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eISSN: 2572-8520

Civil Engineering

Research Article Volume 2 Issue 2

Elastic instability analysis of biaxially compressed flat rectangular isotropic all round clamped (CCCC) plates

David Ogbonna Onwuka, Stanley Emeka Iwuoha

Department of Civil Engineering, Federal University of Technology Owerri, Nigeria

Correspondence: Stanley Emeka Iwuoha, Department of Civil Engineering, Federal University of Technology Owerri, Nigeria

Received: January 28, 2017 | Published: February 23, 2017

Citation: Onwuka DO, Iwuoha SE. Elastic instability analysis of biaxially compressed flat rectangular isotropic all-round clamped (CCCC) plates. MOJ Civil Eng. 2017;2(2):52-56. DOI: 10.15406/mojce.2017.02.00027

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Abstract

The Galerkin’s method was used to carry out the elastic instability analysis of biaxially compressed flat rectangular isotropic all-round clamped plates. The biaxial critical buckling load equation was obtained by substituting the plate deflection equation (obtained via the polynomial series) into the Galerkin’s functional. Throughout the analysis, the aspect ratios (defined as the ratio of length, “b” of the plate on the y axis to the length, “a” of plate on the x- axis) was considered to range from 1 to 2. A linear relationship was obtained for the buckling load on the y axis in terms of that on the x-axis. Results for the critical buckling load were obtained for the various aspect ratios (1 to 2) and “k” (relationship constant between forces on the Y- axis and forces on the X-axis) values (0.1 to 1). A maximum buckling load coefficient of 108.0006 was obtained for a square plate at a “k” value of 0, while the least buckling load was 40.50021, obtained for a rectangular plate of aspect ratio equal to 2 and a “k” value of 1.0. At k equal to zero and for all aspect ratios, the results of the present study showed a maximum percentage difference of 0.69389 with those given by Ibearugbulem et al, which shows that the results for the buckling analysis of biaxially loaded CCCC plates presented in this paper for the given aspect ratios and “k” values are very accurate.

Keywords: elastic instability analysis, thin plates, biaxial forces, galerkin’s method, boundary conditions

Abbreviations

A, coefficient of deflection of the plate; a, length of the plate; b, width of the plate; W, deflection equation of the plate; H, shape function of the plate; D, flexural rigidity of the plate; α  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHjaacckaaaa@3968@ - aspect ratio=b/a; Nx, load applied in the x-direction; Ny, load applied in the y-direction; Nxcr, critical buckling load in the x-direction; h, thickness of the plate; X, primary axis of the plate; Y, secondary axis of the plate; C, clamped support; R, non-dimensional parameter equal to x/a; Q, non-dimensional parameter equal to y/b; F, buckling load coefficient; k, constant, relating ny and nx; Nxi, the critical buckling load coefficients at k=0.i ; w ' R ,w ' Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bGaai4ja8aadaahaaqcfasabeaapeGaamOuaaaajuaG caGGSaGaam4DaiaacEcapaWaaWbaaeqajuaibaWdbiaadgfaaaaaaa@3DBC@ –first derivative of the deflection equation with respect to r and q respectively

Introduction

Thin rectangular plates are used in the construction of thin walled structures for the transmission of both in-plane and lateral loads. The aeronautic and marine industries make particular use of such materials as thin plates. Thin plates had been defined by Szilard1 as one whose ratio of its basic dimension to its thickness falls within the range 8…10 ≤ a h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWccaWdaeaapeGaamyyaaWdaeaapeGaamiAaaaacqGHKjYO aaa@3A7C@ 80…100. Due to the importance and wide application of this structural material, several researches had been carried out with the aim of maximizing its potentials for wider structural applications. Areas of research of plate analysis include the vibration of plates, bending of plates and the buckling of plates. Buckling is the phenomenon in which a material under the action of in-plane compressive loads, begins to move from the state of stable equilibrium to a state of unstable equilibrium at a critical value of the compressive loads even when transverse loads are not applied. According to Ventsel & Krauthermmer2 failure of thin plate elements may be attributed to an elastic instability and not to the lack of their strength” Therefore, determination of the critical buckling loads of a plate, is essential to safe design of a plate for the intended use of the plate for any Engineering purpose within the safe load. Several works on the buckling analysis of plates had been done in the past. Makhtar et al,3 used a first order shear deformation theory to carry out the thermal buckling analysis of simply supported functionally graded plate and showed that when the plate aspect ratio, a b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaamyyaaWdaeaapeGaamOyaaaaaaa@38BF@ is decreased, the critical temperature reduces and the plate becomes thinner. Chajes4 showed that the buckling load of a plate simply supported all round and uniformly compressed in one direction is given by Equation 1

N x = D π 2 b 2 ( mb a + n 2 a mb ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aabeaapeGa eyypa0ZaaSaaa8aabaWdbiaadseacqaHapaCpaWaaWbaaeqajuaiba WdbiaaikdaaaaajuaGpaqaa8qacaWGIbWdamaaCaaabeqcfasaa8qa caaIYaaaaaaajuaGdaqadaWdaeaapeWaaSaaa8aabaWdbiaad2gaca WGIbaapaqaa8qacaWGHbaaaiabgUcaRmaalaaapaqaa8qacaWGUbWd amaaCaaajuaibeqaa8qacaaIYaaaaKqbakaadggaa8aabaWdbiaad2 gacaWGIbaaaaGaayjkaiaawMcaa8aadaahaaqcfasabeaapeGaaGOm aaaaaaa@4E3D@

While for a plate fully clamped on all sides and uniaxially loaded, the buckling load is given by Equation 2.

N x = 10.67  π 2 D a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aabeaapeGa eyypa0ZaaSaaa8aabaWdbiaaigdacaaIWaGaaiOlaiaaiAdacaaI3a GaaiiOaiabec8aW9aadaahaaqcfasabeaapeGaaGOmaaaajuaGcaWG ebaapaqaa8qacaWGHbWdamaaCaaajuaibeqaa8qacaaIYaaaaaaaaa a@45FF@

Singh & Chakrabarti5 developed an efficient CoFE model based on higher zigzag theory for the buckling analysis of a uniaxially loaded simply supported cross ply square plate. Jayashankarbabu et al.6 used the finite element method to obtain the elastic buckling load factor for square plates of different boundary conditions (such as, SCSC, CCCC, SSSS) containing square and circular cutouts, subjected to uniaxial compression, with the loads applied at the simply supported and at the clamped edges. Yao et al,7 proposed a new method which do not require the global stiffness matrix of the system but, reduces the system matrix order and improves the computational efficiency for analyzing plates which are simply supported on all edges. Ezeh et al.8 proposed shape functions based on the characteristic orthogonal polynomial and used them in the Galerkin’s indirect variational principle for carrying out the elastic buckling analysis of a thin plate clamped at all edges, and subjected to axial load in the x-direction. Ventsel & Krauthermmer,2 Iyengar9 & Chajes,4 individually, demonstrated that for a biaxially loaded square SSSS plates subject to uniform pressure on both sides, the critical load Ncr, is given by Equation 3.

N cr = 2  π 2 D a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadogacaWGYbaapaqabaqc fa4dbiabg2da9maalaaapaqaa8qacaaIYaGaaiiOaiabec8aW9aada ahaaqcfasabeaapeGaaGOmaaaajuaGcaWGebaapaqaa8qacaWGHbWd amaaCaaajuaibeqaa8qacaaIYaaaaaaaaaa@43F5@  Eqn 3

Ibearugbulem et al.10 derived a polynomial shape function and used it in the Ritz method to carry out the buckling analysis of plates with boundary conditions (such as the SSSS, CCCC, CSSS, CCSS, CSCS, and the CCCS).From available literature, it will be discovered that works on the buckling analysis of plates had revolved mostly around uniaxially loaded, and square simply supported biaxially loaded plates subject to uniform pressure. To the best of our knowledge, there is extreme dearth of literature on the buckling analysis of all-round clamped thin isotropic square and rectangular plates subject to biaxial loading. The objective of this work, is to fill the gap in literature, by providing solutions to the buckling analysis of all-round clamped thin isotropic rectangular plates subject to biaxial loading (with unequal forces in the both axes) by using the polynomial shape function proposed by Ibearugbulem11 in the Galerkin’s work method to derive the equation for buckling of plates.

Material and method

The method of solution is detailed as presented in the following stages below.

Formulation of the equation of buckling of biaxially compressed thin rectangular isotropic plates

Consider a fully clamped flat isotropic plate under the action of biaxial compressive in-plane loads as shown in Figure 1. Let the thickness of the plate in the z- direction be far smaller than both the length and width of the plate in the x-and y-directions.

Figure 1 A CCCC Plate Under-going Biaxial Compression.

The overall governing differential equations for plates, is given by Ibearugbulem et al.10 as, Equation 4

q N x ( d 2 w x 2 )2 N xy ( 2 w xy ) N y ( d 2 w y 2 )+m λ 2 w=D[ 4 w x 4 +2 4 w x 2 y 2 + 4 w y 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbGaeyOeI0IaamOta8aadaWgaaqcfasaa8qacaWG4baa juaGpaqabaWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaamiza8aada ahaaqcfasabeaapeGaaGOmaaaajuaGcaWG3baapaqaa8qacqGHciIT caWG4bWdamaaCaaajuaibeqaa8qacaaIYaaaaaaaaKqbakaawIcaca GLPaaacqGHsislcaaIYaGaamOta8aadaWgaaqcfasaa8qacaWG4bGa amyEaaWdaeqaaKqba+qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgk Gi2+aadaahaaqcfasabeaapeGaaGOmaaaajuaGcaWG3baapaqaa8qa cqGHciITcaWG4bGaeyOaIyRaamyEaaaaaiaawIcacaGLPaaacqGHsi slcaWGobWdamaaBaaajuaibaWdbiaadMhaaKqba+aabeaapeWaaeWa a8aabaWdbmaalaaapaqaa8qacaWGKbWdamaaCaaajuaibeqaa8qaca aIYaaaaKqbakaadEhaa8aabaWdbiabgkGi2kaadMhapaWaaWbaaKqb GeqabaWdbiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiabgUcaRiaad2 gacqaH7oaBpaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaam4Daiab g2da9iaadseadaWadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aada ahaaqcfasabeaapeGaaGinaaaajuaGcaWG3baapaqaa8qacqGHciIT caWG4bWdamaaCaaajuaibeqaa8qacaaI0aaaaaaajuaGcqGHRaWkca aIYaWaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGin aaaajuaGcaWG3baapaqaa8qacqGHciITcaWG4bWdamaaCaaajuaibe qaa8qacaaIYaaaaKqbakabgkGi2kaadMhapaWaaWbaaKqbGeqabaWd biaaikdaaaaaaKqbakabgUcaRmaalaaapaqaa8qacqGHciITpaWaaW baaKqbGeqabaWdbiaaisdaaaqcfaOaam4DaaWdaeaapeGaeyOaIyRa amyEa8aadaahaaqcfasabeaapeGaaGinaaaaaaaajuaGcaGLBbGaay zxaaaaaa@8D6A@

Where W = AH     (5)

For biaxial buckling q=Nxy=m ? 2 w=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbGaeyypa0JaamOtaiaadIhacaWG5bGaeyypa0JaamyB aiaac+dapaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaam4Daiabg2 da9iaaicdacaGGSaaaaa@434E@ hence, Ventsel & Krauthammer2 gave the buckling equation as; Equation (6)

N x ( 2 w x 2 ) N y ( 2 w y 2 )=D[ 4 w x 4 +2 4 w x 2 y 2 + 4 w y 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aa beaapeWaaeWaa8aabaWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaK qbGeqabaWdbiaaikdaaaqcfaOaam4DaaWdaeaapeGaeyOaIyRaamiE a8aadaahaaqabKqbGeaapeGaaGOmaaaaaaaajuaGcaGLOaGaayzkaa GaeyOeI0IaamOta8aadaWgaaqcfasaa8qacaWG5baajuaGpaqabaWd bmaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaajuaibe qaa8qacaaIYaaaaKqbakaadEhaa8aabaWdbiabgkGi2kaadMhapaWa aWbaaKqbGeqabaWdbiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiabg2 da9iaadseadaWadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaah aaqcfasabeaapeGaaGinaaaajuaGcaWG3baapaqaa8qacqGHciITca WG4bWdamaaCaaajuaibeqaa8qacaaI0aaaaaaajuaGcqGHRaWkcaaI YaWaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGinaa aajuaGcaWG3baapaqaa8qacqGHciITcaWG4bWdamaaCaaajuaibeqa a8qacaaIYaaaaKqbakabgkGi2kaadMhapaWaaWbaaKqbGeqabaWdbi aaikdaaaaaaKqbakabgUcaRmaalaaapaqaa8qacqGHciITpaWaaWba aKqbGeqabaWdbiaaisdaaaqcfaOaam4DaaWdaeaapeGaeyOaIyRaam yEa8aadaahaaqabKqbGeaapeGaaGinaaaaaaaajuaGcaGLBbGaayzx aaaaaa@76C7@

The Equation (6), is the equation of forces acting on the biaxially loaded plate. These forces (both internal and external) acting on the plate, together have the tendency to cause the plate to be deformed. If “w” is the average deformation caused on the plate by the forces, then the work done by the forces on the plates is as given by Equation (7)

N x ( 2 w x 2 )w N y ( 2 w y 2 )w=D[ 2 w x 4 .w+2w. 4 w x 2 y 2 + 4 w y 4 .w ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aa beaapeWaaeWaa8aabaWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaK qbGeqabaWdbiaaikdaaaqcfaOaam4DaaWdaeaapeGaeyOaIyRaamiE a8aadaahaaqcfasabeaapeGaaGOmaaaaaaaajuaGcaGLOaGaayzkaa Gaam4DaiabgkHiTiaad6eapaWaaSbaaKqbGeaapeGaamyEaaqcfa4d aeqaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaa qcfasabeaapeGaaGOmaaaajuaGcaWG3baapaqaa8qacqGHciITcaWG 5bWdamaaCaaajuaibeqaa8qacaaIYaaaaaaaaKqbakaawIcacaGLPa aacaWG3bGaeyypa0Jaamiramaadmaapaqaa8qadaWcaaWdaeaapeGa eyOaIy7damaaCaaabeqcfasaa8qacaaIYaaaaKqbakaadEhaa8aaba WdbiabgkGi2kaadIhapaWaaWbaaKqbGeqabaWdbiaaisdaaaaaaKqb akaac6cacaWG3bGaey4kaSIaaGOmaiaadEhacaGGUaWaaSaaa8aaba WdbiabgkGi2+aadaahaaqcfasabeaapeGaaGinaaaajuaGcaWG3baa paqaa8qacqGHciITcaWG4bWdamaaCaaajuaibeqaa8qacaaIYaaaaK qbakabgkGi2kaadMhapaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqb akabgUcaRmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbi aaisdaaaqcfaOaam4DaaWdaeaapeGaeyOaIyRaamyEa8aadaahaaqc fasabeaapeGaaGinaaaaaaqcfaOaaiOlaiaadEhaaiaawUfacaGLDb aaaaa@7DC7@

The Equation (7) being Galerkin’s expression for the biaxial buckling of plates at any arbitrary point, was obtained by multiplying Equation (5) by the average deformation “w” of the plate.

The entire work done on the plate, obtained by integrating Equation (7) completely along the x-and y-axes, is given by Equation (8).

N x ( 2 w x 2 ).wxy N y ( 2 w y 2 ).wxy=D [ 4 w x 4 .w+2w 4 w x 2 y 2 + 4 w y 4 .w ]xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWGobWdamaaBaaajuaibaWdbiaadIhaa8aabeaa juaGdaqfGaqabeqabaGaaGzaVdqaamXvP5wqSX2qVrwzqf2zLnhary GqHrxyUDgaiuaapeGaa8hlIaaadaqadaWdaeaapeWaaSaaa8aabaWd biabgkGi2+aadaahaaqcfasabeaapeGaaGOmaaaajuaGcaWG3baapa qaa8qacqGHciITcaWG4bWdamaaCaaajuaibeqaa8qacaaIYaaaaaaa aKqbakaawIcacaGLPaaacaGGUaGaam4DaiabgkGi2kaadIhacqGHci ITcaWG5bGaeyOeI0IaamOta8aadaWgaaqcfasaa8qacaWG5baapaqa baqcfa4aaubiaeqabeqaaiaaygW7aeaapeGaa8hlIaaadaqadaWdae aapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGOm aaaajuaGcaWG3baapaqaa8qacqGHciITcaWG5bWdamaaCaaajuaibe qaa8qacaaIYaaaaaaaaKqbakaawIcacaGLPaaacaGGUaGaam4Daiab gkGi2kaadIhacqGHciITcaWG5bGaeyypa0Jaamira8aadaqfGaqabe qabaGaaGzaVdqaa8qacaWFSicaamaadmaapaqaa8qadaWcaaWdaeaa peGaeyOaIy7damaaCaaajuaibeqaa8qacaaI0aaaaKqbakaadEhaa8 aabaWdbiabgkGi2kaadIhapaWaaWbaaKqbGeqabaWdbiaaisdaaaaa aKqbakaac6cacaWG3bGaey4kaSIaaGOmaiaadEhadaWcaaWdaeaape GaeyOaIy7damaaCaaajuaibeqaa8qacaaI0aaaaKqbakaadEhaa8aa baWdbiabgkGi2kaadIhapaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfa OaeyOaIyRaamyEa8aadaahaaqcfasabeaapeGaaGOmaaaaaaqcfaOa ey4kaSYaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaG inaaaajuaGcaWG3baapaqaa8qacqGHciITcaWG5bWdamaaCaaajuai beqaa8qacaaI0aaaaaaajuaGcaGGUaGaam4DaaGaay5waiaaw2faai abgkGi2kaadIhacqGHciITcaWG5baaaa@9CEA@

 For rapid solution of the plate problem, the Cartesian coordinates, are expressed in terms of non-dimensional parameters as;

R= x a andQ= y b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbGaeyypa0ZaaSaaa8aabaWdbiaadIhaa8aabaWdbiaa dggaaaGaamyyaiaad6gacaWGKbGaamyuaiabg2da9maalaaapaqaa8 qacaWG5baapaqaa8qacaWGIbaaaaaa@4183@          (9)

Let the aspect ratio, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHDisTaaa@3824@  be given by the Equation (10)

= b a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHDisTcqGH9aqpdaWcaaWdaeaapeGaamOyaaWdaeaapeGa amyyaaaaaaa@3B45@

Substituting Equations (5), (9) and (10) into equation (8), and simplifying gives Equation (11)

  N x a 2 ( H R ) 2 RQ N y a 2 2 ( H Q ) 2 RQ      MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcGaeyOeI0YaaSaaa8aabaWdbiaad6eapaWaaSbaaKqb GeaapeGaamiEaaWdaeqaaaqcfayaa8qacaWGHbWdamaaCaaabeqcfa saa8qacaaIYaaaaaaajuaGpaWaaubiaeqabeqaaiaaygW7aeaatCvA UfeBSn0BKvguHDwzZbqegiuy0fMBNbacfaWdbiaa=XIiaaWaaeWaa8 aabaWdbmaalaaapaqaa8qacqGHciITcaWGibaapaqaa8qacqGHciIT caWGsbaaaaGaayjkaiaawMcaa8aadaahaaqcfasabeaapeGaaGOmaa aajuaGcqGHciITcaWGsbGaeyOaIyRaamyuaiabgkHiTmaalaaapaqa a8qacaWGobWdamaaBaaajuaibaWdbiaadMhaa8aabeaaaKqbagaape Gaamyya8aadaahaaqcfasabeaapeGaaGOmaaaajuaGcqGHDisTpaWa aWbaaeqajuaibaWdbiaaikdaaaaaaKqba+aadaqfGaqabeqabaGaaG zaVdqaa8qacaWFSicaamaabmaapaqaa8qadaWcaaWdaeaapeGaeyOa IyRaamisaaWdaeaapeGaeyOaIyRaamyuaaaaaiaawIcacaGLPaaapa WaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaeyOaIyRaamOuaiabgkGi 2kaadgfacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaaaa@74D3@

= D a 4 [ ( 4 H R 4 )H+ 2H 2 ( 4 H R 2 Q 2 )+ H 4 ( 4 H Q 4 ) ]RQ   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaWcaaWdaeaapeGaamiraaWdaeaapeGaamyya8aa daahaaqcfasabeaapeGaaGinaaaaaaqcfa4damaavacabeqabeaaca aMb8oabaWexLMBbXgBd9gzLbvyNv2CaeHbcfgDH52zaGqba8qacaWF Sicaamaadmaapaqaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgk Gi2+aadaahaaqcfasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qa cqGHciITcaWGsbWdamaaCaaajuaibeqaa8qacaaI0aaaaaaaaKqbak aawIcacaGLPaaacaWGibGaey4kaSYaaSaaa8aabaWdbiaaikdacaWG ibaapaqaa8qacqGHDisTpaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaK qbaoaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqc fasaa8qacaaI0aaaaKqbakaadIeaa8aabaWdbiabgkGi2kaadkfapa WaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeyOaIyRaamyua8aadaah aaqcfasabeaapeGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaGaey4kaS YaaSaaa8aabaWdbiaadIeaa8aabaWdbiabg2Hi1+aadaahaaqcfasa beaapeGaaGinaaaaaaqcfa4aaeWaa8aabaWdbmaalaaapaqaa8qacq GHciITpaWaaWbaaKqbGeqabaWdbiaaisdaaaqcfaOaamisaaWdaeaa peGaeyOaIyRaamyua8aadaahaaqcfasabeaapeGaaGinaaaaaaaaju aGcaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyOaIyRaamOuaiabgkGi 2kaadgfacaGGGcGaaiiOaaaa@7C9D@

Let the forces in the x- and y- axes of the plates be related by Equation 12.

N y =K N x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadMhaaKqba+aabeaapeGa eyypa0Jaam4saiaad6eapaWaaSbaaKqbGeaapeGaamiEaaqcfa4dae qaaaaa@3E41@

Substituting Equation (12) into (11), and multiplying the resulting equation by a2, yields Equation (13)

N x ( H R ) 2 RQ K N x 2 ( H Q ) 2 RQ= D a 2 [ ( 4 H R 4 )H+ 2H 2 ( 4 H R 2 Q 2 )+ H 4 ( 4 H d Q 4 ) ]RQ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWGobWdamaaBaaajuaibaWdbiaadIhaa8aabeaa juaGdaqfGaqabeqabaGaaGzaVdqaamXvP5wqSX2qVrwzqf2zLnhary GqHrxyUDgaiuaapeGaa8hlIaaadaqadaWdaeaapeWaaSaaa8aabaWd biabgkGi2kaadIeaa8aabaWdbiabgkGi2kaadkfaaaaacaGLOaGaay zkaaWdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkGi2kaadkfa cqGHciITcaWGrbGaeyOeI0YaaSaaa8aabaWdbiaadUeacaWGobWdam aaBaaajuaibaWdbiaadIhaaKqba+aabeaaaeaapeGaeyyhIu7damaa Caaajuaibeqaa8qacaaIYaaaaaaajuaGpaWaaubiaeqabeqaaiaayg W7aeaapeGaa8hlIaaadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi 2kaadIeaa8aabaWdbiabgkGi2kaadgfaaaaacaGLOaGaayzkaaWdam aaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkGi2kaadkfacqGHciIT caWGrbGaeyypa0ZaaSaaa8aabaWdbiaadseaa8aabaWdbiaadggapa WaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqba+aadaqfGaqabeqabaGa aGzaVdqaa8qacaWFSicaamaadmaapaqaa8qadaqadaWdaeaapeWaaS aaa8aabaWdbiabgkGi2+aadaahaaqabKqbGeaapeGaaGinaaaajuaG caWGibaapaqaa8qacqGHciITcaWGsbWdamaaCaaajuaibeqaa8qaca aI0aaaaaaaaKqbakaawIcacaGLPaaacaWGibGaey4kaSYaaSaaa8aa baWdbiaaikdacaWGibaapaqaa8qacqGHDisTpaWaaWbaaKqbGeqaba WdbiaaikdaaaaaaKqbaoaabmaapaqaa8qadaWcaaWdaeaapeGaeyOa Iy7damaaCaaajuaibeqaa8qacaaI0aaaaKqbakaadIeaa8aabaWdbi abgkGi2kaadkfapaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaeyOa IyRaamyua8aadaahaaqcfasabeaapeGaaGOmaaaaaaaajuaGcaGLOa GaayzkaaGaey4kaSYaaSaaa8aabaWdbiaadIeaa8aabaWdbiabg2Hi 1+aadaahaaqabeaapeGaaGinaaaaaaWaaeWaa8aabaWdbmaalaaapa qaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaisdaaaqcfaOaamis aaWdaeaapeGaamizaiaadgfapaWaaWbaaKqbGeqabaWdbiaaisdaaa aaaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faaiabgkGi2kaadkfa cqGHciITcaWGrbaaaa@A37E@

Making N x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aabeaaaaa@397F@  the subject of the Equation (13), gives the general equation of buckling of a biaxially compressed thin rectangular isotropic plate as Equation (14)

N x = D a 2 [ ( 4 H R 4 )H+ 2H 2 ( 4 H R 2 Q 2 )+ H 4 ( 4 H Q 4 ) ]RQ [ ( H R ) 2 + K 2 ( H Q ) 2 ]RQ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhaa8aabeaajuaGpeGa eyypa0JaeyOeI0YaaSaaa8aabaWdbmaaliaapaqaa8qacaWGebaapa qaa8qacaWGHbWdamaaCaaajuaibeqaa8qacaaIYaaaaaaajuaGpaWa aubiaeqabeqaaiaaygW7aeaatCvAUfeBSn0BKvguHDwzZbqegiuy0f MBNbacfaWdbiaa=XIiaaWaamWaa8aabaWdbmaabmaapaqaa8qadaWc aaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI0aaaaKqbak aadIeaa8aabaWdbiabgkGi2kaadkfapaWaaWbaaKqbGeqabaWdbiaa isdaaaaaaaqcfaOaayjkaiaawMcaaiaadIeacqGHRaWkdaWcaaWdae aapeGaaGOmaiaadIeaa8aabaWdbiabg2Hi1+aadaahaaqcfasabeaa peGaaGOmaaaaaaqcfa4aaeWaa8aabaWdbmaalaaapaqaa8qacqGHci ITpaWaaWbaaeqajuaibaWdbiaaisdaaaqcfaOaamisaaWdaeaapeGa eyOaIyRaamOua8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcqGHci ITcaWGrbWdamaaCaaabeqcfasaa8qacaaIYaaaaaaaaKqbakaawIca caGLPaaacqGHRaWkdaWcaaWdaeaapeGaamisaaWdaeaapeGaeyyhIu 7damaaCaaajuaibeqaa8qacaaI0aaaaaaajuaGdaqadaWdaeaapeWa aSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGinaaaaju aGcaWGibaapaqaa8qacqGHciITcaWGrbWdamaaCaaajuaibeqaa8qa caaI0aaaaaaaaKqbakaawIcacaGLPaaaaiaawUfacaGLDbaacqGHci ITcaWGsbGaeyOaIyRaamyuaaWdaeaadaqfGaqabeqabaGaaGzaVdqa a8qacaWFSicaamaadmaapaqaa8qadaqadaWdaeaapeWaaSaaa8aaba WdbiabgkGi2kaadIeaa8aabaWdbiabgkGi2+aadaWgaaqaa8qacaWG sbaapaqabaaaaaWdbiaawIcacaGLPaaapaWaaWbaaKqbGeqabaWdbi aaikdaaaqcfaOaey4kaSYaaSaaa8aabaWdbiaadUeaa8aabaWdbiab g2Hi1+aadaahaaqcfasabeaapeGaaGOmaaaaaaqcfa4aaeWaa8aaba Wdbmaalaaapaqaa8qacqGHciITcaWGibaapaqaa8qacqGHciITcaWG rbaaaaGaayjkaiaawMcaa8aadaahaaqcfasabeaapeGaaGOmaaaaaK qbakaawUfacaGLDbaacqGHciITcaWGsbGaeyOaIyRaamyuaaaaaaa@9D71@

Taylor-McLaurin’s series formulated deflection function of the CCCC isotropic rectangular plate

Since, for an all-round clamped plate, the deflections and rotations, are zeros at all edges of the plate, the boundary conditions of the SSSS plate is as follows:

w( R=0 )=w ' R ( R=0 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bWaaeWaa8aabaWdbiaadkfacqGH9aqpcaaIWaaacaGL OaGaayzkaaGaeyypa0Jaam4DaiaacEcapaWaaWbaaKqbGeqabaWdbi aadkfaaaqcfa4aaeWaa8aabaWdbiaadkfacqGH9aqpcaaIWaaacaGL OaGaayzkaaGaeyypa0JaaGimaaaa@465F@ 15

w( R=1 )=w ' R ( R=1 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bWaaeWaa8aabaWdbiaadkfacqGH9aqpcaaIXaaacaGL OaGaayzkaaGaeyypa0Jaam4DaiaacEcapaWaaWbaaeqajuaibaWdbi aadkfaaaqcfa4aaeWaa8aabaWdbiaadkfacqGH9aqpcaaIXaaacaGL OaGaayzkaaGaeyypa0JaaGimaaaa@4661@  16

w( Q=0 )=w ' Q ( Q=0 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bWaaeWaa8aabaWdbiaadgfacqGH9aqpcaaIWaaacaGL OaGaayzkaaGaeyypa0Jaam4DaiaacEcapaWaaWbaaeqajuaibaWdbi aadgfaaaqcfa4aaeWaa8aabaWdbiaadgfacqGH9aqpcaaIWaaacaGL OaGaayzkaaGaeyypa0JaaGimaaaa@465C@  17

w( Q=1 )=w ' Q ( Q=1 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bWaaeWaa8aabaWdbiaadgfacqGH9aqpcaaIXaaacaGL OaGaayzkaaGaeyypa0Jaam4DaiaacEcapaWaaWbaaeqajuaibaWdbi aadgfaaaqcfa4aaeWaa8aabaWdbiaadgfacqGH9aqpcaaIXaaacaGL OaGaayzkaaGaeyypa0JaaGimaaaa@465E@  18

They w R and w Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbauaapaWaaWbaaKqbGeqabaWdbiaadkfaaaqcfaOa amyyaiaad6gacaWGKbGaaGPaVlqadEhagaqba8aadaahaaqcfasabe aapeGaamyuaaaaaaa@401A@ are the first derivatives of the deflection function, w, in the R-and Q-directions respectively.

Ibearugbulem (2011) assumed the shape function, w, to be continuous and differentiable. He expanded it in Taylor-Mclaurin series and truncated the infinite polynomial series at m = n = 4 and got the general polynomial deflection equation of rectangular plates as follows.

w= m=0 4 n=0 4 a m b n R m . Q n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dai abg2da9maaqahabaWaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGTbaa beaaaeaacaWGUbGaeyypa0JaaGimaaqaaiaaisdaaKqbakabggHiLd aajuaibaGaamyBaiabg2da9iaaicdaaeaacaaI0aaajuaGcqGHris5 aiaadkgadaWgaaqcfasaaiaad6gaaeqaaKqbakaadkfadaahaaqcfa sabeaacaWGTbaaaKqbakaac6cacaWGrbWaaWbaaeqajuaibaGaamOB aaaaaaa@4F55@ (19)

Where a m and b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGHbWdamaaBaaajuaibaWdbiaad2gaa8aabeaajuaGpeGa amyyaiaad6gacaWGKbGaaGzaVlaaygW7caaMb8UaaGzaVlaaygW7ca aMb8UaaGPaVlaadkgapaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqa aaaa@4A05@  are constants and RandQ   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbGaamyyaiaad6gacaWGKbGaaGPaVlaadgfacaGGGcGa aiiOaaaa@3EE6@ are as already defined earlier.

Substituting the first and second boundary conditions (i.e Equations (15) and (16)) into

Equation (19) and solving the resulting simultaneous equations, yields,

a 0 = a 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyypa0JaaGimaa aa@3D05@ , a 3 =2 a 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabg2da9iab gkHiTiaaikdacaWGHbWdamaaBaaaleaapeGaaGinaaWdaeqaaaaa@3CDA@  and a 2 = a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaaI0aaapaqabaaaaa@3B31@ .(20)

In the same way, substituting Equations (17) and Equations (18) into Equation (19), yields,

b 0 = b 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbWdamaaBaaabaWdbiaaicdaa8aabeaapeGaeyypa0Ja amOya8aadaWgaaqaa8qacaaIXaaapaqabaWdbiabg2da9iaaicdaaa a@3D6C@ , b 3 =2 b 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbWdamaaBaaajuaibaWdbiaaiodaa8aabeaajuaGpeGa eyypa0JaeyOeI0IaaGOmaiaadkgapaWaaSbaaKqbGeaapeGaaGinaa qcfa4daeqaaaaa@3EC3@  and b 2 = b 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaapeGa eyypa0JaamOya8aadaWgaaqcfasaa8qacaaI0aaapaqabaaaaa@3C8B@ .(21)

Substituting Equations (20) and (21) into Equation (19), gives the particular deflection equation of an all-round clamped isotropic thin rectangular plate as Equation (22)

w= a 4 b 4 ( R 2 2 R 3 + R 4 )( Q 2 2 Q 3 + Q 4 )     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bGaeyypa0Jaamyya8aadaWgaaqcfasaa8qacaaI0aaa paqabaqcfa4dbiaadkgapaWaaSbaaKqbGeaapeGaaGinaaqcfa4dae qaa8qadaqadaWdaeaapeGaamOua8aadaahaaqcfasabeaapeGaaGOm aaaajuaGcqGHsislcaaIYaGaamOua8aadaahaaqcfasabeaapeGaaG 4maaaajuaGcqGHRaWkcaWGsbWdamaaCaaabeqcfasaa8qacaaI0aaa aaqcfaOaayjkaiaawMcaamaabmaapaqaa8qacaWGrbWdamaaCaaaju aibeqaa8qacaaIYaaaaKqbakabgkHiTiaaikdacaWGrbWdamaaCaaa juaibeqaa8qacaaIZaaaaKqbakabgUcaRiaadgfapaWaaWbaaKqbGe qabaWdbiaaisdaaaaajuaGcaGLOaGaayzkaaGaaiiOaiaacckacaGG GcGaaiiOaaaa@5A7E@

Where,

A= a 4 b 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaeyypa0Jaamyya8aadaWgaaqcfasaa8qacaaI0aaa paqabaqcfa4dbiaadkgapaWaaSbaaKqbGeaapeGaaGinaaqcfa4dae qaaaaa@3DDF@

H=( R 2 2 R 3 + R 4 )( Q 2 2 Q 3 + Q 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGibGaeyypa0ZaaeWaa8aabaWdbiaadkfapaWaaWbaaeqa juaibaWdbiaaikdaaaqcfaOaeyOeI0IaaGOmaiaadkfapaWaaWbaaK qbGeqabaWdbiaaiodaaaqcfaOaey4kaSIaamOua8aadaahaaqabKqb GeaapeGaaGinaaaaaKqbakaawIcacaGLPaaadaqadaWdaeaapeGaam yua8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcqGHsislcaaIYaGa amyua8aadaahaaqcfasabeaapeGaaG4maaaajuaGcqGHRaWkcaWGrb WdamaaCaaajuaibeqaa8qacaaI0aaaaaqcfaOaayjkaiaawMcaaaaa @5040@

Determination of the critical biaxial buckling loads for CCCC Plates

Differentiating Equation (24) with respect to the R-and Q-axes, gave the following results;

( H R ) 2 =( 4 R 2 24 R 3 +52 R 4 48 R 5 +16 R 6 )*( Q 4 4 Q 5 +6 Q 6 4 Q 7 + Q 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2kaadIeaa8aa baWdbiabgkGi2kaadkfaaaaacaGLOaGaayzkaaWdamaaCaaajuaibe qaa8qacaaIYaaaaKqbakabg2da9maabmaapaqaa8qacaaI0aGaamOu a8aadaahaaqcfasabeaapeGaaGOmaaaajuaGcqGHsislcaaIYaGaaG inaiaadkfapaWaaWbaaKqbGeqabaWdbiaaiodaaaqcfaOaey4kaSIa aGynaiaaikdacaWGsbWdamaaCaaajuaibeqaa8qacaaI0aaaaKqbak abgkHiTiaaisdacaaI4aGaamOua8aadaahaaqcfasabeaapeGaaGyn aaaajuaGcqGHRaWkcaaIXaGaaGOnaiaadkfapaWaaWbaaKqbGeqaba WdbiaaiAdaaaaajuaGcaGLOaGaayzkaaGaaiOkamaabmaapaqaa8qa caWGrbWdamaaCaaajuaibeqaa8qacaaI0aaaaKqbakabgkHiTiaais dacaWGrbWdamaaCaaajuaibeqaa8qacaaI1aaaaKqbakabgUcaRiaa iAdacaWGrbWdamaaCaaajuaibeqaa8qacaaI2aaaaKqbakabgkHiTi aaisdacaWGrbWdamaaCaaajuaibeqaa8qacaaI3aaaaKqbakabgUca RiaadgfapaWaaWbaaKqbGeqabaWdbiaaiIdaaaaajuaGcaGLOaGaay zkaaaaaa@6DA6@

( 4 H R 4 )H=24( Q 2 2 Q 3 + Q 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqc fasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacqGHciITcaWGsb WdamaaCaaabeqcfasaa8qacaaI0aaaaaaaaKqbakaawIcacaGLPaaa caWGibGaeyypa0JaaGOmaiaaisdadaqadaWdaeaapeGaamyua8aada ahaaqcfasabeaapeGaaGOmaaaajuaGcqGHsislcaaIYaGaamyua8aa daahaaqcfasabeaapeGaaG4maaaajuaGcqGHRaWkcaWGrbWdamaaCa aabeqcfasaa8qacaaI0aaaaaqcfaOaayjkaiaawMcaaaaa@4FB0@

( 4 H Q 4 )H=24( R 2 2 R 3 + R 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqc fasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacqGHciITcaWGrb WdamaaCaaajuaibeqaa8qacaaI0aaaaaaaaKqbakaawIcacaGLPaaa caWGibGaeyypa0JaaGOmaiaaisdadaqadaWdaeaapeGaamOua8aada ahaaqcfasabeaapeGaaGOmaaaajuaGcqGHsislcaaIYaGaamOua8aa daahaaqcfasabeaapeGaaG4maaaajuaGcqGHRaWkcaWGsbWdamaaCa aabeqcfasaa8qacaaI0aaaaaqcfaOaayjkaiaawMcaaaaa@4FB2@

( 4 H R 2 Q 2 )H=( 2 R 2 16 R 3 +38 R 4 35 R 5 +12 R 6 )( 2 Q 2 16 Q 3 +38 Q 4 35 Q 5 +12 Q 6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqc fasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacqGHciITcaWGsb WdamaaCaaabeqcfasaa8qacaaIYaaaaKqbakabgkGi2kaadgfapaWa aWbaaeqajuaibaWdbiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiaadI eacqGH9aqpdaqadaWdaeaapeGaaGOmaiaadkfapaWaaWbaaKqbGeqa baWdbiaaikdaaaqcfaOaeyOeI0IaaGymaiaaiAdacaWGsbWdamaaCa aabeqcfasaa8qacaaIZaaaaKqbakabgUcaRiaaiodacaaI4aGaamOu a8aadaahaaqcfasabeaapeGaaGinaaaajuaGcqGHsislcaaIZaGaaG ynaiaadkfapaWaaWbaaKqbGeqabaWdbiaaiwdaaaqcfaOaey4kaSIa aGymaiaaikdacaWGsbWdamaaCaaabeqcfasaa8qacaaI2aaaaaqcfa OaayjkaiaawMcaamaabmaapaqaa8qacaaIYaGaamyua8aadaahaaqa bKqbGeaapeGaaGOmaaaajuaGcqGHsislcaaIXaGaaGOnaiaadgfapa WaaWbaaKqbGeqabaWdbiaaiodaaaqcfaOaey4kaSIaaG4maiaaiIda caWGrbWdamaaCaaajuaibeqaa8qacaaI0aaaaKqbakabgkHiTiaaio dacaaI1aGaamyua8aadaahaaqabKqbGeaapeGaaGynaaaajuaGcqGH RaWkcaaIXaGaaGOmaiaadgfapaWaaWbaaKqbGeqabaWdbiaaiAdaaa aajuaGcaGLOaGaayzkaaaaaa@77D3@

Integrating the derivatives of Equations (25) to (2.9) with respect to R and Q from 0-1, yields the results given as Equations (30) to (34):

0 1 0 1 ( H R ) 2 RQ=[ 4 3 24 4 + 52 5 48 6 + 16 7 ]*[ 1 5 4 6 + 6 7 4 8 + 1 9 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8aabaWd biabgUIiYdaadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8 aabaWdbiabgUIiYdaadaqadaWdaeaafaqabeGabaaabaWdbiabgkGi 2kaadIeaa8aabaWdbiabgkGi2kaadkfaaaaacaGLOaGaayzkaaWdam aaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkGi2kaadkfacqGHciIT caWGrbGaeyypa0ZaamWaa8aabaWdbmaalaaapaqaa8qacaaI0aaapa qaa8qacaaIZaaaaiabgkHiTmaalaaapaqaa8qacaaIYaGaaGinaaWd aeaapeGaaGinaaaacqGHRaWkdaWcaaWdaeaapeGaaGynaiaaikdaa8 aabaWdbiaaiwdaaaGaeyOeI0YaaSaaa8aabaWdbiaaisdacaaI4aaa paqaa8qacaaI2aaaaiabgUcaRmaalaaapaqaa8qacaaIXaGaaGOnaa WdaeaapeGaaG4naaaaaiaawUfacaGLDbaacaGGQaWaamWaa8aabaWd bmaalaaapaqaa8qacaaIXaaapaqaa8qacaaI1aaaaiabgkHiTmaala aapaqaa8qacaaI0aaapaqaa8qacaaI2aaaaiabgUcaRmaalaaapaqa a8qacaaI2aaapaqaa8qacaaI3aaaaiabgkHiTmaalaaapaqaa8qaca aI0aaapaqaa8qacaaI4aaaaiabgUcaRmaalaaapaqaa8qacaaIXaaa paqaa8qacaaI5aaaaaGaay5waiaaw2faaaaa@6D6E@ =3.023431587* 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIZaGaaiOlaiaaicdacaaIYaGaaG4maiaaisda caaIZaGaaGymaiaaiwdacaaI4aGaaG4naiaabQcacaaIXaGaaGima8 aadaahaaqabeaapeGaeyOeI0scfaIaaGynaaaaaaa@4401@

0 1 0 1 ( H Q ) 2 RQ=[ 1 5 4 6 + 6 7 4 8 + 1 9 ][ 4 3 24 4 + 52 5 48 6 + 16 7 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8aabaWd biabgUIiYdaadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8 aabaWdbiabgUIiYdaadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi 2kaadIeaa8aabaWdbiabgkGi2kaadgfaaaaacaGLOaGaayzkaaWdam aaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkGi2kaadkfacqGHciIT caWGrbGaeyypa0ZaamWaa8aabaWdbmaalaaapaqaa8qacaaIXaaapa qaa8qacaaI1aaaaiabgkHiTmaalaaapaqaa8qacaaI0aaapaqaa8qa caaI2aaaaiabgUcaRmaalaaapaqaa8qacaaI2aaapaqaa8qacaaI3a aaaiabgkHiTmaalaaapaqaa8qacaaI0aaapaqaa8qacaaI4aaaaiab gUcaRmaalaaapaqaa8qacaaIXaaapaqaa8qacaaI5aaaaaGaay5wai aaw2faamaadmaapaqaa8qadaWcaaWdaeaapeGaaGinaaWdaeaapeGa aG4maaaacqGHsisldaWcaaWdaeaapeGaaGOmaiaaisdaa8aabaWdbi aaisdaaaGaey4kaSYaaSaaa8aabaWdbiaaiwdacaaIYaaapaqaa8qa caaI1aaaaiabgkHiTmaalaaapaqaa8qacaaI0aGaaGioaaWdaeaape GaaGOnaaaacqGHRaWkdaWcaaWdaeaapeGaaGymaiaaiAdaa8aabaWd biaaiEdaaaaacaGLBbGaayzxaaaaaa@6CE1@ =3.023431587*?10 ? (5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIZaGaaiOlaiaaicdacaaIYaGaaG4maiaaisda caaIZaGaaGymaiaaiwdacaaI4aGaaG4naiaacQcacaGG=aGaaGymai aaicdacaGG=aWdamaaCaaabeqaa8qacaGGOaGaeyOeI0scfaIaaGyn aKqbakaacMcaaaaaaa@476F@

0 1 0 1 ( 4 H R 4 )HRQ=[ 24( 1 3 2 4 + 1 5 )( 1 5 4 6 + 6 7 4 8 + 1 9 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8aabaWd biabgUIiYdaadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8 aabaWdbiabgUIiYdaadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi 2+aadaahaaqabKqbGeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacq GHciITcaWGsbWdamaaCaaajuaibeqaa8qacaaI0aaaaaaaaKqbakaa wIcacaGLPaaacaWGibGaeyOaIyRaamOuaiabgkGi2kaadgfacqGH9a qpdaWadaWdaeaapeGaaGOmaiaaisdadaqadaWdaeaapeWaaSaaa8aa baWdbiaaigdaa8aabaWdbiaaiodaaaGaeyOeI0YaaSaaa8aabaWdbi aaikdaa8aabaWdbiaaisdaaaGaey4kaSYaaSaaa8aabaWdbiaaigda a8aabaWdbiaaiwdaaaaacaGLOaGaayzkaaWaaeWaa8aabaWdbmaala aapaqaa8qacaaIXaaapaqaa8qacaaI1aaaaiabgkHiTmaalaaapaqa a8qacaaI0aaapaqaa8qacaaI2aaaaiabgUcaRmaalaaapaqaa8qaca aI2aaapaqaa8qacaaI3aaaaiabgkHiTmaalaaapaqaa8qacaaI0aaa paqaa8qacaaI4aaaaiabgUcaRmaalaaapaqaa8qacaaIXaaapaqaa8 qacaaI5aaaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@69BD@ =1.269841257*?10 ? (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaaiOlaiaaikdacaaI2aGaaGyoaiaaiIda caaI0aGaaGymaiaaikdacaaI1aGaaG4naiaacQcacaGG=aGaaGymai aaicdacaGG=aWdamaaCaaabeqaa8qacaGGOaGaeyOeI0scfaIaaG4m aKqbakaacMcaaaaaaa@4776@

0 1 0 1 ( 4 H Q 4 )HRQ=[ 24( 1 5 4 6 + 6 7 4 8 + 1 9 )( 1 3 2 4 + 1 5 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8aabaWd biabgUIiYdaadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8 aabaWdbiabgUIiYdaadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi 2+aadaahaaqcfasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacq GHciITcaWGrbWdamaaCaaabeqcfasaa8qacaaI0aaaaaaaaKqbakaa wIcacaGLPaaacaWGibGaeyOaIyRaamOuaiabgkGi2kaadgfacqGH9a qpdaWadaWdaeaapeGaaGOmaiaaisdadaqadaWdaeaapeWaaSaaa8aa baWdbiaaigdaa8aabaWdbiaaiwdaaaGaeyOeI0YaaSaaa8aabaWdbi aaisdaa8aabaWdbiaaiAdaaaGaey4kaSYaaSaaa8aabaWdbiaaiAda a8aabaWdbiaaiEdaaaGaeyOeI0YaaSaaa8aabaWdbiaaisdaa8aaba WdbiaaiIdaaaGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaa iMdaaaaacaGLOaGaayzkaaWaaeWaa8aabaWdbmaalaaapaqaa8qaca aIXaaapaqaa8qacaaIZaaaaiabgkHiTmaalaaapaqaa8qacaaIYaaa paqaa8qacaaI0aaaaiabgUcaRmaalaaapaqaa8qacaaIXaaapaqaa8 qacaaI1aaaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@69BC@ =1.269841257*?10 ? (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaaiOlaiaaikdacaaI2aGaaGyoaiaaiIda caaI0aGaaGymaiaaikdacaaI1aGaaG4naiaacQcacaGG=aGaaGymai aaicdacaGG=aWdamaaCaaabeqaa8qacaGGOaGaeyOeI0scfaIaaG4m aKqbakaacMcaaaaaaa@4776@

0 1 0 1 ( 4 H R 2 Q 2 )HRQ= [ 2 3 16 4 + 38 5 36 6 + 12 7 ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8aabaWd biabgUIiYdaadaGfWbqab8aabaWdbiaaicdaa8aabaWdbiaaigdaa8 aabaWdbiabgUIiYdaadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi 2+aadaahaaqcfasabeaapeGaaGinaaaajuaGcaWGibaapaqaa8qacq GHciITcaWGsbWdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkGi 2kaadgfapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaaqcfaOaayjkai aawMcaaiaadIeacqGHciITcaWGsbGaeyOaIyRaamyuaiabg2da9maa dmaapaqaa8qadaWcaaWdaeaapeGaaGOmaaWdaeaapeGaaG4maaaacq GHsisldaWcaaWdaeaapeGaaGymaiaaiAdaa8aabaWdbiaaisdaaaGa ey4kaSYaaSaaa8aabaWdbiaaiodacaaI4aaapaqaa8qacaaI1aaaai abgkHiTmaalaaapaqaa8qacaaIZaGaaGOnaaWdaeaapeGaaGOnaaaa cqGHRaWkdaWcaaWdaeaapeGaaGymaiaaikdaa8aabaWdbiaaiEdaaa aacaGLBbGaayzxaaWdamaaCaaajuaibeqaa8qacaaIYaaaaaaa@65DA@ = 3.628117911*?10 ? (4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIZaGaaiOlaiaaiAdacaaIYaGaaGioaiaaigdacaaIXaGa aG4naiaaiMdacaaIXaGaaGymaiaacQcacaGG=aGaaGymaiaaicdaca GG=aWdamaaCaaabeqaa8qacaGGOaGaeyOeI0scfaIaaGinaKqbakaa cMcaaaaaaa@466B@

Substituting the numerical values obtained from Equations (30) – (34) into Equation (14), gave

N x = D a 2 [ 1.26984125* 10 3 + 2 2 ( 3.628117911* 10 4 )+ 1 4 ( 1.269841257* 10 3 ) ] [ 3.023431587* 10 5 + k 2 ( 3.02341587* 10 5 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhaaKqba+aabeaapeGa eyypa0JaeyOeI0YaaSaaa8aabaWdbmaalaaapaqaa8qacaWGebaapa qaa8qacaWGHbWdamaaCaaajuaibeqaa8qacaaIYaaaaaaajuaGdaWa daWdaeaapeGaaGymaiaac6cacaaIYaGaaGOnaiaaiMdacaaI4aGaaG inaiaaigdacaaIYaGaaGynaiaabQcacaaIXaGaaGima8aadaahaaqa beaapeGaeyOeI0scfaIaaG4maaaajuaGcqGHRaWkdaWcaaWdaeaape GaaGOmaaWdaeaapeGaeyyhIu7damaaCaaajuaibeqaa8qacaaIYaaa aaaajuaGdaqadaWdaeaapeGaaG4maiaac6cacaaI2aGaaGOmaiaaiI dacaaIXaGaaGymaiaaiEdacaaI5aGaaGymaiaaigdacaqGQaGaaGym aiaaicdapaWaaWbaaeqabaWdbiabgkHiTKqbGiaaisdaaaaajuaGca GLOaGaayzkaaGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiab g2Hi1+aadaahaaqcfasabeaapeGaaGinaaaaaaqcfa4aaeWaa8aaba WdbiaaigdacaGGUaGaaGOmaiaaiAdacaaI5aGaaGioaiaaisdacaaI XaGaaGOmaiaaiwdacaaI3aGaaeOkaiaaigdacaaIWaWdamaaCaaabe qaa8qacqGHsisljuaicaaIZaaaaaqcfaOaayjkaiaawMcaaaGaay5w aiaaw2faaaWdaeaapeWaamWaa8aabaWdbiaaiodacaGGUaGaaGimai aaikdacaaIZaGaaGinaiaaiodacaaIXaGaaGynaiaaiIdacaaI3aGa aiOkaiaaigdacaaIWaWdamaaCaaabeqaa8qacqGHsisljuaicaaI1a aaaKqbakabgUcaRmaalaaapaqaa8qacaWGRbaapaqaa8qacqGHDisT paWaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqbaoaabmaapaqaa8qaca aIZaGaaiOlaiaaicdacaaIYaGaaG4maiaaisdacaaIXaGaaGynaiaa iIdacaaI3aGaaiOkaiaaigdacaaIWaWdamaaCaaabeqaa8qacqGHsi sljuaicaaI1aaaaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faaaaa aaa@9751@

The Equation (35) is the expression for the critical buckling load of a biaxially loaded CCCC plate. While the Equation (36)

F= [ 1.26984125* 10 3 + 2 2 ( 3.628117911* 10 4 )+ 1 4 ( 1.269841257* 10 3 ) ] [ 3.023431587* 10 5 + k 2 ( 3.02341587* 10 5 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbGaeyypa0JaeyOeI0YaaSaaa8aabaWdbmaadmaapaqa a8qacaaIXaGaaiOlaiaaikdacaaI2aGaaGyoaiaaiIdacaaI0aGaaG ymaiaaikdacaaI1aGaaeOkaiaaigdacaaIWaWdamaaCaaabeqaa8qa cqGHsislcaaIZaaaaiabgUcaRmaalaaapaqaa8qacaaIYaaapaqaa8 qacqGHDisTpaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqbaoaabmaa paqaa8qacaaIZaGaaiOlaiaaiAdacaaIYaGaaGioaiaaigdacaaIXa GaaG4naiaaiMdacaaIXaGaaGymaiaabQcacaaIXaGaaGima8aadaah aaqabeaapeGaeyOeI0scfaIaaGinaaaaaKqbakaawIcacaGLPaaacq GHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaapeGaeyyhIu7damaaCaaa beqcfasaa8qacaaI0aaaaaaajuaGdaqadaWdaeaapeGaaGymaiaac6 cacaaIYaGaaGOnaiaaiMdacaaI4aGaaGinaiaaigdacaaIYaGaaGyn aiaaiEdacaqGQaGaaGymaiaaicdapaWaaWbaaeqabaWdbiabgkHiTK qbGiaaiodaaaaajuaGcaGLOaGaayzkaaaacaGLBbGaayzxaaaapaqa a8qadaWadaWdaeaapeGaaG4maiaac6cacaaIWaGaaGOmaiaaiodaca aI0aGaaG4maiaaigdacaaI1aGaaGioaiaaiEdacaGGQaGaaGymaiaa icdapaWaaWbaaeqabaWdbiabgkHiTiaaiwdaaaGaey4kaSYaaSaaa8 aabaWdbiaadUgaa8aabaWdbiabg2Hi1+aadaahaaqcfasabeaapeGa aGOmaaaaaaqcfa4aaeWaa8aabaWdbiaaiodacaGGUaGaaGimaiaaik dacaaIZaGaaGinaiaaigdacaaI1aGaaGioaiaaiEdacaGGQaGaaGym aiaaicdapaWaaWbaaeqabaWdbiabgkHiTKqbGiaaiwdaaaaajuaGca GLOaGaayzkaaaacaGLBbGaayzxaaaaaaaa@9003@

Is the expression for the biaxial coefficient of an all-round CCCC plate.

Results and discussion

The values of the plate aspect ratios (1-2) and the constant “k” varying from 0-1, at intervals of 0.1, were substituted into Equation (36). The results of the critical buckling load coefficients for a biaxially compressed all-round clamped (CCCC) isotropic thin rectangular plate were obtained and presented in Table 1.

Table 1 Critical Buckling Load Coefficients for biaxially loaded CCCC Plates

Where N xi ( i=1,2,310 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaaBaaajuaibaWdbiaadIhacaWGPbaajuaGpaqa baWdbmaabmaapaqaa8qacaWGPbGaeyypa0JaaGymaiaacYcacaaIYa
GaaiilaiaaiodacqGHMacVcaaIXaGaaGimaaGaayjkaiaawMcaaaaa @44B1@ are the critical buckling load coefficients at k=0.i. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaeyypa0JaaGimaiaac6cacaWGPbGaaiOlaaaa@3BA7@

(where k=Constant, relating Ny and Nxi: and Nxi The Critical buckling load coefficients at various fractions of k )

The results presented in Table 1 were plotted as shown on Figure 2. From Figure 2 and Table 1, it could be observed that, as the aspect ratios increase from one to two, the buckling load coefficients of the CCCC plates reduces. This is because as the aspect ratio increases, the plate begins to behave as a slender column and hence loses its ability to resist load in the y- direction, which causes it to buckle faster; hence, the reduction in the buckling load. It is also observed that as the forces in the y-direction increased (i.e. as “k” increases), the buckling coefficient reduces, this is because as the loads in the y-direction of the plate, increased (with loads applied on the x-axis), the plate becomes weaker and less resistant to applied loads and hence, it buckles faster.

Figure 2 Graph of Critical Buckling Load Coefficients Versus Aspect Ratios of a Biaxially Loaded all round clamped Plate.

Biaxial Buckling Load Coefficient for CCCC Plate

Aspect Ratios

Ibearugbulem et al. (2014)

Present Study

Percentage Difference

1

108.667

108.001

0.61298

1.1

91.082

90.5218

0.61511

1.2

79.415

78.9217

0.62117

1.3

71.3565

70.9069

0.63005

1.4

65.5979

65.1782

0.63984

1.5

61.3621

60.9633

0.64995

1.6

58.167

57.784

0.65847

1.7

55.706

55.3335

0.66876

1.8

53.773

53.4086

0.67766

1.9

52.229

51.8713

0.68491

2

50.979

50.6253

0.69389

Table 2 Comparison of the Biaxial Buckling Load Coefficients Obtained in this work for CCCC Rectangular Plates under Uniform Unilateral Pressure (i.e. at K= 0), with those of Ibearugbulem et al8

The results of the present study were compared with that obtained by Ibearugbulem et al12 at k=0 (uniaxial buckling only, of CCCC plates) as presented on Table 2. From the Table 2, it is seen that, the results of this present work agrees very closely with established results of uniaxially loaded CCCC plates subject to uniform pressure along the x-axis. (i.e. at k = 0), for different aspect ratios. This therefore, validates the results of the critical buckling load coefficients for the other k values (presented on Table 1) for which there are no existing results in literature.

Conclusion

From the study, the following conclusions have been, drawn: The equation for the determination of the critical buckling loads for a biaxially loaded all- round clamped thin rectangular isotropic plate, for all aspect ratios and k-values, has been derived in this work. The critical buckling load coefficients for a biaxially loaded thin rectangular isotropic plates, have been determined for different aspect ratios and k-values. Given that the results obtained from this work agrees with the results of Ibearugbulem11 at k = 0, it therefore follows that, the results obtained in this work for other k values (for which there are no other existing results to compare with, in literature), are also correct.

The polynomial shape function used in this work (based on the Tailor-Mclaurin series) can be said to have accurately defined the plate’s deformed shape, given the high accuracy of the buckling load coefficients obtained. Given the difficulty in the use of the trigonometric shape functions to determine the critical buckling load coefficients of an all-round clamped thin rectangular isotropic plate, the use of the equations and tables developed in this work is recommended for a quick and easy analysis of the buckling loads of such plates, for other aspect ratios which are not covered in this work.

Acknowledgements

None.

Conflict of interest

The author declares no conflict of interest.

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