THÔNG TIN TÀI LIỆU
International Journal of Mechanical Sciences 100 (2015) 322–327 Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci A semi-analytical study on static behavior of thin skew plates on Winkler and Pasternak foundations Amin Joodaky n, Iman Joodaky Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Islamic Republic of Iran art ic l e i nf o a b s t r a c t Article history: Received 19 April 2012 Received in revised form March 2015 Accepted 28 June 2015 Available online 17 July 2015 This study presents a semi analytical closed-form solution for governing equations of thin skew plates with various combination of clamp, free and simply supports subjected to uniform loading rested on the elastic foundations of Winkler and Pasternak The governing forth-order partial differential equation (PDE) of two-variable function of deflection, w(X,Y), is defined in Oblique coordinates system Application of EKM together with the idea of weighted residual technique, converts the forth-order governing equation to two ODEs in terms of X and Y in Oblique coordinates Both resulted ODEs, are then solved iteratively in a closed-form manner with a very fast convergence Finally deflection function is obtained It is shown that some parameters such as angle of skew plate and stiffness of elastic foundation have an important effect on the results Also it is investigated that shear stresses exist considerably in skew plates comparing to the corresponding rectangular plates Comparisons of the deflection and stresses at the various points of the plates show very good agreement with results of other analytical and numerical analyses & 2015 Elsevier Ltd All rights reserved Keywords: Bending Skew plate PDE Galerkin Extended Kantorovich Pasternak foundation Introduction Kerr [1] developed the idea of the well-known Kantorovich method [2] to obtain highly accurate approximate closed-form solution for torsion of prismatic bars with rectangular cross-section The method employs the novel idea of Kantorovich to reduce the governing partial differential equation of a two-dimensional (2D) elasticity problem to a double set of ordinary differential equations Since then, the Extended Kantorovich Method (EKM) extensively has been applied for various 2D elasticity problems in Cartesian coordinates system Among these applications, one can refer to eigenvalue problems [3], buckling [4] and free vibrations [5] of thin rectangular plates, bending of thick rectangular isotropic [6,7] and laminated composite [8] plates and free-edge strength analysis [9] Most recent EKM based articles include vibration of variable thickness [10] and buckling of symmetrically laminated [11] rectangular plates Accuracy of the results and rapid convergence of the method together with possibility of obtaining closed-form solutions for ODE systems have been discussed in these articles and others [12] Finally, a few research consider polar coordinates e.g using EKM for sector plates [13] All these applications of EKM, are devoted and restricted to the problems in the Cartesian and polar coordinate systems The authors of the n Corresponding author E-mail address: aminjoodaky@gmail.com (A Joodaky) http://dx.doi.org/10.1016/j.ijmecsci.2015.06.025 0020-7403/& 2015 Elsevier Ltd All rights reserved present paper, for the first time applied EKM in Oblique coordinate system for bending of skew plates under clamp boundary conditions without considering foundations and stress analysis [14] Based on the other solution methods, several research have studied bending, buckling, vibration and other analysis for skew plates in term of Oblique coordinate system [15–20] Winkler and Pasternak foundations are considered in the design of structures rested on elastic mediums Winkler model considers the foundation as a series of springs which not have any interaction with each other More advanced models like Pasternak simulate the coupling between these springs too [21,22] This study aims to examine the applicability of the EKM to obtain highly accurate approximate closed-form solutions for 2D elasticity problems in Oblique coordinate system Applying Extended Kantorovich Method (EKM) with the aid of a weighted residual technique (Galerkin method), the governing PDE, is converted to two uncoupled ordinary differential equations (ODE) of f(X) and g(Y) Then an initial guess function is considered for one of those functions to obtain the constants of the ODE of the other function After solving the first ODE, constants of the second ODE are achieved Then the second ODE is solved for obtaining the first ODE's constants These iterations continues unless a good convegence is achieved In every iteration step, exact closed-form solutions are obtained for two ODE systems Deflection and stress analysis of thin isotropic skew plates with a various combinations of clamp, free and simply supports subjected to uniform loading and resting on the Winkler and A Joodaky, I Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327 323 Pasternak foundations as Fig 1, is considered Comparisons of the deflections and stresses at the various points of the skew plate show very good agreement with the results of other valid literatures and FEM analysis of ANSYS code Discussions reveals the existence of shear stresses in skew plates comparing to the corresponding rectangular plates Fig Skew plate in Oblique coordinate (X,Y) resting on the elastic foundations with the stiffness of k Governing equations If no axial force exists, differential equation of motion is expressed as [23] ∂2 M xy ∂2 M y ∂2 M x ỵ ỵ2 ỵ qx; yị k0 wx; yị ỵk1 wx; yị ẳ xy x2 y2 ð1Þ and in terms of w as ∇4 wðx; yÞ ẳ qx; yị ỵ k0 wx; yị k1 wðx; yÞ D multiplication of different single variable functions as wij ðX; YÞ ffif i ðXÞ Ug j ðYÞ where f i ðXÞ and g j ðYÞ are unknown functions to be determined and subscripts i and j denote number of iterations Using Eq (6), expanding of Eq (7) is ð2Þ Eq (2) is the governing equation for a thin plate, in which w(x, y) is the deflection function, q is the applied distributed load, D is flexural rigidity for isotropic plates, k0 and k1 are the stiffness of Winkler and Pasternak foundation respectively Having just Winkler foundation it is enough to equal k1 to zero Now, consider a thin skew plate with dimensions of 2a 2b as Fig For a clamp–support, deflection (w) and its first derivative with respect to the normal direction of the boundary must be vanished For a simply-support, deflection and its second derivative with respect to the normal direction of the boundary must be vanished Considering Fig 1, for example for SSSC boundary conditions (S and C represent simply and clamp respectively) we have 2 ð3Þ Governing Eqs (1) and (2) must be converted from Cartesian coordinates system (x,y) to Oblique coordinates system (X,Y) as it is shown in Fig The relations between Cartesian(x,y) and Oblique(X,Y) are X ¼ x y tan φ and Y ¼ y= cos φ Also, ∇4 becomes ∂4 ẳ ỵ ỵ sin φ 4 cos φ ∂X ∂X ∂Y ỵ þ sin φ ∂X ∂Y ∂X∂Y ∂Y 4 k0 cos φ k1 cos φ d f ðXÞ ðf ðXÞ U g ðYÞÞ g ðYÞ D D dX ! dg ðYÞ df ðXÞ d g ðYÞ cos ỵ f Xị qX; Yị ịẳ sin dY dX D dY ỵ ð10Þ For Eq (7), according to the Galerkin weighted residual method, we have [13] Z 2a Z 2b D w q ỵ k0 w k1 wịwdXdY ẳ 11ị 0 Now, for a prescribed function of g j Yị, jẳ0 and referred to Eq (8), w becomes w ẳ g0 Yị U f i ð6Þ For the governing Eq (2) in Oblique coordinates system we have D wX; Yị ỵ k0 wX; Yị k1 wX; Yị ẳ qX; Yị 9ị By considering assumption of Eq (8), we have φ φ ∂2 ðf ðXÞ:g ðYÞÞ f ðXÞ:g ðYÞ k1 cos cos φ U f Xị:g Yịị ỵ k0 cos D D ∂X ! 2 ∂ f ðXÞ:g Yị f Xị:g Yị ỵ sin φ ∂X∂Y ∂Y ( d2 g Yị d2 f Xị d f Xị ỵ ỵ sin ẳ g Yị dX dY dX ) ! 3 dg ðYÞ d f ðXÞ d g Yị df Xị d g Yị ỵ f Xị þ sin φ dY dX dX dY dY ð4Þ Consequently operator ∇2 in Cartesian coordinates could be converted to Oblique coordinates as ∇ 2 ỵ ¼ sin φ ð5Þ cos φ ∂X XY Y D w ỵ k0 wX; Yị k1 wX; Yị ẳ qX; Yị w D w ỵ ỵ sin φ cos φ ∂X ∂X ∂Y ∂ w ∂4 w w ỵ k0 w ỵ sin φ ∂X ∂Y ∂X∂Y ∂Y k1 w w w ỵ sin ẳ qX; Yị XY Y cos φ ∂X w ¼ d w=dx2 ¼ for x ¼ 0; x ¼ 2a w ¼ d w=dy2 ¼ for y ¼ and w ¼ dw=dy ẳ for y ẳ 2b 8ị 12ị Substitution of Eq (8) into Eq (11) in conjunction with Eq (12) leads to # Z 2a "Z 2b D∇ ðf i U g Þ q ỵ k0 f i Ug ị k1 ∇ ðf i Ug Þ g dY δf i dX ẳ 0 7ị 13ị Based on the existing rules in the Variational principle, Eq (13) is satisfied if the expression in the bracket is vanished Iterative solution by EKM According to the Extended Kantorovich Method (EKM) [1], the two-variable-function of the plate deflection, w(X,Y) is assumed as Z 2b D∇ f i Ug ị q ỵ k0 f i Ug Þ k1 ∇ ðf i Ug ị g dY ẳ 14ị 324 A Joodaky, I Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327 Now referred to Eq (10), Eq (14) becomes k1 cos φ ∂2 f i ðXÞ:g ðYÞ k0 cos φ f i ðXÞ U g ðYÞ cos φ U ∇ f i Xị U g Yị ỵ D D ∂X !# ∂2 f i ðXÞ:g ðYÞ f Xị:g Yịị ỵ i Y sin φ g dY ¼ ∂X∂Y ! ! ! Z 2b Z 2b d2 g ðYÞ k cos φ 2 f i ðXÞ d f i ðXÞ g g 20 ðYÞdY d dX ỵ2 sin Yị g Yị dY ỵ ỵ 0 dX D dY 0 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} R 2b 4 A4 A2 ! dg ðYÞ f i ðXÞ sin φ g ðYÞ dY d dX ỵ dY |{z} Z 2b ð15Þ A3 ! ! Z 2b k1 cos φ dgðYÞ d g ðYÞ df i Xị g sin YịdY ỵ g Yị dY f i Xị ỵ 0 dX dY D dY dY 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z 2b sin dg Yị ỵ A1 Z 2b Z 2b ! A0 ! k0 cos φ k1 cos φ d g ðYÞ cos φ g ðYÞdY g ðYÞ dY f Xị ẳ i D D D dY 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z 2b A5 A6 Using an arbitrary prescribed function for g ðYÞ, as initial guess, the constants of Ai (i¼ 0–6), could be calculated and Eq (15) turns into a forth order ODE as A4 d f i ðXÞ dX þ A3 d f i ðXÞ dX þ A2 d f i Xị dX ỵ A1 dX ỵ A3 d f i Xị A2 d f i Xị A1 df i Xị A0 ỵ A5 ị A6 ỵ ỵ ỵ f i Xị ẳ A4 A4 dX A4 dX A4 dX A4 ð17Þ The corresponding characteristic equation related to Eq (17), is m4 ỵ A3 A2 A1 A0 ỵ A5 ị m ỵ m ỵ mỵ ẳ0 A4 A4 A4 A4 ð18Þ Eq (18) may has four complex roots as: mr ¼ a1 7b1 i that, r ¼1–4 and first description of fi (X) as f1(X) is expressed as f Xị ẳ ea1 X C cos b1 Xị ỵ C sin b1 Xịị ỵ e a1 X C cos b1 Xị ỵC sin b1 Xịị ỵ C 19aị or Eq (18) has four real roots as: m1;2 ¼ a1 ; m3;4 ¼ b1 , then f1(X) is considered as f Xị ẳ C ea1 X þ C e a1 X þC eb1 X ỵ C e b1 X ỵ C 19bị where C ẳ A6 =A0 ỵ A5 Þ It should be noted that after substituting the double-term function of w(X,Y) with the multiplied single-term functions of f(X) and g(Y), the new forms of boundary conditions must be considered in terms of single-term functions For example new forms of boundary conditions of SSSC in Eq (3), are f Xị ẳ d f Xị=dX ẳ for X ẳ and X ẳ 2a gYị ẳ d gYị=dY ẳ for Y ẳ 0; and gYị ẳ dgYị=dY ẳ for Y ẳ 2b 20ị Solving Eqs (19) in conjunction with the new boundary data leads to the first estimate of the function f ðXÞ Similarly, it is possible to continue the procedure by introducing the obtained function f ðXÞ to Eq (8) The new form of δw is δw ¼ f ðXÞ U δgj Similarly Galerkin equation is obtained as "Z # 2a D∇ ðf U g j ị ỵ k0 f U g j Þ k1 ∇ ðf U g j Þ q f dX δg j dY ¼ 2b ð22Þ Dividing both sides by A4, yields d f i Xị Z df i Xị ỵ A0 ỵ A5 ịf Xị ẳ A6 dX 16ị q Ug ðYÞ dY ð21Þ Again, in order to satisfy Eq (22), the bracket must be vanished Using the already obtained f ðXÞ for the expression in the bracket and integration with respect to X leads to the second forth-order ODE in terms of gðYÞ for obtaining g1(Y) in B4 d g ðYÞ dY þ B3 d g1 ðYÞ dY þ B2 d g Yị dY ỵ B1 dg Yị ỵ B0 ỵ B5 ịg Yị ẳ B6 dY ð23Þ The corresponding characteristic equation related to Eq (23) is n4 ỵ B3 B2 B1 B0 ỵ B5 ị n ỵ n ỵ nỵ ẳ0 B4 B4 B4 B4 ð24Þ Again Eq (24) either has four complex roots as nk ¼ a2 b2 i and kẳ 14, so g1(Y) is shown as g Yị ẳ ea2 Y D1 cos b2 Yị ỵ D2 sin b2 Yịị ỵ ea2 Y D3 cos b2 Yị ỵ D4 sin b2 Yịị ỵ D5 25aị or Eq (24) has four real roots as: n1;2 ¼ a2 ; n3;4 ¼ b2 , so g(Y) could be expressed as g Yị ẳ D1 ea2 X ỵ D2 e a2 X ỵ D3 eb2 X ỵ D4 e b2 X ỵ D5 25bị where D5 ẳ B6 =B0 ỵ B5 ị The process is continued by solving Eqs (25) together with the new boundary data and obtaining the new prediction for g ðYÞ This finishes the first iteration for determination of deflection function of w(X,Y) Eqs (19) and (25) are then solved iteratively and new updated estimates for functions f i ðXÞ and g j ðYÞ are determined Iteration procedure continues in the Table Sample mechanical properties for the skew plate on Winkler and Pasternak foundations Properties 2a 1m 2b 1.5 m h 0.01 m ν 0.3 φ 151 k0 1e6 Pa/m k1 1e5 N/m q 1e4 Pa E 70 GPa Table Convergence of the deflection, in the center of the skew plate (a,b) on the Winkler– Pasternak foundation for different combination of boundary conditions Boundary conditions Central deflections (m) during iterations (i¼ to 4) CCCC SSSS SCFC SCSC CFCF SCCC 0.202e–2 0.334e–2 0.466e–2 0.334e–2 0.202e–2 0.261e–2 0.1882e–2 0.3719e–2 0.4647e–2 0.3181e–2 0.2131e–2 0.2377e–2 0.1881e–2 0.3734e–2 0.4637e–2 0.3179e–2 0.2172e–2 0.2375e–2 0.1881e–2 0.3734e–2 0.4637e–2 0.3179e–2 0.2172e–2 0.2375e–2 Deflection of the skew Plate, z Axis (meter) A Joodaky, I Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327 3.50E-03 k0=0, k1=0 3.00E-03 k0=1e6 , k1=1e5 2.50E-03 k0=1e5 , k1=1e6 2.00E-03 k0=1e7 , k1=1e7 1.50E-03 1.00E-03 5.00E-04 0.00E+00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x Axis (meter) along y=0 Fig Deflection of the CCCF skew plate, for various foundation stiffness of k0 and k1 along the Y ¼b and X axes 4.00E-03 of the skew Plate (Pa) 2.00E+07 3.50E-03 3.00E-03 2.50E-03 2.00E-03 1.50E-03 CCCC 1.00E-03 SSSS CCSS CCSF 5.00E-04 0.00E+00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 σ Deflection of the skew plate, Z Axis (meter) 325 1.00E+07 0.00E+00 -1.00E+07 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -2.00E+07 phi=15 phi=30 -5.00E+07 0.8 0.9 1.0 phi=0 -3.00E+07 -4.00E+07 0.7 0.9 x Axis (meter) along y=b 1.0 Fig Stress of σ X along the Y ¼ b and X axes for the CCSS plate in Table with different skew angles X Axis (meter) along Y=b 2.5E-03 phi=0 2.0E-03 phi=15 1.5E-03 phi=30 phi=45 1.0E-03 σ of the skew Plate (Pa) Deflection of the skew plate, Z Axis (meter) Fig Deflection of the skew plate, for various boundary conditions 1.20E+07 1.00E+07 8.00E+06 6.00E+06 4.00E+06 2.00E+06 0.00E+00 -2.00E+06 0.0 -4.00E+06 -6.00E+06 phi=0 phi=15 phi=30 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 x Axis (meter) along y=b 5.0E-04 0.0E+00 0.4 Fig Stress of σ XY along the Y ¼ b and X axes for the CCSS plate in Table with different skew angles X Axis (meter) along Y=b same way until a reasonable level of convergence achieves After obtaining the deflection function, one can determine all other mechanical parameters in terms of deflection (w), i.e stresses and moments using well-known expressions presented elsewhere, see for instance, [23] One can consider φ equals to zero and develop all above equations and results to rectangular plates σ of the skew Plate (Pa) Fig Deflection of the CSCS skew plate, for various skew angles of φ 1.50E+07 1.00E+07 5.00E+06 0.00E+00 -5.00E+06 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1.00E+07 0.9 1.0 phi=15 -1.50E+07 -2.00E+07 0.8 phi=0 phi=30 x Axis (meter) along y=b Fig Stress of σ Y along the Y ¼b and X axes for the CCSS plate in Table with different skew angles Results and discussion Consider a skew plate resting on the Winkler–Pasternak foundation as Fig The plate is subjected to a uniform loading and different combinations of clamp, free and simply supports The initial guess of g Yị ẳ Y b ị2 that does not satisfy all boundary conditions necessarily, is considered As it was mentioned, this g0(Y) is applied to obtain f1(X)'s constants in Eqs (19) and w0(X,Y)¼f1(X)g0(Y) is the result for the iteration #0 Then the obtained f1(X) is used to calculate g1(Y) and w1(X,Y) ¼ f1(X)g1(Y) completes the iteration #1 This procedure continues up to less than four iterations when a high convergence is achieved In Table 1, the sample's mechanical properties and geometry are presented although in some cases a parameter may be considered as a variable for studying its effects on the results Considering Fig and downward pressure loading of q, it is clear that plate also deflects downward although, for brevity, the signs of minus ( ) for deflection results are neglected and all diagrams are shown inversely along z direction When the skew angle is equaled to zero, the results are developed to rectangular plates Table shows deflection convergence in the center of the skew plate for different boundary conditions It reveals that the convergence of the method is very fast as there are no major changes after the second iteration It also shows that different boundary conditions lead to obtain different results for the center deflection of the skew plate It is clear that more clamp supports decreases deflection 326 A Joodaky, I Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327 Table Comparison of the maximum deflection and stresses in the isotropic 151 skew plate (a,b) on Winkler foundation in Table but E¼ 380 GPa, q¼ 1e6 Pa, k0 ¼ 1e8 Pa/m with the results of ANSYS code Deflections (m) and stresses (Pa) w σX σY σ XY Boundary conditions Present ANSYS Present ANSYS Present ANSYS Present ANSYS CCCC CSCS 0.548e–4 0.555e–4 0.415e10 0.404e10 0.151e10 0.146e10 0.762e9 0.718e09 0.609e–4 0.623e–4 0.450e10 0.438e10 0.164e10 0.158e10 0.827e9 0.779e09 Table Comparison of deflection, w, in the center of the SSSS isotropic rectangular plate on Winkler foundation, which is obtained from a plate by considering φ¼ 0, q ¼1000 and k0 ¼ 1e6 with a similar plate from [23] for different plate dimensions Source Present [23] of a simply supported isotropic rectangular plate on Winkler foundation with the properties in Table but φ ¼0, q¼ 1000 and k0 ¼1e6 are compared with the bending results of a similar SSSS plate in the reference [23] Deflections (m) in (a, b) 2a¼ 1, 2b¼ 2a¼ 1, 2b¼ 1.5 2a ¼1, 2b¼ 0.44780e–3 0.44842e–3 0.66567e–3 0.66679e–3 0.74845e–3 0.74969e–3 Fig shows the deflection diagrams of the skew plate with various boundary conditions, which are not symmetric in some cases Note that maximum deflection occurs in the center, if the plate is under symmetric boundary conditions e.g SSSS By increasing the angle of φ edges of the skew plate become closer to each other, so deflections decrease Fig shows the effect of φ on the deflection results of CSCS skew plate Fig for CCCF boundary conditions shows that foundation stiffness of ki changes the deflection diagram considerably In addition, k1 changes the results much more than k0 In other word, Pasternak's foundation limits deflection more than Winkler does with the same stiffness For this sample, these changes are not considerable when the stiffness factor of ki is lesser than 1e4 Also, if the foundation is rigid enough, for example for this case larger than 1e7, the plate does not deflect at all The functions of stresses and moments are obtained in terms of lateral deflection of w [23] Therefore, one can achieve stress components for every point of the skew plate Figs 5–7 show amounts of the stress components of σ X ; σ Y ; σ XY of the CCSS skew plate along the Y ¼b and X axes respectively for different skew angle of φ The effect of the skew angle is more tangible in σ XY Since the first and second boundaries are clamp in this case, the beginning points of the skew plate have primary stresses For the ending edge of the plate, stresses are zero since boundaries are simply and not have resistance to bending Increasing the angle of the skew plate causes a considerable change in the stress components particularly for σ XY For the rectangular plates (φ ¼0), this stress is zero in the edges and shows limited variations along the plate The shear stresses that are a consequence of the skew angle, need to be considered during the applied designs calculations of skew structures In Table 3, bending results of an isotropic skew plate on Winkler foundation with the plate's properties in Table but for q ¼1e6 Pa, k0 ¼1e8 Pa/m, E ¼380 GPa, for two boundaries of CCCC and CSCS are compared with the similar modeled plate in ANSYS code It shows that there is a good agreement between the results Considering φ ¼0 the skew plate is converted to a rectangular plate In Table 4, for different plate dimensions, the bending results Concluding remarks Application of EKM based on Galerkin method successfully obtains a highly accurate approximate closed-form solution for deflection and stress analysis of the skew plates resting on Winkler–Pasternak foundation and subjected to uniform loadings with various boundary conditions EKM extracts two sets of decoupled ordinary differential equations in terms of X and Y in Oblique coordinates from the forth-order partial differential governing equation of the main problem The solution procedure then completes by presenting an exact approximate closed-form solution for two sets of ODE systems in an iterative scheme The method provides very fast convergence and highly accurate predictions Angle of skew plates has an important role on the deflection function and consequently on stresses diagrams, that are completely different from corresponding rectangular plate In other words, even a small angle of φ that makes a deviation from rectangular shape, causes a change in the stress distributions that need to be considered in the designing calculations of these structures Pasternak foundation imposes a larger deflection limitation on the plate comparing to Winkler foundation Comparing to the results of the other valid literatures and FEM analysis of ANSYS code, there are very good agreements with the results of the present study in every case Finally, by equaling inclination angle of the skew plate to zero, the present study could be developed and compared to the studies for rectangular plates References [1] Kerr AD 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