Microsoft Word - 2355 Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 208 Mechanical stability investigation of advanced composite plates resting on elastic foundations using a new four-unknown refined theory Yassine Khalfi Université de Sidi Bel Abbes, BP 89 Cité Ben M’hidi 22000 Sidi Bel Abbes, Algérie khalfiyassinesba@yahoo.fr Aboubakar Seddik Bouchikhi, Yassine Bellebna Université de Sidi Bel Abbes, Ecole National Polytechnique D’oran asbouchiki@yahoo.fr, yassinebellebna@yahoo.fr ABSTRACT. A refined and simple shear deformation theory for mechanical buckling of composite plate resting on two-parameter Pasternak’s foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces.Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories.It is assumed that the warping of the cross sections generated by transverse shear is presented by a hyperbolic function. The stability equations are determined using the present theory and based on the existence of material symmetry with respect to the median plane.The nonlinear strain-displacement of Von Karman relations are also taken into consideration. The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical load of mechanical buckling, which are useful for engineers in design. The effects of the foundation parameters, side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are presented comprehensively for the mechanical buckling of rectangular composite plates. KEYWORDS. Rectangular composite plate; Critical buckling load; Refined shear theory; Elastic foundation; Non-linear relations of Vont Karman. Citation: Khalfi, Y, Bouchiki, A, S., Bellebna, Y., Mechanical stability investigation of advanced composite plates resting on elastic foundations using a new four-unknown refined theory, Frattura ed Integrità Strutturale, 48 (2019) 208-221. Received: 20.01.2019 Accepted: 12.02.2019 Published: 01.04.2019 Copyright: © 2019 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. http://www.gruppofrattura.it/VA/48/2355.mp4 Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 209 INTRODUCTION omposite materials have important advantages over traditional materials. They bring many functional advantages: lightness, mechanical and chemical resistance, reduced maintenance, freedom of forms. They make it possible to increase the lifespan of certain equipment thanks to their mechanical and chemical properties. They contribute to the reinforcement of safety thanks to a better resistance to shocks and fire. They offer better thermal or sound insulation and, for some of them, good electrical insulation. They also enrich the design possibilities by lightening structures and making complex shapes, able to fulfill several functions. In each of the application markets (automotive, building, electricity, industrial equipment, etc.), these remarkable performances are at the origin of innovative technological solutions [1]. They constitute one of the most advanced class of materials whose popularity in industrial applications keeps growing exponentially [2]. Their advent has been aided by the development of new processing methods, theoretical approaches of homogenization [3, 4] and numerical simulations of heterogeneous materials [5]. This class of materials is commonly divided into three categories [6] : (i) fibrous composites consisting of continuous fibers embedded in a matrix, (ii)laminated composites consisting of various stacked layers, and (iii) particle-reinforced composites composed of particles in a matrix. The buckling of rectangular plates has been a subject of study in solid mechanics for more than a century. Many exact solutions for isotropic and orthotropic plates have been developed, most of them can be found in Timoshenko and Woinowsky-Krieger [7], Timoshenko and Gere [8], Bank and Jin [9], Kang and Leissa [10], Aydogdu and Ece [11], and Hwang and Lee [12]. In company with studies of buckling behavior of plate, many plate theories have been developed. The simplest one is the classical plate theory (CPT) which neglects the transverse normal and shear stresses. This theory is not appropriate for the thick and orthotropic plate with high degree of modulus ratio. In order to overcome this limitation, the shear deformable theory which takes account of transverse shear effects is recommended. The Reissner [13] and Mindlin [14] theories are known as the first-order shear deformation theory (FSDT), and account for the transverse shear effects by the way of linear variation of in-plane displacements through the thickness. However, these models do not satisfy the zero traction boundary conditions on the top and bottom faces of the plate, and need to use the shear correction factor to satisfy the constitutive relations for transverse shear stresses and shear strains. For these reasons, many higher-order theories have been developed to improve in FSDT such as Levinson [15] and Reddy [16]. Shimpi and Patel [17] presented a four variable refined plate theory (RPT) for orthotropic plates.This theory which looks like higher-orde theory uses only four unknown functions in order to derive two governing equations for orthotropic plates. The most interesting feature of this theory is that it does not require shear correction factor, and has strong similarities with the CPT in some aspects such as governing equation, boundary conditions and moment expressions. The accuracy of this theory has been demonstrated for static bending and free vibration behaviors of plates by Shimpi and Patel [17], therefore, it seems to be important to extend this theory to the static buckling behavior. In this paper, the four variable RPT developed by Shimpi and Patel [17] has been extended to the buckling behavior of isotropic and orthotropic plate resting on two-parameter Pasternak’s foundations subjected to the in-plane loading. Using the Navier method, the closed-form solutions have been obtained. Numerical examples involving side-to-thickness ratio, effects of the foundation parameters and modulus ratio are presented to illustrate the accuracy of the present theory in predicting the critical buckling load of isotropic and orthotropic plates. The numerical results obtained by the present theory are compared with solutions of classical theory (CPT) and solutions of first order shear deformation theory (FSDT) and high order shear theory (HSDT). MATHEMATICAL FORMULATION onsider a rectangular composite plate of thickness h, length a and width b, referred to the rectangular Cartesian coordinate system (x, y, z), as shown in Fig 1. Since in this type of plates there is material symmetry with respect to the median plane (the origin of the coordinate system is appropriately chosen in the direction of the thickness of the composite plate so that it will be confused with the neutral surface) the equations of membranes and bending will be decoupled and therefore equilibrium equations [18]. Based on the refined theory of shear deformation [19], the displacement field can be written as: 0( , , ) ( , ) ( )        b sw wu x y z u x y z f z x x (1a) 0( , , ) ( , ) ( )        b sw wv x y z v x y z f z y y (1b) C C Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 210 ( , ) ( , ) ( , ) b sw x y w x y w x y (1c) where u0 and v0 are the mid-plane displacements of the plate in the x and y direction, respectively; wb and ws are the bending and shear components of transverse displacement, respectively. Figure 1: Coordinate system and geometry for rectangular composite plate on Pasternak elastic foundation. This displacement field verifies the nullity of traction boundary conditions on the top and bottom faces of the plate, and leads to a quadratic variation of transverse shear deformations (and therefore stresses) across the thickness. Thus, it is not necessary to use shear correction factors. The nonlinear deformation-displacement equations of Von Karman are as follows: 0 0 0 ( ) , ( ) b s x x x sx yzyzb s y y y y s xz yzb s xy xy xy xy k k z k f z k g z k k                                                                                (2) where 2 2 2 0 0 2 2 1 , , 2                 b sb s b s x x x u w w w w k k x x x x x (3a) 2 2 2 0 0 2 2 1 , , 2                    b sb s b s y x x v w w w w k k y y y y y (3b) 0 0 0                       b s b s xy u v w w w w y x x x y y 2 2 2 2 2 , 2         b sb s xy xy w w k k y y (3c)  / sinh( ) , , ( ) cosh( / ) 1              s ss s yz xz h z z w w hf z y x h ( ) ( ) 1 ( ), ( ) f z g z f z f z z       (3d) The linear constitutive relations of a composite plate can be written as Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 211 12 22 12 22 66 44 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x y y xy xy yz yz xz xz Q Q Q Q Q Q Q                                                    (4) where (σx, σy, τxy, τyz, τyx) and (εx, εy, γxy, γyz, γyx) are the stress and strain components, respectively. Qij are the elements of the reduced stiffness matrix that are defined as follows: 1 12 2 2 11 12 22 66 12 44 23 55 13 12 21 12 21 12 21 , , , , , 1 1 1 E E E Q Q Q Q G Q G Q G                 (5) Based on the present refined shear plate theory, the stress resultants are related to the stresses by the equations [19]   / 2 / 2 1 , , ( )                          x y xy h b b b x y xy x y xy h s s s x y xy N N N M M M z dz f zM M M (6a) / 2 / 2 ( , ) ( , ) ( )     h s s xz yz xz yz h S S g z dz (6b) Using Eq. (4) in Eq. (6), the stress resultants of the can be related to the total strains by 0 0 , s b s b s s s s s s N A B M D D k S A M B D H k                              (7) where      , , , , , , ,t tt b b b b s s s sx y xy x y xy x y xyN N N N M M M M M M M M   (8a)      0 0 0, , , , , , , ,t t tb b b b s s s sx y xy x y xy x y xyk k k k k k k k      (8b) 11 12 11 12 12 22 12 22 66 66 0 0 0 , 0 0 0 0 0                    A A D D A A A D D D A D (9a) 11 12 11 12 11 12 12 22 12 22 12 22 66 66 66 0 0 0 0 , 0 , 0 0 0 0 0 0 0                                      s s s s s s s s s s s s s s s s s s B B D D H H B B B D D D H H H B D H (9b)     44 55 0 , , , , 0              s t ts s s yz xz yz xz s A S S S A A (9c) Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 212 where Aij,Dij , etc., are the plate stiffness, defined by 2 2 11 11 11 11 11 11 / 2 2 2 12 12 12 12 12 12 / 2 2 2 66 66 66 66 66 66 22 22 22 22 22 11 (1, , ( ), ( ), ( )) (1, , ( ), ( ), ( )) (1, , ( ), ( ), ( )) ( , , , , ) ( ,                          s s s h s s s h s s s s s s A D B D H Q z f z zf z f z A D B D H Q z f z zf z f z dz A D B D H Q z f z zf z f z A D B D H A D11 11 11 11, , , ) s s sB D H   2/ 2 44 55 / / 2 44 ( )   h s s h A A Q g z dz (10) EQUILIBRIUM AND STABILITY EQUATIONS he equilibrium equations of the rectangular composite plate resting on the Pasternak elastic foundation under mechanical loadings may be derived on the basis of the stationary potential energy. The total potential energy of the plate, V, may be written in the form V=U+UF (11) Here, U is the total strain energy of the plate, and is calculated as / 2 0 0 / 2 1 2 a b h x x y y xy xy yz yz xz xz h U dzdydx                    (12) and Uf is the strain energy due to the Pasternak elastic foundation, which is given by [20] 0 0 1 ( ) 2 a b f e b sU f w w dydx   (13) where fe is the density of reaction force of foundation. For the Pasternak foundation model: 2( ) ( )e W b s g b sf K w w K w w     (14) where KW is the Winkler foundation stiffness and Kg is a constant showing the effect of the shear interactions of the vertical elements. Using Eqs. (2), (3), and (7) and employing the virtual work principle to minimize the functional of total potential energy function result in the expressions for the equilibrium equations of plate resting on two parameters elastic foundation as 0      xyx NN x y 0       xy yN N x y 2 22 2 2 2 0           b bb xy yx e M MM N f x yx y T Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 213 2 22 2 2 2 0                 s s ss s xy y yzx xz e M M SM S N f x y x yx y (15) where 2 2 2 2 2 ( ) ( ) ( ) 2b s b s b sxy w w w w w w N Nx N Ny x yx y                 (16) The stability equations in terms of displacement can be obtained by substituting equation (7) in equation (15). The equations obtained based on the present theory of refined shear deformation of the composite plates resting on two-parameter elastic foundation are four in number and are as follows: 2 2 2 3 3 0 0 0 11 66 12 66 11 12 662 2 3 2 ( ) ( 2 ) 0                   s s ss su u v w wA A A A B B B x yx y x x y 2 2 2 3 3 0 0 0 12 66 66 22 22 12 662 2 3 2 ( ) ( 2 ) 0                    s s ss su v v w wA A A A B B B x y x y y x y 4 4 4 4 4 4 11 12 66 22 11 12 66 224 2 2 4 4 2 2 4 2( 2 ) 2( 2 )                        s s s sb b b s s sw w w w w wD D D D D D D D x x y y x x y y 2( ) ( ) 0      W b s g b sK w w K w w N 3 3 3 3 4 4 0 0 0 0 11 12 66 12 66 22 11 12 663 2 2 3 4 2 2 4 4 4 4 2 2 22 11 12 66 22 55 444 4 2 2 4 2 2 ( 2 ) ( 2 ) 2( 2 ) 2( 2 ) ( )                                               s s s s s s s s sb b s s s s s s sb s s s s s W b s g u u v v w w B B B B B B D D D x x y x y y x x y w w w w w w D H H H H A A K w w y x x y y x y K 2 ( ) 0   b sw w N (17) TRIGONOMETRIC SOLUTION TO MECHANICAL BUCKLING ectangular plates are generally classified in accordance with the type of support. We are here concerned with the exact solution of Eq. (17) for a simply supported rectangular composite plate (Figure 2a). The following boundary conditions are imposed for the present refined shear deformation theory at the side edges [21]: 0 0,          b ss b s x x x w v w w N M M at x a y (18a) 0 0,          b ss b s y y y w u w w N M M at y b y (18b) The following approximate solution is seen to satisfy both the differential equation and the boundary conditions 0 0 1 1 cos( ) sin( ) cos( ) sin( ) sin( ) sin( ) sin( ) sin( ) mn mn b bmnm n s smn u U x y v V x y w W x y w W x y                                         (19) where Umn, Vmn, Wbmn, and Wsmn are arbitrary parameters to be determined and λ = mπ/a and μ = nπ/b. Substituting Eq. (19) into Eq. (17), one obtains R Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 214    0K   (20) where {Δ} denotes the column    , , , tmn mn bmn smnU V W W  (21) and [K] is the symmetric matrix given by   11 12 13 14 12 22 23 24 13 23 33 34 14 24 34 44 a a a a a a a a K a a a a a a a a             (22) Figure 2: Rectangular plate: (a) boundary condition and (b) in-plane forces. in which 2 211 11 66( )   a A A 12 12 66( )  a A A 13 0a 2 214 11 12 66( 2 )       s s sa B B B 2 222 66 22( )   a A A 23 0a 2 224 12 66 22( 2 )       s s sa B B B 4 2 2 4 0 2 0 2 2 233 11 12 66 22( 2( 2 ) ( ) )                x y g Wa D D D D N N K K 4 2 2 4 0 2 0 2 2 234 11 12 66 22( 2( 2 ) ( ) )                 s s s s x y g Wa D D D D N N K K Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 215 4 2 2 4 2 2 0 2 0 2 2 2 44 11 12 66 22 55 44( 2( 2 ) ( ) )                     s s s s s s x y g Wa H H H H A A N N K K (23) By applying the static condensation approach to eliminate the coefficients associated with the in-plane displacements, Eq. (20) can be rewritten as 11 22 12 1 212 0 0T K K K K                                  (24) where 33 3411 12 1411 12 22 34 4412 22 24 0 , , 0                            a aa a a K K K a aa a a (25a) 1 2,                mn bmn mn smn U W V W (25b) Equation (24) represents a pair of two matrix equations: 11 1 12 2 0         K K (26a) 12 1 22 2 0          T K K (26b) Solving Eq. (26a) for Δ1 and then substituting the result into Eq. (26b), the following equation is obtained: 22 2 0K     (27) where 122 33 3422 12 11 12 34 44 T a a K K K K K a b                            (28a) and 33 3433 34, a a a a 1 243 34 44 44 14 24 0 0 ,    b b a a b a a a b b 20 11 22 12 1 14 22 12 24 2 11 24 12 14, , ,     b a a a b a a a a b a a a a (28b) For nontrivial solution, the determinant of the coefficient matrix in Eq. (27) must be zero. This gives the following expression for the mechanical buckling load 2 0 0 33 44 34 2 2 33 44 34 1 2 x y a b a N N a a a       (29) Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 216 Clearly, when the effect of transverse shear deformation is neglected, the Eq.(29) yields the result obtained using the classical plate theory [22]. It indicates that transverse shear deformation has the effect of reducing the buckling load. For each choice of m and n, there is a corresponsive unique value of N0. The critical buckling load is the smallest value of N0(m, n). RESULTS AND DISCUSSION o illustrate the proposed theory, a simply supported rectangular plate subjected to the different types of loading (or even Figure.2), is considered to verify the accuracy of the current theory in the prediction of the critical loads of the mechanical buckling of rectangular composite plates. Comparisons are made with different plate theories available in the literature The description of the different displacement models is given in Table 1. In order to study the effects of the parameters of the foundation, side-to-thickness ratio (a/h) and the modulus ratios (E1/E2), isotropic square plates and orthotropes are considered. The shear correction factor (k=5/6) and also used for the first order shear deformation theory (FSDT) and a comparison with the current theory is established. Figure 3: The loading conditions of square plate for (a) uniaxial compression, (b) biaxial compression. Model Theory Unknown functions CPT FSDT HSDT Present Classical plate theory First-order shear deformation theory [23] Higer order shear deformation theory [24] Present refined plate theory 3 5 5 4 Table 1: Displacement models. It is assumed that the thickness and properties of materials for all laminates are the same. The following engineer constants are used [25]:  for isotropic rectangular plates: E1 = E2= E, G12 = G13= G23 = G = E/2(1+ ν), ν12 = ν13 = ν23= ν = 0.3 (30)  For orthotropic rectangular plates: E1 = E2 varied, G12/E2 = G13/E2 = 0.5, G23/E2 = 0.2, ν12 = 0.25, ν21:=( ν12 E2)/E1 (31) For convenience, the following nondimensional buckling load is used: T Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 217 2 3 2 crN aN E h  (32) where a is the length of the square plate and h is the thickness of the plate. The following dimensionless of Winkler’s and Pasternak’s elastic foundation parameters, as well as the critical buckling temperature difference are used in the present analysis 4 4 1 2,W g a a k K k K D D   (33) where 3 2 2 / 12(1 )D E h     (34) a/h Theories k1=0, k2=0 k1=10, k2=0 k1=10, k2=10 k1=0, k2=0 k1=10, k2=0 k1=10, k2=10 Isotropic v = 0.3 Orthotrophic E1/E2=10 25 40 Orthotrophic E1/E2=10 25 40 Orthotrophic E1/E2=10 25 40 5 present CPT FSDT HSDT 2.9512 3.0440 4.8755 3.6152 3.7080 5.5395 2.9498 3.0042 3.9345 2.9512 3.0440 4.8755 6.3487 9.1039 10.578 11.1628 23.4949 35.8307 6.1804 8.2199 9.1085 6.3487 9.1039 10.578 6.4378 9.1939 10.6685 11.2528 23.5849 35.9207 6.2265 8.2666 9.1564 6.4378 9.1939 10.6685 8.2156 10.9717 12.4463 13.0306 25.3627 37.6986 7.0759 9.1356 10.0504 8.2156 10.9717 12.4463 10 present CPT FSDT HSDT 3.4224 3.5151 5.3466 3.6152 3.7080 5.5395 3.4222 3.5088 5.1996 3.4224 3.5151 5.3466 9.3732 16.7719 22.2581 11.1628 23.4949 35.8307 9.2733 15.8736 20.3044 9.3732 16.7719 22.2581 9.4632 16.8620 22.3482 11.2528 23.5849 35.9207 9.3531 15.9501 20.3789 9.4632 16.8620 22.3482 11.2410 18.6397 24.1260 13.0306 25.3627 37.6986 10.9144 17.4460 21.8356 11.2410 18.6397 24.1260 20 present CPT FSDT HSDT 3.5650 3.6578 5.4893 3.6152 3.7080 5.5395 3.5650 3.6569 5.4711 3.5650 3.6578 5.4893 10.6534 21.3479 31.0685 11.1628 23.4949 35.8307 10.6199 20.9528 30.0139 10.6534 21.3479 31.0685 10.7435 21.4380 31.1586 12.0634 25.7465 39.4333 10.7085 21.0405 30.1009 10.7435 21.4380 31.1586 12.5212 23.2158 32.9364 13.0306 25.3627 37.6986 12.4555 22.7709 31.8171 12.5212 23.2158 32.9364 50 present CPT FSDT HSDT 3.6071 3.6999 5.5314 3.6152 3.7080 5.5395 3.6071 3.6998 5.5303 3.6071 3.6999 5.5314 11.0780 23.1225 34.9717 11.1628 23.4949 35.8307 11.0721 23.0464 34.7487 11.0780 23.1225 34.9717 11.1681 23.2125 35.0618 11.2528 23.5849 35.9207 11.1621 23.1360 34.8386 11.1681 23.2125 35.0618 12.9458 24.9903 36.8396 13.0306 25.3627 37.6986 12.9379 24.9105 36.6118 12.9458 24.9903 36.8396 100 present CPT FSDT HSDT 3.6132 3.7060 5.5375 3.6152 3.7080 5.5395 3.6132 3.7060 5.5373 3.6132 3.7060 5.5375 11.1415 23.4007 35.6120 11.1628 23.4949 35.8307 11.1400 23.3810 35.5538 11.1415 23.4007 35.6120 11.2315 23.4907 35.7021 11.2528 23.5849 35.9207 11.2300 23.4711 35.6438 11.2315 23.4907 35.7021 13.0093 25.2685 37.4798 13.0306 25.3627 37.6986 13.0076 25.2484 37.4210 13.0093 25.2685 37.4798 Table 2: Comparison of nondimensional critical buckling load of square plates subjected to uniaxial compression. In order to verify the mechanical buckling solutions determined in this work, the results of composite plates under uniaxial and biaxial loading are obtained and compared with those predicted by CPT, FSDT, and HSDT as indicated in Tables 2 and 3.It is clear that the results show significant differences between the shear deformation theories and the classical plate theory, due to the shear deformation effect. In addition, an excellent agreement is obtained between the current theory and the HSDT for all side-to-thickness ratio values a/h and the modulus ratiosE1/E2. The disagreement between the present theory RPT and HSDT on the one hand and the first order shear theory FSDT on the other hand increases as the side-to- thickness a/h and the modulus ratios E1/E2 increases. It can also be noted that the dimensionless critical load of buckling increases rapidly with the increase of the side-to-thickness ratio a/h, while this dimensionless load ceases to increase when Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 218 this ratio exceeds 25 (the plate becomes thinner more and more), and the results obtained by shear deformation theories (current RPT theory and first-order shear theory FSDT and high order theory HSDT) become identical to that obtained by conventional CPT theory, implying that the effect transverse shear becomes useless. Indeed, the non dimensional critical load of the buckling does not depend on the variation of side-to-thickness according to the classical theory which neglects the effect of the transverse shear. It is clear to note the considerable increase in dimensionless critical load of buckling when the plate rests on an elastic foundation. It should be noted that the unknown function in present theory is 4, while the unknown function in FSDT and HSDT is 5. It can be concluded that the present theory is not only accurate but also simple in predicting the critical buckling load of rectangular composite plates. a/h Theories k1=0, k2=0 k1=10, k2=0 k1=10, k2=10 k1=0, k2=0 k1=10, k2=0 k1=10, k2=10 Isotropic v = 0.3 Orthotrophic E1/E2=10 25 40 Orthotrophic E1/E2=10 25 40 Orthotrophic E1/E2=10 25 40 5 present CPT FSDT HSDT 1.4756 1.5220 2.4378 1.8076 1.8540 2.7698 1.4749 1.5070 2.1138 1.4756 1.5220 2.4378 2.8549 3.3309 3.4800 5.5814 8.4069 10.8715 2.8319 3.1422 3.2822 2.8549 3.3309 3.4800 2.8729a 3.3489a 3.6905a 5.6264 8.4249a 10.8895a 2.8341a 3.1453 a 3.2859a 2.8729a 3.3489a 3.6905a 4.1078 4.2378a 4.5794a 6.5153 9.3138a 11.7783a 2.9976 a 3.3390a 3.4994a 4.1078 4.2378a 4.5794a 10 present CPT FSDT HSDT 1.7112 1.7576 2.6733 1.8076 1.8540 2.7698 1.7111 1.7552 2.6208 1.7112 1.7576 2.6733 4.6718 6.0646a 7.2536 5.5814 8.4069 10.8715 4.6367 5.8370 6.6325 4.6718 6.0646a 7.2536 4.6898a 6.0826a 7.2716a 5.6264 8.4249a 10.8895a 4.6765 5.8491 a 6.6444a 4.6898a 6.0826a 7.2716a 5.5787a 6.9714 a 8.1604a 6.5153 9.3138a 11.7783a 5.2882 a 6.4367 a 7.2181a 5.5787a 6.9714 a 8.1604a 20 present CPT FSDT HSDT 1.7825 1.8289 2.7446 1.8076 1.8540 2.7698 1.7825 1.8286 2.7380 1.7825 1.8289 2.7446 5.3267 7.6643a 9.6614a 5.5814 8.4069 10.8715 5.3100 7.5546 9.3049 5.3267 7.6643a 9.6614a 5.3717 7.6823a 9.6794a 5.6264 8.4249a 10.8895a 5.3542 7.5716a 9.3217a 5.3717 7.6823a 9.6794a 6.2606 8.5711a 10.5682a 6.5153 9.3138a 11.7783a 6.2277 8.4083 a 10.1518a 6.2606 8.5711a 10.5682a 50 present CPT FSDT HSDT 1.8036 1.8500 2.7657 1.8076 1.8540 2.7698 1.8036 1.8499 2.7653 1.8036 1.8500 2.7657 5.5390 8.2784a 10.6576a 5.5814 8.4069 10.8715 5.5361 8.2566 10.5810 5.5390 8.2784a 10.6576a 5.5840 8.2964a 10.6756a 5.6264 8.4249a 10.8895a 5.5810 8.2744a 10.5989a 5.5840 8.2964a 10.6756a 6.4729 9.1853a 11.5645a 6.5153 9.3138a 11.7783a 6.4689 9.1597 a 11.4835a 6.4729 9.1853a 11.5645a 100 present CPT FSDT HSDT 1.8066 1.8530 2.7687 1.8076 1.8540 2.7698 1.8066 1.8530 2.7687 1.8066 1.8530 2.7687 5.5707 8.3744a 10.8172a 5.5814 8.4069 10.8715 5.5700 8.3687 10.7972 5.5707 8.3744a 10.8172a 5.6158 8.3924a 10.8352a 5.6264 8.4249a 10.8895a 5.6150 8.3867 a 10.8151a 5.6158 8.3924a 10.8352a 6.5046 9.2813a 11.7241a 6.5153 9.3138a 11.7783a 6.5038 9.2752a 11.7035a 6.5046 9.2813a 11.7241a a Mode for plate is (m, n) = (1,2). Table 3: Comparison of nondimensional critical buckling load of square plates subjected to biaxial compressive load Figures 4 show the effect of the side-to-thickness ratio (a/h) on the dimensionless critical buckling N when the square plate (a/b = 1) without elastic foundation or resting on Winkler’s or Pasternak’s elastic foundations using the present refined shear deformation theory. It is noted that N increases rapidly with increasing side-to-thickness ratio. However, for the plate without elastic foundation or resting on one parameter Winkler’s foundation, the variation of the dimensionless critical buckling N is almost independent of the side-to-thickness ratio (a/h) when this latter is higher than 25 . It can be also seen that the presence of elastic foundations lead to an increase of the dimensionless critical buckling N . Figures 5 and 6 show the variation of the critical load of the dimensionless buckling N of the rectangular composite plates without elastic foundation or resting on Winkler’s or Pasternak’s elastic foundations as a function of the modulus ratio (E1/E2).The plate is assumed to be subjected to axial loading shown in Fig. 3 (uniaxial compression and biaxial compression). It is found that the critical load of dimensionless buckling increases monotonically as the the modulus ratio (E1/E2) increases. It is also noted that the critical dimensionless load N of the rectangular composite plates under unaxial compression is greater than that of a plate under biaxial compression. Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 219 Figure 4: The effect of side-to-thickness and modulus ratios on the critical buckling load of square plate with or without elastic foundations subjected to uniaxial compression: (a)isotropic,(b)E1/E2 =10, (c)E1/E2 =25 and (d)E1/E2= 40. Figure 5: The effect of modulus ratio on the critical buckling load of square plate with or without elastic foundations subjected to uniaxial compression : (a)a =10h and (b)a =20h. Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22 220 Figure 6: The effect of modulus ratio on the critical buckling load of square plate with or without elastic foundations subjected to biaxial compression: (a)a =10h and (b)a =20h. CONCLUSION refined and simple shear deformation theory is presented for mechanical buckling of rectangular composite plates in contact with two-parameter elastic foundation. Unlike the conventional shear deformation theories, the proposed refined shear deformation theory contains only four unknowns and has strong similarities with the CPT in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. To clarify the effect of shear deformation on the critical buckling, the results obtained by the present theory as well as HSDT, S and FSDT are compared with those obtained by CPT. It is shown through the numerical examples that the results of the shear deformation plate theories are lower than those of the CPT, indicating the shear deformation effect. All comparison studies show that the critical buckling mechanical obtained by the proposed theory with four unknowns are almost identical with those predicted by other shear deformation theories containing five unknowns. It can be concluded that the proposed theory is accurate and efficient in predicting the mechanical buckling responses of rectangular composite plates resting on two parameter (Pasternak’s model) elastic foundations. Due to the interesting features of the present theory, the present findings will be a useful benchmark for evaluating the reliable of other future plate theories. REFERENCES [1] Seyvet, J. (2002). The French composite materials industry (Louis Berreur, Bertrand de Maillard & Stanislas Nösperger., Paris). [2] Gay, D. (2014). Design and Applications (Composite Materials third edition, CRC Press) . [3] Ponte Castañeda, P and Suquet, P. (1997). Nonlinear composites, Adv. Appl. Mech., 34, pp. 171–302. [4] Milton, G. W. (2002). The Theory of Composites Cambridge University Press. [5] Moulinec, H. and Suquet, P. (1998). A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Meth-ods Appl. Mech. Eng. 157, pp. 69–94. 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Mechanics of laminated composite plate: theory and analysis, New York, CRC Press. << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice