J Appl Mech, Vol. 18, pp. 31-38. Naghdi, P.M. (1972): The Theory of Shells,inHandbuch der Physik VIa/2, C. Truesdell, ed., Springer-Verlag, Berlin, pp. 425–640. Qian,L.F.;Batra,R.C.;Chen,L.M.(2003): Elasto- static Deformations of a Thick Plate by using a Higher- Order Shear and Normal Deformable Plate Theory and two Meshless Local Petrov-Galerkin (MLPG) Methods, CMES: Computer Modeling in Engineering & Sciences, Vol. 4, No.1, pp.161-175. Reddy, J. N. (1984): A Simple High-Order Theory for Laminated Composite Plates, J Appl Mech, Vol. 45, pp. 745-752. Rehfield,L.W.;Murthy,P.L.N.(1982): Toward a New Engineering Theory of Bending: Fundamentals, AIAA J, Vol. 20, No. 5, pp. 693-699. Rehfield,L.W.;Valisetty,R.R.(1984): A Simple, Refined Theory for Bending and Stretching of Homoge- neous Plates”, AIAA J, Vol. 22, No.1, pp. 90-95. Reissner, E. (1945): The Effect of Transverse Shear De- formation on the Bending of Elastic Plates, J Appl Mech, Vol. 12, pp. A69-A77. Reissner, E. (1947): On Bending of Elastic Plates, Quart Appl Mech, Vol. 5, pp. 55-68. Reissner, E. (1975): On Transverse Bending of Plates including the Effects of Transverse Shear Deformation, Int J Solids Struct, Vol. 11, pp. 569-573. Reissner, E. (1983): A Twelfth Order Theory of Trans- verse Bending of Transversely Isotropic Plates, ZAMM, Vol. 63, pp. 285-289. Suetake, Y.; Tomoda, T. (2004): A Consideration of Transverse Lord Effect in Thick Plate Analyses, JAp- plied Mechanics (JSCE), Vol. 7, pp. 47-56. (in Japanese)