Hindawi Mathematical Prolems in Engineering Volume 27, Article ID 7539276, 6 pages https://doi.org/.55/27/7539276 Research Article Analysis Bending Solutions of lamped Rectangular Thick Plate Yang Zhong and Qian Xu School of ivil & Hydraulic Engineering, Dalian University of Technology, Dalian, hina orrespondence should e addressed to Yang Zhong; zhongy@dlut.edu.cn Received January 27; Accepted April 27; Pulished 24 May 27 Academic Editor: Xiao-Qiao He opyright 27 Yang Zhong and Qian Xu. This is an open access article distriuted under the reative ommons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. The ending solutions of rectangular thick plate with all edges clamped and supported were investigated in this study. The asic governing equations used for analysis are ased on Mindlin s higher-order shear deformation plate theory. Using a new function, the three coupled governing equations have een modified to independent partial differential equations that can e solved separately. These equations are coded in terms of deflection of the plate and the mentioned functions. By solving these decoupled equations, the analytic solutions of rectangular thick plate with all edges clamped and supported have een derived. The proposed method eliminates the complicated derivation for calculating coefficients and addresses the solution to prolems directly. Moreover, numerical comparison shows the correctness and accuracy of the results.. Introduction The ending prolem of rectangular thick plates with various cominations of oundary conditions is sparingly common in many engineering fields, such as aerospace, concrete pavements, and mechanical and structural engineering. Moreover with the development of modern industry, relatively more accurate and practical studies on ending plate are required. Prolems involving rectangular plates fall into three distinct categories []: (a) plates with all edges simply supported; () plates with a pair of opposite edges simply supported; (c) plates which do not fall into any of the aove categories. The classical plate theory (PT) is frequently used to analyze thin plates. This theory works on the assumption to ignore the transverse shear deformation and assumes that the normal to the middle plane efore deformation remains straight and normal to the middle surface after deformation. Therefore, utilizing classical plate theory to analyze thick plates leads to somehow inaccurate and even wrong results. Following classical plate theory, a series of theories have een developed y many researchers to analyze thick plates y taking account of the shear deformation, such as Mindlin s first-order, Reddy s third-order, and Reissner s higher-order shear deformation plate theory. The couples governing differential equations of higher ordercouldeotainedthroughtheanaloguetheorymentioned aove such as Mindlin s first-order, Reddy s thirdorder, and Reissner s higher-order shear deformation plate theory which have two more unknowns variales in comparison with the classical plate theory. The following three types of methods can e used to solve the governing equation which are numerical methods including finite element method [2], Ritz energy method [3], and superposition method [4] and semianalytical methods which include Levy method [5], Navier method [5, 6], and the exact analytical methods which including symplectic geometry method [7, 8], and integral transform method [9]. The imposition of oundary conditions on the governing equations increased the mathematical complexity of the solution procedure. Therefore, the analytic ending solutions of rectangular thick plates are hard to solve. Furthermore, numerical methods could e used to solve the ending prolems of plate. However, only the analytical method can give the exact solution, which is used to verify the results otained from various numerical methods. There are two common ways to deal with the plate prolem. First way is to find new plate theories [ 5] which can reducethenumerofunknownequations.houarietal.[] use the new simple higher-order shear deformation theory
2 Mathematical Prolems in Engineering to analyze ending and free viration of functionally graded plates. Tounsi et al. [] perform the new 3-unknown nonpolynomial shear deformation theory for the uckling and viration analyses of functionally graded material (FGM) sandwich plates. The aovementioned theories only dealt with three unknowns as the classical plate theory. Similarly, Beldjelili et al. [2] employ a four-variale refined plate theory to discuss the hygrothermomechanical ending ehavior of sigmoid functionally graded material (S-FGM) plate resting on variale two-parameter elastic foundations. The second way is to find method to simplify the coupled governing differential equations of high order. For this purpose, decouplingmethodisusedtohandlesuchkindofprolem. In this context, the study focused on the improvement of modified Navier method to solve ending prolem of rectangular plates with all edges clamped and supported. By using decoupling, modified Navier method has een modified into a new simple approach to solve the partial differential equations for Mindlin plate. In Section 2., the multiple differential equations have een decoupled while adding a new variale and ascending the equations order. Two of the otained equations are independent which can e solved directly, and another two equations are also much simpler than the already developed equations. In Section 2.2, generalized displacement variales in governing equations are otained y using independent equations in the modified Navier method, and other solutions of the prolem, namely, ending moments, have een otained through related expressions of variales. In the end numerical comparison studies are shown to verify the results. 2. Solution for Rectangular Thick Plate 2.. Decouple Mindlin Equations. The governing equations for ending prolem of rectangular thick plates are given y 2 w ( φ x + φ y y ) + q = () 2 φ x 2 = 2 φ y y 2 =, + μ 2 φ x 2 y 2 + μ 2 φ y 2 2 + 2 + 2 2 φ y y + D ( w φ x) 2 φ x y + D ( w y φ y) where 2 = 2 / 2 + 2 / y 2, D is the flexural rigidity of the plate and its expression is D=Eh 3 /2( μ 2 ), is the shearing stiffness of the plate and its expression is = 5Eh/2( + μ), and E, μ, and h are the elastic module, Poisson s ratio and the thickness of the plate, respectively. w is the transverse deflection of the middle surface. φ x and φ y are the rotations of a normal line due to plate ending. q is the load distriution (2) (3) function. The resultant ending moments, M x and M y,the twisting moments M xy caneotained;namely, M x = D( φ x +μ φ y y ) (4) M y = D( φ y y +μ φ x ) (5) M xy = D( μ) 2 ( φ y y + φ x ). (6) Another new variale M is given for decoupling the governing equations () (3). According to the left expression of (), let and()caneexpressedas M= φ x + φ y y 2 w M+ q =. (8) Taking partial derivative of (2) and (3) with respect to x and y, respectively, then considering (7), we can otain (7) 2 M+ D ( 2 w M) =. (9) Sustituting (8) into (9), the independent differential equation aout M yieldisotainedasfollows: 2 M q =. () D Based on (8) and (), the independent equation aout w is otained as follows: 4 w= q D 2 q. () Taking partial derivative of () with respect to x and y, respectively,isasfollows: 2 w φ x ( 2 2 + 2 φ y y )+ 2 w φ y y ( 2 y 2 + 2 φ x y )+ q = (2) q =. (3) y Multiply (2) y coefficient ( + μ)/2, and then sutract with (2) to eliminate φ y in the equation. The expression etween w and φ x yields 2 φ x Qφ x = Q w μ ( 2 w+ q ). (4) The same as the derivation for (4), the expression etween w and φ y can e otained as 2 φ y Qφ y = Q w y μ y ( 2 w+ q ). (5)
Mathematical Prolems in Engineering 3 According to (8), further simplifying (4) and (5) yields 2 φ x Qφ x = Q w μ 2 φ y Qφ y = Q w y μ y. The asic governing equations are reexpressed as follows: (6) where α n =nπ/aand β m =mπ/. Based on the definition of M as (7), the oundary condition for M is otained as follows: x=,a: M= y=,: M=. The expression of M isalsoassumedasdoulesineseries: (22) M(x,y)= M nm sin α n x sin β m y. (23) 2 M q D = 4 w= q D 2 q 2 φ x Qφ x = Q w μ 2 φ y Qφ y = Q w y μ y. (7a) (7) (7c) (7d) Expanding q in the form of doule sine series, q= Q nm sin α n x sin β m y, (24) where Q nm is defined as Q nm = 4 a a q(x,y)sin α n x sin β m ydxdy. (25) Sustituting (23) and (24) into (7a) gives 2.2. Solution Method of Decoupled Equation. onsidering the example of rectangular thick plates with all edges clamp supported, the solution of asic governing equations () (3) is otained through the modified Navier method. First, the oundary condition equations for plates are given y M nm (α 2 n +β 2 m ) sin α n x sin β m y = Q nm D sin α nx sin β m y. (26) x=,a: According to the uniqueness theorem of Fourier expansion, equating the coefficient M nm in (26), y=,: w=; φ x =; φ y = w=; φ x =; φ y = (8) M nm = a Q nm D(α n 2 +β m 2 ) = 4 ad (α n 2 +β m 2 ) q(x,y)sin α n x sin β m ydxdy. (27) Sustituting (27) into (23), the expression for M is otained as follows: M(x,y)= 4 [ ad (α 2 n +β 2 m ) and (8) shows the asic form of oundary condition. The expressions of w, φ x,andφ y areassumedasdoulesineseries w(x,y)= W nm sin α n x sin β m y (9) a q(x,y)sin α n x sin β m ydxdy]sin α n x sin β m y. (28) Sustituting (9) and (24) into (7) yields the following result: φ x (x, y) = X nm sin α n x sin β m y (2) φ y (x, y) = Y nm sin α n x sin β m y, (2) W nm (α 4 n +β 4 m +2α 2 n β 2 m ) sin α n x sin β m y = ( D + α n +β m )Q nm sin α n x sin β m y. (29)
4 Mathematical Prolems in Engineering According to the uniqueness theorem of Fourier expansion, W mn is otained as W nm =( D(α 2 n +β 2 m ) + ) ( 4 a a (α n 2 +β m 2 ). q(x,y)sin α n x sin β m ydxdy) (3) Sustituting (3) into (9), the expression of w is otained as follows: w(x,y)= ( 4 a a sin α n x sin β m y. α n +β ( m D(α 2 n +β 2 m ) + ) q(x,y)sin α n x sin β m ydxdy) And sustituting (29) and (3) into (7c) and (7d) yielded X nm (α 2 n +β 2 m +Q)sin α n x sin β m y = [M nm μ +W nmq] α n cos α n x sin β m y Y nm (α 2 n +β 2 m +Q)sin α n x sin β m y = [M nm μ +W nmq] β m sin α n x cos β m y. (3) (32) Thus M nm and W nm otained are shown as (27) and (3). Equating the unknown coefficients X nm and Y nm in (32), the expressions of φ x and φ y will e yielded as well. First unify theseriescoreinamannerasfollows: cos α n x= cos β m y= r= s= h rn sin α r x; h sm sin β s y; <x<a < y <, (33) where 4r n+r=odd, n r = odd h rn = π(r 2 n 2 ) n+r = even, n r = even (34) 4s s+m=odd, m s = odd h sm = π(s 2 m 2 ) m s = even, m s = even asedon(33),expandcosα n x and cos β m y in the form of sin Fourier series, and reset the dummy variales: sin α n x sin β m yx nm (α 2 n +β 2 m +Q) = m= h nr α r n= sin α n x sin β m y [M rm μ +W rmq] r= sin α n x sin β m yy nm (α 2 n +β 2 m +Q) = sin α n x sin β m y [M ns μ +W nsq] h ms β s. s= (35) Finally, equating the coefficients in (35) directly according to the uniqueness theorem of Fourier expansion, X nm = (α 2 n +β 2 m +Q) [M rm r= μ +W rmq] (36) h nr α r Y nm = (α 2 n +β 2 m +Q) [M ns s= μ +W nsq] h ms β s. (37) Sustituting (36) and (37) into (2) and (2), respectively, the expressions of φ x and φ y are otained as follows: φ x (x, y) = α 2 n +β 2 m +Q α r +β m r= Q rm φ y (x, y) = α 2 n +β 2 m +Q α n +β s s= Q ns D( μ) +[ D(α 2 r +β 2 m ) + ]Q}h nrα r sin α n x sin β m y (38) D( μ) +[ D(α 2 n +β 2 s ) + ]Q}h msβ s sin α n x sin β m y. (39)
Mathematical Prolems in Engineering 5 Tale : Nondimensional deflection and moment of a square plate, under uniform pressure and clamped () oundary conditions. a/ m, n w x=a/2, y=/2 (qa 4 D ) M x x=, y=/2 (qa2 ) m, n =.365.4225 3 m, n = 5.36.4295 m, n = 2.36.4353 FEM results.36.435 m, n =.26.48 5 m, n = 5.24.432 m, n = 2.24.4589 FEM results.2.458 m, n =.483.432 m, n = 5.483.4739 m, n = 2.483.54 FEM results.48.5 Similarly sustituting (38) and (39) into (4) and (5), the expressions of M x and M y caneotainedasfollows: M x = D α n cos α n x sin β m y(α 2 n +β 2 m +Q) Q [ rm D(α 2 r +β 2 m ) μ +Q(Q rm D r= + α r +β m r +β 4 m +2α 2 r β 2 m ) ]h nr α r +μβ m sin α n x cos β m y (α 2 n +β 2 m +Q) Q [ ns s= D(α 2 n +β 2 s ) μ +Q(Q ns D + α n +β s n +β 4 s +2α 2 n β 2 s ) ]h ms β s } M y = D β m sin α n x cos β m y(α 2 n +β 2 m +Q) (4) Q [ ns D(α 2 n +β 2 s ) μ +Q(Q ns D + α n +β s n +β 4 s +2α 2 n β 2 s ) ]h ms β s +μα n cos α n x sin β m y s= (α 2 n +β 2 m +Q) Q [ rm r= D(α 2 r +β 2 m ) μ +Q(Q rm D + α r +β m r +β 4 m +2α 2 r β 2 m ) ]h nr α r }. 3. Numerical Example A thick plate with all edges clamped () has een taken as a numerical example to justify the correctness of the aove solution. The length and width of the plate are a=,withthe Poisson ratio of μ =.3. Figure shows the change in deflection of the plate. Tale highlights the comparison of nondimensional deflection results with solutions given y FEM, which shows that the results otained are in accordance with the ones given efore, and proves the correctness of the aove method and the derivations. 4. onclusion The decoupling method and the modified Navier s solution have een used together in this study for a simple analysis of rectangular thick plates with all edges clamped and supported. Unlike the original modified Navier method, the proposed approach does not need complicated matrix derivations for calculating the coefficients. The procedure for solving the ending rectangular thick plates with all edges clamped is made simpler than efore. Moreover the proposed method can e further extended to address the prolem
6 Mathematical Prolems in Engineering 4 2 3 4 8 6 4 2 2 4 6 8 Figure : The deflections of a square plate. of rectangular thick plates with other cominations of free and simply supported oundary conditions. The proposed method has many practical applications and can e used in foundation design of high-rise uilding and rigid pavements of highway and airport. Additionally, the plate support prolems such as point supports and spring supports can e solved well analytically y utilizing similar approach, which would expectantly develop inspiring extensions in the field. Moreover, the results otained from numerical example validate the precision and correctness of method and derivations. onflicts of Interest The authors declare that they have no conflicts of interest. Acknowledgments The work descried in this paper was supported y the National Natural Science Foundation of hina no. 78239. References [] K. Bhaskar and B. Kaushik, Simple and exact series solutions for flexure of orthotropic rectangular plates with any comination of clamped and simply supported edges, omposite Structures, vol. 63, no., pp. 63 68, 24. [2] S. Weiming and Y. Guangsong, Rational finite element method for elastic ending of reissner plates, Applied Mathematics and Mechanics, vol. 2, no. 2, pp. 93 99, 999. [3] A. W. Leissa and F. W. Niedenfuhr, A study of the cantilevered square plate sujected to a uniform loading, the Aero/Space Sciences, vol. 29, no. 2, 962. [4] M. K. Huang and H. D. onway, Bending of a uniformly loaded rectangular plate with two adjacent edges clamped and the others either simply supported or free, Apply Mechanics, vol. 9, pp. 45 46, 952. [5] S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 959. [6] H. R. H. Kair and R. A. haudhuri, Boundary-continuous fourier solution for clamped Mindlin plates, Engineering Mechanics, vol. 8, no. 7, pp. 457 467, 992. [7] Y. Zhong and R. Li, Exact ending analysis of fully clamped rectangular thin plates sujected to aritrary loads y new symplectic approach, Mechanics Research ommunications, vol. 36, no. 6, pp. 77 74, 29. [8] R. Li, B. Wang, and P. Li, Hamiltonian system-ased enchmark ending solutions of rectangular thin plates with a corner point-supported, International Mechanical Sciences, vol. 85, pp. 22 28, 24. [9] Y. Zhong and J.-H. Yin, Free viration analysis of a plate on foundation with completely free oundary y finite integral transform method, Mechanics Research ommunications, vol. 35, no. 4, pp. 268 275, 28. [] M. S. Houari, A. Tounsi, A. Bessaim, and S. Mahmoud, A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates, Steel and omposite Structures, vol. 22, no. 2, pp. 257 276, 26. [] A. Tounsi, M. S. Houari, and A. Bessaim, A new 3-unknowns non-polynomial plate theory for uckling and viration of functionally graded sandwich plate, Structural Engineering and Mechanics, vol. 6, no. 4, pp. 547 565, 26. [2] Y. Beldjelili, A. Tounsi, and S. Mahmoud, Hygro-thermomechanical ending of S-FGM plates resting on variale elastic foundations using a four-variale trigonometric plate theory, Smart Structures and Systems, vol. 8, no. 4, pp. 755 786, 26. [3] H. Heali, A. Tounsi, M. S. A. Houari, A. Bessaim, and E. A. A. Bedia, New quasi-3d hyperolic shear deformation theory for the static and free viration analysis of functionally graded plates, ASE, Engineering Mechanics, vol. 4, no. 2, pp. 374 383, 24. [4] A. Hamidi, M. S. A. Houari, S. R. Mahmoud, and A. Tounsi, A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical ending of functionally graded sandwich plates, Steel and omposite Structures, vol. 8, no., pp. 235 253, 25. [5] A. Mahi, E. A. Adda Bedia, and A. Tounsi, A new hyperolic shear deformation theory for ending and free viration analysis of isotropic, functionally graded, sandwich and laminated composite plates, Applied Mathematical Modelling. Simulation and omputation for Engineering and Environmental Systems, vol. 39, no. 9, pp. 2489 258, 25.
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