Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theory

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1 J. Appl. Comput. ech., 4(3) (018) DOI: /JAC ISSN: jacm.scu.ac.ir Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor Atteshamuddin S. Saad1, Yuwaraj. Ghugal 1 Department of Civil Engineering, SRES s Sanjivani College of Engineering, Savitribai Phule Pune Universit, Kopargaon-43601, aharashtra, India attu_saad@ahoo.co.in, Department of Applied echanics, Government College of Engineering, Karad-41514, Shivaji Universit, Kolhapur, aharashtra State, India ghugal@rediffmail.com Received August 0 017; Revised October ; Accepted for publication October Corresponding author: Atteshamuddin S. Saad, attu_saad@ahoo.co.in Copright 018 Shahid Chamran Universit of Ahvaz. All rights reserved. Abstract. A trigonometric plate theor is assessed for the static bending analsis of plates resting on Winkler elastic foundation. The theor considers the effects of transverse shear and normal strains. The theor accounts for realistic variation of the transverse shear stress through the thickness and satisfies the traction free conditions at the top and bottom surfaces of the plate without using shear correction factors. The governing equations of equilibrium and the associated boundar conditions of the theor are obtained using the principle of virtual work. A closed-form solution is obtained using double trigonometric series. The numerical results are obtained for fleure of simpl supported plates subjected to various static loadings. The displacements and stresses are obtained for three different values of foundation modulus. The numerical results are also generated using higher order shear deformation theor of Redd, first order shear deformation theor of indlin, and classical plate theor for the comparison of the present results. Kewords: Shear deformation; Normal strain; Shear stress; Shear correction factor; Winkler elastic foundation; Foundation modulus. 1. Introduction The bending of plates resting on elastic foundation has man practical applications in civil, mechanical, aerospace, marine, offshore, and automotive structures. Therefore, various analtical and numerical methods have been emploed to stud bending analsis of plates resting on elastic foundation. Winkler foundation is the most commonl used elastic foundation. The well-known classical plate theor (CPT) is based on the Kirchhoff [1] hpothesis that straight lines perpendicular to the midplane before deformation remain straight and perpendicular to the midplane after deformation. The theor assumes linear variation of inplane displacements through the thickness and neglects the effects of transverse shear deformation and normal strain. In the indlin [] first-order shear deformable theor (FSDT), it is assumed that straight lines perpendicular to the midsurface before deformation remain straight but not perpendicular to the mid-surface after deformation. The theor removes some of the defects of the classical plate theor. Although this theor considers the effect of transverse shear deformation, it neglects the effect of transverse normal strain. Although the effect of transverse shear deformation is considered in this theor, the effect of transverse normal strain is neglected. This theor predicts the constant shear strain through the plate thickness and requires a problem dependent shear correction coefficient. These limitations of CPT and FSDT have been rectified in recent ears b introducing higher-order functions in the displacement field which has led to the development of higher-order plate theories. The third order theor of Redd [3] is a well-known higher order theor but neglects the effect of transverse normal Published Online: June 1 018

2 188 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 strain. atsunaga [4] obtained natural frequencies and buckling stresses of a thick isotropic plate resting on two-parameter elastic foundations considering the effect of shear deformation, thickness change, and rotator inertia. A higher order theor based on power series epansion of the displacement components is used. The bending analsis of a plate resting on Winkler elastic foundation is studied b Huang and Thambiratnam [5]. Chen and Bian [6] applied the methods of state space and the differential quadrature for bending and free vibration of arbitraril thick beams resting on a Pasternak elastic foundation. A new shear deformation theor for the free vibration analsis of simpl supported functionall graded plates resting on a Winkler-Pasternak elastic foundation has been presented b Atmane et al. [7]. The closed-form solutions were obtained b using Navier technique. Thai et al. [8] proposed a refined shear deformation theor for bending, buckling, and vibration of thick plates resting on elastic foundation. The elastic foundation is modeled as two-parameter Pasternak foundation. The Levtpe solution is presented for rectangular plates with two opposite simpl-supported edges and the other two edges having arbitrar boundar conditions. Zenkour [9] carried out bending analsis of orthotropic plates resting on Pasternak's foundations b using the mied first-order shear deformation theor. Zenkour et al. [10] also presented mied first-order theories for plates resting on elastic foundations. Saad [11] has developed the eponential shear deformation theor for the bending of orthotropic plates. Akbas [1] presented free vibration and static bending of a simpl-supported functionall graded plate with the porosit effect using the first-order shear deformation theor. Akbas [13, 14] also presented static and stabilit analsis of a nanoplate using the generalized differential quadrature method based on the classical plate model. Civalek [15] and Gurses et al. [16, 17] developed the discrete singular convolution method, for the static analsis of thick smmetric cross-pl laminated composite plates based on the first-order shear deformation theor. Recentl, Saad and Ghugal [18, 19] presented a detailed review of displacement-based shear deformation theories for the analsis of beams and plates otivation and contribution 1. It is highl recommended that an refinements of classical models are meaningless in general, unless the effects of transverse shear and normal deformations are both taken into account. But, in a whole lot of literature on this subject man researchers neglected this effect in view of minimizing the number of unknown variables. In the present stud, authors have developed a trigonometric shear deformation theor for the bending of shear fleible plates considering both transverse shear and normal deformations.. The kinematics of the present theor is much richer than those of the other higher order shear deformation theorie s, because if the trigonometric term is epanded in power series, the kinematics of higher order theories are implicit l taken into account to a great etent. oreover, it needs to be noted that ever additional power of thickness coordinate in the displacement field of higher-order theories introduces additional unknown variables in those theories which are difficult to interpret phsicall. Thus, the use of the sinusoidal function in terms of thickness coordinate enhances the richness of the theor and also results in reduction of the number of unknown variables as compared with other competitive displacementbased higher-order theories without loss of the phsics of the problem in modeling. 3. In this stud, a bending analsis of thick plates resting on Winkler elastic foundation has been carried out. A uniforml distributed laer of Winkler s springs usuall represents the elastic foundation. A trigonometric shear and normal deformation theor developed b Ghugal and Saad [0-3] is applied for the analtical solutions. The theor has four unknown variables and includes the effect of transverse shear and transverse normal strain. The theor developed is based on certain kinematical and phsical assumptions. The governing equations and boundar conditions are obtained using the principle of virtual work. A closed-form solution is obtained using Navier s solution technique. The results obtained for various loading cases are compared with those of the higher order theor of Redd [3], the first-order shear deformation theor of indlin [], and the classical plate theor [1]. 4. The effect of local stress concentration due to concentrated load has assessed effectivel b the present theor which has not addressed b the other researchers..1. Plate under consideration Consider a rectangular plate of length a, width b, and total thickness h made up of linearl elastic isotropic material. The plate occupies a region (0 a, 0 b, -h/ z h/) in Cartesian coordinate sstem. The plate is resting on Winkler elastic foundation. The geometr and coordinate sstem of the plate is shown in Fig. 1. Fig. 1. The geometr and coordinate sstem of plate Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

3 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor The displacement field, strains and stresses Based upon the aforementioned assumptions, the displacement field of the present theor is given as below: w, h z u,,z z sin, h w, h z v,,z z sin, h h z w,,z w,, h cos where u, v, and w are the displacements in,, and z directions, respectivel. The trigonometric functions in in-plane displacements are associated with the cosine distribution of transverse shear stress through the thickness of the plate. The, and are the unknown functions to be determined. The strains associated with the displacement field given b Eq. (1) are obtained within the framework of linear theor of elasticit. u w h z z sin h v w h z z sin h (1) w z z sin z h z u v w h z z sin h u w z h cos z h () v w z h cos z h The three-dimensional stress-strain relationships for the isotropic plate given b Timoshenko and Goodier [15] are used as follows: G, where z G, z G, z z G, G, G z z and G are the Lame s constants as given below: E and The moment and shear force resultants are obtained b integrating stresses through the thickness. The moment resultants, and analogues to classical plate theor are as follows: h/ G h/ w w z dz L1 L L 3 L4 L5 E (3) (4) h/ w w z dz L L 1 L 4 L3 L5 h/ (5) h/ w z dz L6 L7 h/ The refined moment resultants s s s and s,, z corresponding to transverse shear deformation are defined as follows: Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

4 190 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 h/ s h z w w sin dz L3 L 4 L 8 L9 L10 h h/ h/ s h z w w sin dz L4 L 3 L 9 L8 L10 h h/ h/ s h z w w z z sin dz L5 L 10 L11 h/ h h/ s h sin z w dz L7 L 1 h h/ The shear force resultants Q and Q corresponding to transverse shear deformation are as follows: h/ z h Q z cos dz L13 h h/ h/ z h Q cos dz L13 h/ h and V and V The constants L 1 to L 1 appeared in stress resultants are as follows: h h h h h h L1 G, L, L3 G, L 3 4, L 3 5, L 6 G, h h h h h h h 7, 3 8, 9, 10, 11, 1, 13 L G L G L L L G L G L G (6) (7) (8) (9).3. Governing differential equations and boundar conditions for present theor The governing differential equations and boundar conditions associated with the present theor are obtained using the principle of virtual work given below where is the variational operator, and k f is Winkler modulus. h/ b a b a σ δ +σ δ +σz δ z δ + τz δ z + τ δ d d dz q k f wd d = 0 (10) -h/ B substituting the strains from Eq. () and the stresses from Eq. (3) into Eq. (10), the principle of virtual work can be rewritten as: b a b a w w w s s d d Q d d b a b a b a s s h h s Q d d Q Q z d d q k f wd d B integrating Eq. (11) b parts and collecting coefficients of w,, and, one can obtain the following variationall consistent governing equations and associated boundar conditions. w : q k 0 f s s : Q 0 s s : Q 0 Q Q s : z 0 h The boundar conditions obtained at = 0 and = a are of the following form: Journal of Applied and Computational echanics, Vol. 4, No. 3, (018), (11) (1)

5 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor 191 w Either 0 or is prescribed Either V 0 or w is prescribed s Either 0 or is prescribed s Either 0 or is prescribed Either Q 0 or is prescribed The boundar conditions obtained at = 0 and = b are of the following form: w Either 0 or is prescribed Either V 0 or w is prescribed s Either 0 or is prescribed s Either 0 or is prescribed Either Q 0 or is prescribed The governing equations of the plate in terms of displacement variables can be written as follows: w w w D D D q k f (13) (14) (15) w w D D D D D D w w D D D D D D w w D D D D where D 1 through D 9 are the stiffnesses as given below: h h h h D1 G, D G, D 3 3, D 4 G, h h h h h D G, D G, D G, D G, D G (16) (17) (18) (19) 3. Analtical solution using double trigonometric series (The Navier solution) To prove the efficienc and validit of the presented theor, the bending analsis of a simpl supported plate resting on Winkler elastic foundation is considered. The plate is subjected to a transverse load, q, on the top surface of the plate (i.e. z h / ). Si different tpes of static loading conditions are considered for the detailed numerical stud. Navier s solution technique is emploed to determine numerical solution for the simpl supported plate. The following are the boundar conditions of the simpl supported plate. At edges = 0, = a: s (0) w 0 At edges = 0, = b: s w 0 (1) According to Navier solution technique, the transverse load can be epressed in double trigonometric series as: m n q, q sin sin m 1,3,5 n 1,3,5 a b () where q is the coefficient of Fourier epansion of load. The values of this coefficient for various loading cases are obtained using Eq. (3) and are given in Table 1. Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

6 19 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 b a 4 m n q q d d ab a b, sin sin (3) 0 0 Table 1. Coefficients (q) in the double trigonometric series epansion of loads in the Navier method Load Sinusoidall distributed load q, q0 q q q 0 m 1, n 1 Uniforml distributed load q, q0 q 16 q 0 m 5, n 5 Linearl varing or hdrostatic load q a 0 q, q 8q0 cos m m 5, n 5 Parabolic load q, q0 a 4 q0 q I, m 5, n 5 sin m cos m m m I 1 cos n cos m m Central patch load q, q0 q 16 q m n sin sin m 5, n 5, 0.0 a, 0.0b 0 0 Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

7 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor 193 Central concentrated load q, P q 4 P m n sin sin ab m 5, n 5 The following is the solution form assumed for unknown variables in the displacement field which eactl satisfies simpl supported boundar conditions: m n w, w sin sin a b m 1,3,5 n 1,3,5 m n, cos sin a b m 1,3,5 n 1,3,5 m n, sin cos a b m 1,3,5 n 1,3,5 m n, sin sin a b m 1,3,5 n 1,3,5 where w,, and are the unknown coefficients of the respective Fourier epansions and m, n are positive integers. Substitution of the load q, from Eq. () and solution form from Eq. (4) into the Eqs. (15) - (18) ields the four algebraic simultaneous equations in the following matri form. where elements of stiffness matri [K] are as follows: K11 K1 K13 K14 w q K K K K K31 K3 K33 K 34 0 K41 K4 K43 K m m n n m m n n n m K11 D1 k 4 4 f, K1 D, K 3 13 D, 3 a a b b a ab b b a m n m n m n m K14 D 3, K D4 D 5 D 6, K3 D 7, K4 D 8, a b a b ab a n m K33 D4 D 5 D 6, b a n m n K34 D 8, K44 D5 D 9. b a b Solving Eq. (5) gives the values of w,, and. Having these values, one can then calculate all the displacement and stress components within the plate using Eqs. (1) - (3). Transverse shear stresses can be obtained b either constitutive relations or integrating equilibrium equations of the theor of elasticit. These stresses are indicated b and when obtained b constitutive relations and are indicated b equilibrium equations of the theor of elasticit are given below. z (4) (5) (6) and when obtained b using equilibrium equations. The z Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

8 194 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 z z z 0, 0, 0 z z z (7) 4. Numerical results and discussion The numerical results are obtained for isotropic square (b = a) plates subjected to different static loadings. The following material properties are used for the plate. E E 10 GPa, 0.3 and G = 1 where E is Young s modulus, is Poisson s ratio, and G is Shear modulus. The transverse deflection (w), in-plane stresses (,, ), and transverse shear stresses ( z, ) are presented in the following normalized form for the purpose of presenting the results. K 4 4 k, f a a b 100 E w h, w,,0,, 0,0,, D q hs q S a b h a z b,,, z,0,0, 0,,0 q S q0 S q0 S 0 Here, S is the aspect ratio (a/h) of the plate. In case of central point load q 0 becomes P. Since an eact threedimensional elasticit solution for bending of the isotropic plate resting on Winkler elastic foundation is not available in the literature, the present results are compared and discussed with the corresponding results of the classical plate theor (CPT), first-order shear deformation theor (FSDT), and higher order shear deformation theor of Redd (HSDT). The numerical results are obtained for different values of foundation modulus (K = 1, 3, 5) and aspect ratios (S = 4, 10, 100). (8) (9) 4.1. A simpl supported isotropic plate subjected to sinusoidall distributed load The normalized transverse displacement and stresses for the isotropic square plate subjected to sinusoidall distributed load are shown in Table. The eamination of this table reveals that the results of displacement and stresses obtained b the present theor are in good agreement with those of Redd s theor. CPT underestimates the value of transverse displacement for aspect ratios 4 and 10. The in-plane stresses predicted b FSDT and CPT are closer to each other. 4.. A simpl supported isotropic plate subjected to uniforml distributed load Table 3 shows the comparison of transverse displacement and stresses for the isotropic square plate subjected to uniforml distributed load. The through-thickness distributions of in-plane normal stress and transverse shear stress for different foundation modulus are plotted in Figs. and 3, respectivel. Table 3 shows that the transverse displacement and stresses obtained b using the present theor are in ecellent agreement with those of Redd s theor. It is also observed that the transverse displacements and in-plane stresses are decreased with an increase in aspect ratio as well as foundation modulus. The numerical results obtained b all the theories are more or less the same for the thin plate (S = 100) z / h S = K = 1 K = 3 K = Fig.. Through thickness distribution of in-plane normal stress ( ) for isotropic square plate subjected to uniforml distributed load Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

9 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor S = 4 K = 1 K = K = 5 z / h z Fig. 3. Through thickness distribution of transverse shear stress ( 4.3. A simpl supported isotropic plate subjected to linearl varing (Hdrostatic) load z ) for isotropic square plate subjected to uniforml distributed load The comparison of transverse displacement and stresses for the isotropic square plate subjected to a linearl (Hdrostatic) load is reported in Table 4. The intensit of load (q 0 ) is zero at edge = 0 and maimum at = a. From Table 4 it is figured out that the present theor predicts the transverse displacement and stresses ver accuratel compared with those of Redd s theor. FSDT and CPT show more or less the same values for stresses A simpl supported isotropic plate subjected to parabolic load In this eample, bending analsis of the isotropic square plate is carried out for a parabolic load. The intensit of load (q 0 ) is zero at edge = 0 and maimum at edge = a. The numerical results are reported in Table 5 and found to agree well with those of Redd s theor. FSDT and CPT underestimate the values of transverse displacement and in-plane stresses, whereas the overestimate the values of transverse shear stresses A simpl supported isotropic plate subjected to central patch load In this eample, bending response of the isotropic square plate is eamined under the patch load. The load is centrall applied with size 0 0. The value of 0 =0.0a and 0 =0.0b. The transverse displacement and stresses for various aspect ratios and foundation moduli are reported in Table 6. The eamination of Table 6 reveals that as compared with other loading cases discussed in the preceding sections, the stresses developed due to central patch load are ver small. It is also seen that the present results are in ecellent agreement with those of Redd s theor z / h S = K = 1 K = 3 K = Fig. 4. Through thickness distribution of in-plane normal stress ( ) for isotropic square plate subjected to central concentrated load 4.6. A simpl supported isotropic plate subjected to central concentrated load The displacements and stresses for the isotropic square plate when subjected to a central concentrated load are shown in Table 7. The through-thickness distributions of in-plane normal stress are shown in Fig. 4 for S = 4 and K = 1, 3, 5. The variation of this stress is non-linear through the thickness, which ma be the effect of local stress concentration due to concentrated load, Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

10 196 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 that cannot be captured b the classical theories. The through-thickness distributions of transverse shear stress ( z ) when obtained b using equation of equilibrium is shown in Fig. 5. These distributions are plotted for the aspect ratio 4 and foundation modulus 1, 3, and 5. The distributions show that the maimum transverse shear stress occurs at z = ± 0.35 h instead of at neutral plane. However, these distributions show the negative values at neutral plane. This anomalous behavior of transverse shear stress ma be attributed to the effect of local stress concentration due to concentrated load z / h S = 4 K = 1 K = 3 K = z Fig. 5. Through thickness distribution of transverse shear stress ( Journal of Applied and Computational echanics, Vol. 4, No. 3, (018), z ) for isotropic square plate subjected to central concentrated load Table. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to sinusoidall distributed load S K Theor w z z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3]

11 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor 197 FSDT [] CPT [1] Table 3. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to uniforml distributed load S K Theor w z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Table 4. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to linearl varing (Hdrostatic) load S K Theor w z z z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

12 198 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Table 5. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to parabolic load S K Theor w Journal of Applied and Computational echanics, Vol. 4, No. 3, (018), z z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3]

13 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor 199 FSDT [] CPT [1] Table 6. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to central patch load S K Theor w z z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Table 7. Comparison of non-dimensional deflection and stresses in isotropic square plate subjected to central concentrated load S K Theor w z 4 1 Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] z Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

14 00 A.S. Saad and Y.. Ghugal, Vol. 4, No. 3, 018 FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Present HSDT [3] FSDT [] CPT [1] Conclusions A trigonometric plate theor is assessed for bending analsis of square isotropic plates resting on Winkler elastic foundation subjected to various static loadings. The theor is variationall consistent and obviates the need of shear correction factor. The displacements and stresses are obtained for different foundation moduli. The present results are compared with those generated using other theories. From the numerical results and discussion, the following conclusions are drawn. 1. The results of displacements and stresses predicted b the present theor are in ecellent agreement with those of Redd s theor.. The transverse displacements and stresses are found to decrease with an increase in foundation modulus. The present theor can predict the effect of local stress concentration on the in-plane and transverse shear stress distributions due to concentrated load more prominentl compared with other higher order theories. References [1] Kirchhoff, G.R., Uber das gleichgewicht und die bewegung einer elastischen Scheibe, Journal for Pure and Applied athematics, 40, 1850, [] indlin, R.D., Influence of rotator inertia and shear on fleural motions of isotropic, elastic plates, Journal of Applied echanics, 18, 1951, [3] Redd, J.N., A simple higher order theor for laminated composite plates, Journal of Applied echanics, 51, 1984, [4] atsunaga, H., Vibration and stabilit of thick plates on elastic foundations, Journal of Engineering echanics, 16(1), 000, [5] Huang,.H., Thambiratnam, D.P., Analsis of plate resting on elastic supports and elastic foundation b finite strip method, Computers and Structures, 79(9-30), 001, [6] Chen, W.Q., Bian, Z.G., A mied method for bending and free vibration of beams resting on a Pasternak elastic foundation, Applied athematical odelling, 8(10), 004, [7] Atmane, H.A., Tounsi, A., echab, I., Bedia, E.A.A., Free vibration analsis of functionall graded plates resting on Winkler Pasternak elastic foundations using a new shear deformation theor, International Journal of echanics and aterials in Design, 6(), 010, [8] Thai, H.T., Park,., Choi, D.H., A simple refined theor for bending, buckling, and vibration of thick plates resting on elastic foundation, International Journal of echanical Science, 73, 013, [9] Zenkour, A.., Bending of orthotropic plates resting on Pasternak's foundations b mied shear deformation theor, Acta echanica Sinica, 7(6), 011, [10] Zenkour, A.., Allam,.N.., Shaker,.O., Radwan, A.H., On the simple and mied first-order theories for plates resting on elastic foundations, Acta echanica, 0(1-4), 011, [11] Saad, A.S., Fleure of thick orthotropic plates b eponential shear deformation theor, Latin American Journal of Solids and Structures, 10, 013, Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),

15 Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor 01 [1] Akbas, S.D., Vibration and static analsis of functionall graded porous plates, Journal of Applied and Computational echanics, 3(3), 017, [13] Akbas, S.D., Stabilit of a non-homogenous porous plate b using generalized differantial quadrature method, International Journal of Engineering & Applied Sciences, 9(), 017, [14] Akbas, S.D., Static analsis of a nano plate b using generalized differential quadrature method, International Journal of Engineering & Applied Sciences, 8(), 016, [15] Civalek, O., Analsis of thick rectangular plates with smmetric cross-pl laminates based on first-order shear deformation theor, Journal of Composite aterials, 4(6), 008, [16] Gurses,., Civalek, O., Korkmaz, A.K., Erso, H., Free vibration analsis of smmetric laminated skew plates b discrete singular convolution technique based on first-order shear deformation theor, International Journal for Numerical ethods in Engineering, 79, 009, [17] Saad, A.S., Ghugal, Y.., On the free vibration analsis of laminated composite and sandwich plates: A review of recent literature with some numerical results, Composite Structures, 19, 015, [18] Saad, A.S., Ghugal, Y.., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures, 171, 017, [19] Ghugal, Y.., Saad, A.S., A static fleure of thick isotropic plates using trigonometric shear deformation theor, Journal of Solid echanics, (1), pp , 010. [0] Ghugal, Y.., Saad, A.S., Free vibration of thick isotropic plates using trigonometric shear deformation theor, Journal of Solid echanics, 3(), 011, [1] Ghugal, Y.., Saad, A.S., Static fleure of thick orthotropic plates using trigonometric shear deformation theor, Journal of Structural Engineering, 39(5), 013, [] Ghugal, Y.., Saad, A.S., Free vibration of thick orthotropic plates using trigonometric shear deformation theor, Latin American Journal of Solids and Structures, 8, 011, [3] Timoshenko, S.P., Goodier, J.., Theor of Elasticit, cgraw-hill, Singapore, Journal of Applied and Computational echanics, Vol. 4, No. 3, (018),