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Aeronautics and Aerospace Open Access Journal

Technical Paper Volume 2 Issue 1

Effect of elastic foundations on stochastic thermally induced post buckling response of functionally graded materials plates with uncertain material properties

Rajesh Kumar

School of Mechanical Engineering, Jimma University, Ethiopia

Correspondence: Rajesh Kumar, Professor, School of Mechanical Engineering, Jimma Institute of Technology, Jimma University, Ethiopia, Tel +251909462675

Received: May 11, 2017 | Published: January 30, 2018

Citation: Kumar R. Effect of elastic foundations on stochastic thermally induced post buckling response of functionally graded materials plates with uncertain material properties. Aeron Aero Open Access J. 2018;2(1):11-28. DOI: 10.15406/aaoaj.2018.02.00024

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Abstract

This paper presents, the effect of elastic foundations on stochastic thermally induced post buckling response of functionally graded materials plates with uncertain material properties. Foundation stiffness parameters, material properties of each constituent's material, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C0 continuity. A direct iterative based nonlinear finite element method (DISFEM) is successfully applied for FGMs plate to compute the mean and COV of the post buckling response of the FGM plates resting on elastic foundations. The results obtained by present outlined approach have been validated with the results available in literature and independent Monte Carlo simulation (MCS).

Keywords: fgm plates, elastic foundations, stochastic finite element method, uncertain material properties

Introduction

The abrupt change in material properties across the interface between different materials can result in large inter-laminar stresses leading to delimitation, cracking, and other damage mechanisms which result from the abrupt change of the mechanical properties at the interface between the layers. Initially the uncertainty in material properties at micro level is finally reflected as macro level. Existing uncertain variations in material parameters may have significantly affects on fundamental structural characteristic and consequently these must affect the final design. Therefore, for accurate prediction of system behavior required for sensitive application favours a probabilistic/stochastic approach. The stochastic approach is a mathematical tool capable of handing uncertainties in the material properties to compute the statistic of structural response for example, post buckling response. FGMs are composite and microscopically heterogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other.1 This is achieved by gradually varying the volume fraction of the constituent materials. Typically, these materials are made from a mixture of ceramics and metal or a combination of different materials. The ceramic constituent of the material provides the high-temperature resistance due to its low thermal conductivity and protects the metal from oxidation. The structures subjected to thermal loading with spatial gradients in composition and microstructure is of considerable interest in numerous technological areas such as tribiology, optoelectronics, biomechanics, nanotechnology and high temperature technology. The graded transition in composition across an interface of two materials (metal and ceramics) can essentially reduce the thermal stresses and stress concentration at intersection with free surfaces. Due to the boundary constraints, compressive stresses acting on edges of the component arise due to thermal loading are induced, which may cause buckling, especially in thin walled structures. Buckling and post-buckling characteristics are one of the major design criteria for plates/panels for their optimal usage. Hence, it is important, to study the buckling and post-buckling characteristics of FGM plates under thermal loading for accurate, efficient and reliable design.

Large number of literature is available on deterministic analysis of FGM plate with mechanical thermal and thermo-mechanical buckling. In this direction, Feldman & Aboudi2 studied the elastic buckling of functionally graded plate under in-plane compressive loadings with both simply supported and clamped edges. Praveen & Reddy3 investigated the response of functionally graded ceramic-metal plate, using finite element method. Reddy & Chin4 investigated the thermoelastical analysis of functionally graded cylinders and plates. Reddy5 presented the theoretical and finite element formulations for linear and nonlinear thermo-mechanical response of FGM plates employing higher order shear deformation theory. Vel & Batra6 presented an exact solution for the thermo-elastic deformation of functionally graded thick rectangular simply supported plates subjected to thermo-mechanical loadings. Javaheri & Eslami7,8 obtained the buckling of the FGM plates for uniform in plane compressive loading and thermal loading using variational approach using classical plate theory (CLT). Najafizadeh & Eslami9 studied buckling analysis of circular plates of functionally graded materials under uniform radial compression. Liew et al.10 presented post-buckling behavior of the piezoelectric FGM plate using Galerkin-differential quadrature iteration method based on Reddy’s higher order shear deformation theory (HSDT).

Shen11 presented the post-buckling analysis of functionally graded cylindrical panel under lateral pressure and temperature using Reddy’s higher order shear deformation theory and von-Karman-Donnell type nonlinearity. Liew et al.12 presented the post-buckling behavior of the piezoelectric FGM plate using Fourier series using FSDT. Liew et al.13 studied the post buckling of piezoelectric FGM plates subjected to thermo-electro-mechanical loading. Shen14 presented the post-buckling response of axially loaded FG cylindrical panels subjected to thermal loading using semi-analytical approach based on Reddy’s HSDT. Yang & Shen15 evaluated nonlinear bending and post-buckling response of the FG rectangular plate using a semi-analytical approach with Reddy’s HSDT. Woo et al.16 examined the post-buckling of functionally graded material plates and shallow cylindrical shells under thermo-mechanical loading using an analytical solution with mixed Fourier series. Ma & Wang17,18 presented bending and buckling of FGM circular plate based on CLPT and third order plate theory. Lanhe19 presented a semi-analytical solution for the thermal buckling of FGM rectangular simply supported plate subjected to uniform temperature rise and gradient through the thickness of plate, using the first-order shear deformation theory. Die20 investigated the thermo mechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method. Bhangale et al.21 studied the linear thermo-elastic buckling and free vibration behavior of functionally graded truncated conical shells Wu et al.22 evaluated the post buckling response of functionally graded materials plates based on the CLT with von-Karman nonlinear kinematics with analytically approach and Chebyshev polynomials.

All the above literatures are based on the assumptions of the complete determinacy of structural parameters. In the deterministic analysis of structures, the variations in the material parameters are neglected and mean value of system parameters are used in the analysis, which gives only mean response and unaccounted the deviation caused by inherent random material properties.23,24 Roshan studied the buckling and vibration of functionally graded non-uniform circular plates resting on Winkler foundation. A part from other side, the mechanical properties depends on a wide variety of variable at micro level, for example the material properties of constituent materials, variation in FGM thickness, volume fraction index etc., These variables are statistical in nature; therefore, the mechanical properties of a FGM structures should be quantified probabilistically. The influence of these micro level variables on the macro level effective properties of FGM has been studied numerically using proposed methods. These variations ultimately lead to the variation in the response of the structure. Relatively little efforts have been made in the past by the researchers and investigators on the prediction of response statistics of the structures made of laminated composites and FGMs plates having random system properties. In this direction, Singh et al.25‒34 invested the post buckling response and natural frequencies of laminated composite plate with random system properties with or without elastic foundation.

Further, Yang et al.27,28 evaluated the second-order statistics for elastic buckling of FGM plates with randomness in the material properties using stochastic finite element method via first-order shear deformation theory. Onkar et al.29‒31 investigated the generalized buckling of laminated composite plate with random material properties using classical plate theory (CLT) combined with FOPT. Kirtipornchai et al.30 studied the random vibration of functionally graded laminates in thermal environments composed by the third order shear deformation theory with random thermo mechanical properties using the first order perturbation technique incorporating mixed type and semi-analytical approach to derive the standard eigen value problem. Lal et al.32‒39 investigated the effect of random system properties on buckling of laminated composite plates supported with/without elastic foundation with and without subjected to thermal environment using HSDT based C0 linear and nonlinear FEM combined with and without direct iterative method in conjunction with FOPT. Lal A et al.40 investigated the post buckling response of functionally graded plate subjected to mechanical and thermal loadings with random material properties Jagtap et al.41 investigated the thermo mechanical elastic post buckling of functionally graded material plate with random material properties Lal A & Singh42 analysed the Stochastic mechanical and thermal post buckling response of functionally graded material plates with circular and square holes having material randomness.

It is evident from the available literatures that the studies of stochastic buckling response of geometrically nonlinear FGM plates resting on elastic foundations, subjected to thermal loading involving randomness in material properties of constituent materials and foundation stiffness parameters using computationally efficient direct iterative based C0 nonlinear FEM in combined with mean cantered FOPT not dealt by the researchers. In the present study, the paper is organised as follows. Section 1 gives the brief introduction of related problem with justification and literature surveys based on deterministic and probabilistic analysis. The brief description of the geometric configuration, material properties, foundation stiffness parameters, coefficients of thermal expansion and theoretical formulations of the problem using finite element methods are given in section 2 and 3. In section 4 governing equation of the problem is presented. The solution methodology of probabilistic finite element is presented in section5.Section 6 explains the results and discussion followed by validation studies and section 7 accomplished the conclusions. The numerical results are presented in terms of tables which can suit the benchmark of future work.43‒46

Formulations

Consider a square FGM plate consist of metal and ceramic at the top and bottom layer of length a, width b, and total thickness h, defined in (x, y, z) system with x- and -y axes located in the middle plane and its origin placed at the corner of the plate. Let ( u ¯ , v ¯ , w ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqaqpepC0xbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGabmyDayaaraGaaiilaiaaysW7ceWG2bGbaebacaGGSaGa aGjbVlqadEhagaqeaaGccaGLOaGaayzkaaaaaa@3FC3@ be the displacement parallel to the (x, y, z) axes respectively as shown in Figure 1. The properties of the FGM plate are assumed to be vary through the thickness of the plate only, such that the top surface z=h/2 is ceramic-rich and the bottom surface z=-h/2 is metal reach. The effective thermal properties of the FGMs plate.47

E( z,T )= E b ( T )+[ E t ( T ) E b ( T ) ] V c ( z ) α( z,T )= α b ( T )+[ α t ( T ) α b ( T ) ] V c ( z ) ρ( z,T )= ρ b +[ ρ t ρ b ] V c ( z ) k( z,T )= k b +[ k t k b ] V c ( z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aadweajuaGdaqadaGcbaqcLbsacaWG6bGaaiilaiaadsfaaOGaayjk aiaawMcaaKqzGeGaeyypa0JaamyraSWaaSbaaeaajugWaiaadkgaaS qabaqcfa4aaeWaaOqaaKqzGeGaamivaaGccaGLOaGaayzkaaqcLbsa cqGHRaWkjuaGdaWadaGcbaqcLbsacaWGfbqcfa4aaSbaaSqaaKqzGe GaamiDaaWcbeaajuaGdaqadaGcbaqcLbsacaWGubaakiaawIcacaGL PaaajugibiabgkHiTiaadwealmaaBaaabaqcLbmacaWGIbaaleqaaK qbaoaabmaakeaajugibiaadsfaaOGaayjkaiaawMcaaaGaay5waiaa w2faaKqzGeGaamOvaSWaaSbaaeaajugWaiaadogaaSqabaqcfa4aae WaaOqaaKqzGeGaamOEaaGccaGLOaGaayzkaaaabaqcLbsacqaHXoqy juaGdaqadaGcbaqcLbsacaWG6bGaaiilaiaadsfaaOGaayjkaiaawM caaKqzGeGaeyypa0JaeqySde2cdaWgaaqaaKqzadGaamOyaaWcbeaa juaGdaqadaGcbaqcLbsacaWGubaakiaawIcacaGLPaaajugibiabgU caRKqbaoaadmaakeaajugibiabeg7aHTWaaSbaaeaajugWaiaadsha aSqabaqcfa4aaeWaaOqaaKqzGeGaamivaaGccaGLOaGaayzkaaqcLb sacqGHsislcqaHXoqylmaaBaaabaqcLbmacaWGIbaaleqaaKqbaoaa bmaakeaajugibiaadsfaaOGaayjkaiaawMcaaaGaay5waiaaw2faaK qzGeGaamOvaSWaaSbaaeaajugWaiaadogaaSqabaqcfa4aaeWaaOqa aKqzGeGaamOEaaGccaGLOaGaayzkaaaabaqcLbsacqaHbpGCjuaGda qadaGcbaqcLbsacaWG6bGaaiilaiaadsfaaOGaayjkaiaawMcaaKqz GeGaeyypa0JaeqyWdi3cdaWgaaqaaKqzadGaamOyaaWcbeaajugibi abgUcaRKqbaoaadmaakeaajugibiabeg8aYTWaaSbaaeaajugWaiaa dshaaSqabaqcLbsacqGHsislcqaHbpGClmaaBaaabaqcLbmacaWGIb aaleqaaaGccaGLBbGaayzxaaqcLbsacaWGwbWcdaWgaaqaaKqzadGa am4yaaWcbeaajuaGdaqadaGcbaqcLbsacaWG6baakiaawIcacaGLPa aaaeaajugibiaadUgajuaGdaqadaGcbaqcLbsacaWG6bGaaiilaiaa dsfaaOGaayjkaiaawMcaaKqzGeGaeyypa0Jaam4AaSWaaSbaaeaaju gWaiaadkgaaSqabaqcLbsacqGHRaWkjuaGdaWadaGcbaqcLbsacaWG RbWcdaWgaaqaaKqzadGaamiDaaWcbeaajugibiabgkHiTiaadUgalm aaBaaabaqcLbmacaWGIbaaleqaaaGccaGLBbGaayzxaaqcLbsacaWG wbWcdaWgaaqaaKqzadGaam4yaaWcbeaajuaGdaqadaGcbaqcLbsaca WG6baakiaawIcacaGLPaaaaaaa@CF20@     (1)

Where, t and b represent to the ceramic and metal constituents, respectively. With E, α, ρ and k are the effective young modulus, thermal expansion coefficient, density and thermal conductivity respectively. The ceramic volume fraction index VC is the function of coordinate in the thickness direction, z and is given by

V C ( z )= ( 0.5+ z h ) n ,h/2zh/2,0n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb WcdaWgaaqaaKqzadGaam4qaaWcbeaajuaGdaqadaGcbaqcLbsacaWG 6baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaabmaakeaajugibi aaicdacaGGUaGaaGynaiabgUcaRKqbaoaalaaabaqcLbsacaWG6baa juaGbaqcLbsacaWGObaaaaGccaGLOaGaayzkaaWcdaahaaqabeaaju gWaiaad6gaaaqcLbsacaGGSaGaaGjbVlaaysW7caaMe8UaaGPaVlab gkHiTiaadIgacaGGVaGaaGOmaiabgsMiJkaadQhacqGHKjYOcaWGOb Gaai4laiaaikdacaGGSaGaaGPaVlaaysW7caaIWaGaeyizImQaamOB aiablQNiWjabg6HiLcaa@66DD@      (2)

Where, n is the power index law and is always positive. For n =0, the plate is fully ceramic. With n = 1, the composition of ceramic and metal is linear. The Poisson’s ratio v depends weakly on temperature change and is assumed to constant.

Displacement field model

In the present study the higher order shear deformation theory with C° continuity (modified of C1 continuity proposed by Reddy’s [1] with extra degree of freedom) has been employed,25‒33

u ¯ =u+z ψ x z 3 4/3 h 2 ( ψ x +w/x)=u+ f 1 (z) ψ x + f 2 (z)w/x, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamyDaaaacqGH9aqpcaWG1bGaey4kaSIaamOEaiabeI8a 5TWaaSbaaOqaaKqzadGaamiEaaWcbeaajugibiabgkHiTiaadQhalm aaCaaabeqaaKqzadGaaG4maaaajugibiaaisdacaGGVaGaaG4maiaa dIgalmaaCaaabeqaaKqzadGaaGOmaaaajugibiaacIcacqaHipqElm aaBaaakeaajugWaiaadIhaaSqabaqcLbsacqGHRaWkcqGHciITcaWG 3bGaai4laiabgkGi2kaadIhacaGGPaGaeyypa0JaamyDaiabgUcaRi aadAgalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaaiikaiaadQha caGGPaGaeqiYdK3cdaWgaaGcbaqcLbmacaWG4baaleqaaKqzGeGaey 4kaSIaamOzaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacaGGOaGa amOEaiaacMcacqGHciITcaWG3bGaai4laiabgkGi2kaadIhacaGGSa aaaa@723C@

v ¯ =v+z ψ y z 3 4/3 h 2 ( ψ y +w/y)=v+ f 1 (z) ψ y + f 2 (z)w/y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamODaaaacqGH9aqpcaWG2bGaey4kaSIaamOEaiabeI8a 5TWaaSbaaOqaaKqzadGaamyEaaWcbeaajugibiabgkHiTiaadQhalm aaCaaabeqaaKqzadGaaG4maaaajugibiaaisdacaGGVaGaaG4maiaa dIgalmaaCaaabeqaaKqzadGaaGOmaaaajugibiaacIcacqaHipqElm aaBaaakeaajugWaiaadMhaaSqabaqcLbsacqGHRaWkcqGHciITcaWG 3bGaai4laiabgkGi2kaadMhacaGGPaGaeyypa0JaamODaiabgUcaRi aadAgalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaaiikaiaadQha caGGPaGaeqiYdK3cdaWgaaGcbaqcLbmacaWG5baaleqaaKqzGeGaey 4kaSIaamOzaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacaGGOaGa amOEaiaacMcacqGHciITcaWG3bGaai4laiabgkGi2kaadMhacaGGSa aaaa@7244@ and w ¯ =w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaam4DaaaacqGH9aqpcaWG3baaaa@3A2C@     (3)

Where u ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadwhaaaaaaa@383B@ , v ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadAhaaaaaaa@383C@ and w ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiaadEhaaaaaaa@383D@ denote the displacements of a point along the (x, y, z) coordinates, u, v, and w are corresponding displacements of a point on the mid plane. ψ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK xcfa4aaSbaaSqaaKqzadGaamiEaaWcbeaaaaa@3B56@ and ψ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK 3cdaWgaaqaaKqzadGaamyEaaWcbeaaaaa@3AC9@ are the rotations of normal to the mid plane about the y-axis and x-axis respectively, with θ x = w/ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeea0dXdh9vqai=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeI 7aXTWaaSbaaeaajugWaiaadIhaaSqabaqcLbsacqGH9aqpjuaGdaWc gaGcbaqcLbsacqGHciITcaWG3baakeaajugibiabgkGi2kaadIhaaa aaaa@4382@ and

θ y = w/ y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeea0dXdh9vqai=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeI 7aXTWaaSbaaeaajugWaiaadMhaaSqabaqcLbsacqGH9aqpjuaGdaWc gaGcbaqcLbsacqGHciITcaWG3baakeaajugibiabgkGi2kaadMhaaa aaaa@4384@ f 1 ( z )= C 1 z C 2 z 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=NqzGeGaam OzaSWaaSbaaeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqzGeGa amOEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaWGdbWcdaWgaaqaaK qzadGaaGymaaWcbeaajugibiaadQhacqGHsislcaWGdbWcdaWgaaqa aKqzadGaaGOmaaWcbeaajugibiaadQhalmaaCaaajyaGbeqaaKqzad GaaG4maaaaaaa@4CA2@ f 2 ( z )= C 4 z 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=NqzGeGaam OzaSWaaSbaaeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqzGeGa amOEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcqGHsislcaWGdbWcda WgaaqaaKqzadGaaGinaaWcbeaajugibiaadQhalmaaCaaabeqaaKqz adGaaG4maaaaaaa@47A1@ with C 1 =1, C 2 = C 4 = 4 h 2 /3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiaado ealmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0JaaGymaiaa cYcacaaMb8UaaGjcVlaaykW7caWGdbWcdaWgaaqaaKqzadGaaGOmaa Wcbeaajugibiabg2da9iaadoealmaaBaaabaqcLbmacaaI0aaaleqa aKqzGeGaeyypa0tcfa4aaSGbaOqaaKqzGeGaaGinaiaadIgalmaaCa aabeqaaKqzadGaaGOmaaaaaOqaaKqzadGaaG4maaaaaaa@525A@ .

The displacement vector for the modified models is

{ Λ }= [ u v w θ y θ x ψ y ψ x ] T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiabfU5ambGccaGL7bGaayzFaaqcLbsacqGH9aqpjuaG daWadaGcbaqcLbsafaqabaqahaaaaOqaaKqzGeGaamyDaaGcbaqcLb sacaWG2baakeaajugibiaadEhaaOqaaKqzGeGaeqiUde3cdaWgaaqa aKqzadGaamyEaaWcbeaaaOqaaKqzGeGaeqiUde3cdaWgaaqaaKqzad GaamiEaaWcbeaaaOqaaKqzGeGaeqiYdK3cdaWgaaqaaKqzadGaamyE aaWcbeaaaOqaaKqzGeGaeqiYdK3cdaWgaaqaaKqzadGaamiEaaWcbe aaaaaakiaawUfacaGLDbaalmaaCaaabeqaaKqzadGaamivaaaaaaa@5AC5@  (4)

Strain displacement relations

For the structures considered here, the relevant strain vector consisting of strains in terms of mid-plane deformation, rotation of normal and higher order terms associated with the displacement for isotropic layer are as.35

{ ε }={ ε l }+{ ε nl }{ ε ¯ t } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiabew7aLbGccaGL7bGaayzFaaqcLbsacqGH9aqpjuaG daGadaGcbaqcLbsacqaH1oqzlmaaBaaabaqcLbmacaWGSbaaleqaaa GccaGL7bGaayzFaaqcLbsacqGHRaWkjuaGdaGadaGcbaqcLbsacqaH 1oqzlmaaBaaabaqcLbmacaWGUbGaamiBaaWcbeaaaOGaay5Eaiaaw2 haaKqzGeGaeyOeI0scfa4aaiWaaOqaaKqzGeGafqyTduMbaebalmaa BaaabaqcLbmacaWG0baaleqaaaGccaGL7bGaayzFaaaaaa@57F4@     (5)

where { ε l } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiabew7aLTWaaSbaaeaajugWaiaadYgaaSqabaaakiaa wUhacaGL9baaaaa@3E98@ , { ε nl } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiabew7aLTWaaSbaaeaajugWaiaad6gacaWGSbaaleqa aaGccaGL7bGaayzFaaaaaa@3F8B@ and { ε ¯ t } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiqbew7aLzaaraWcdaWgaaqaaKqzadGaamiDaaWcbeaa aOGaay5Eaiaaw2haaaaa@3EB8@ are the linear and nonlinear strain (von-Karman sense) vectors, thermal strain vector, respectively. The thermal strain vector { ε ¯ t } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiqbew7aLzaaraWcdaWgaaqaaKqzadGaamiDaaWcbeaa aOGaay5Eaiaaw2haaaaa@3EB8@ given in Equation (5) is expressed as.23

{ ε ¯ t }={ { ε ¯ x } { ε ¯ y } { ε ¯ xy } { ε ¯ yz } { ε ¯ zx } }=ΔT{ α 1 α 2 α 12 0 0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiqbew7aLzaaraWcdaWgaaqaaKqzadGaamiDaaWcbeaa aOGaay5Eaiaaw2haaKqzGeGaeyypa0tcfa4aaiWaaOqaaKqzGeqbae qabqqaaaaakeaajuaGdaGadaGcbaqcLbsacuaH1oqzgaqeaKqbaoaa BaaaleaajugWaiaadIhaaSqabaaakiaawUhacaGL9baaaeaajuaGda GadaGcbaqcLbsacuaH1oqzgaqeaSWaaSbaaeaajugWaiaadMhaaSqa baaakiaawUhacaGL9baaaeaajuaGdaGadaGcbaqcLbsacuaH1oqzga qeaSWaaSbaaeaajugWaiaadIhacaWG5baaleqaaaGccaGL7bGaayzF aaaajugibqaabeGcbaqcfa4aaiWaaOqaaKqzGeGafqyTduMbaebaju aGdaWgaaWcbaqcLbmacaWG5bGaamOEaaWcbeaaaOGaay5Eaiaaw2ha aaqaaKqbaoaacmaakeaajugibiqbew7aLzaaraqcfa4aaSbaaSqaaK qzadGaamOEaiaadIhaaSqabaaakiaawUhacaGL9baaaaaaaiaawUha caGL9baajugibiabg2da9iabfs5aejaadsfajuaGdaGadaqcLbsaea qabOqaaKqzGeqbaeqabqqaaaaakeaajugibiabeg7aHTWaaSbaaeaa jugWaiaaigdaaSqabaaakeaajugibiabeg7aHTWaaSbaaeaajugWai aaikdaaSqabaaakeaajugibiabeg7aHTWaaSbaaeaajugWaiaaigda caaIYaaaleqaaaGcbaqcLbsacaaIWaaaaaGcbaqcLbsacaaIWaaaaO Gaay5Eaiaaw2haaaaa@86DF@  (6)

The linear { ε l } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=Nqbaoaacm aakeaajugibiabew7aLTWaaSbaaeaajugWaiaadYgaaSqabaaakiaa wUhacaGL9baaaaa@3E98@ strain vector can be obtained by32 and the nonlinear strain can be written as.35

ε nl = 1 2 [ A nl ]{ ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaba9FcLbsacq aH1oqzlmaaBaaabaqcLbmacaWGUbGaamiBaaWcbeaajugibiabg2da 9KqbaoaalaaabaGaaGymaaqaaiaaikdaaaqcLbsacaaMc8Ecfa4aam WaaOqaaKqzGeGaamyqaSWaaSbaaeaajugWaiaad6gacaWGSbaaleqa aaGccaGLBbGaayzxaaqcLbsacaaMc8Ecfa4aaiWaaOqaaKqzGeGaeq y1dygakiaawUhacaGL9baaaaa@515B@    (7)

where

A nl = 1 2 [ w ,x 0 0 w ,y w ,x w ,y 0 0 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaba9FcLbsaca WGbbWcdaWgaaqaaKqzadGaamOBaiaadYgaaSqabaqcLbsacqGH9aqp juaGdaWcaaqaaiaaigdaaeaacaaIYaaaamaadmaakeaajugibuaabe qafiaaaaGcbaqcLbsacaWG3bqcfa4aaSbaaSqaaKqzGeGaaiilaKqz adGaamiEaaWcbeaaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaadEhajuaGdaWgaaWcbaqcLbsacaGGSaqcLbmacaWG5baa leqaaaGcbaqcLbsacaWG3bqcfa4aaSbaaSqaaKqzGeGaaiilaKqzad GaamiEaaWcbeaaaOqaaKqzGeGaam4DaKqbaoaaBaaaleaajugibiaa cYcajugWaiaadMhaaSqabaaakeaajugibiaaicdaaOqaaKqzGeGaaG imaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaaaakiaawUfacaGL Dbaaaaa@619B@ and ϕ={ w ,x w ,y }, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaba9FcLbsacq aHvpGzcqGH9aqpjuaGdaGadaGcbaqcLbsafaqabeGabaaakeaajugi biaadEhajuaGdaWgaaWcbaqcLbsacaGGSaqcLbmacaWG4baaleqaaa GcbaqcLbsacaWG3bqcfa4aaSbaaSqaaKqzGeGaaiilaKqzadGaamyE aaWcbeaaaaaakiaawUhacaGL9baajugibiaacYcaaaa@4B35@ (8)

where α1, α2 and α12 are coefficients of thermal expansion in the x, y, z direction respectively which can be obtained from the thermal coefficients in the longitudinal (αl) and transverse (α2) directions of the ceramic and metal using transformation matrix and ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=NqzGeGaeu iLdqKaamivaaaa@39E7@ is the uniform temperature change.

The temperature field for uniform temperature change is expressed as

T(z)= T 0 +( T t T b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9yqqrpepu0dbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaba9FcLbsaca WGubGaaiikaiaadQhacaGGPaGaeyypa0JaamivaSWaaSbaaeaajugW aiaaicdaaSqabaqcLbsacqGHRaWkcaGGOaGaamivaSWaaSbaaeaaju gWaiaadshaaSqabaqcLbsacqGHsislcaWGubWcdaWgaaqaaKqzadGa amOyaaWcbeaajugibiaacMcaaaa@4A2E@     (9)

where T (z) is expressed as47

T(z)= T b +( T t T b )η(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9yqqrpepu0dbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaba9FcLbsaca WGubGaaiikaiaadQhacaGGPaGaeyypa0JaamivaSWaaSbaaeaajugW aiaadkgaaSqabaqcLbsacqGHRaWkcaGGOaGaamivaSWaaSbaaeaaju gWaiaadshaaSqabaqcLbsacqGHsislcaWGubGcdaWgaaWcbaqcLbma caWGIbaaleqaaKqzGeGaaiykaiabeE7aOjaacIcacaWG6bGaaiykaa aa@4E69@    (10)

For uniform temperature rise, the initial (T0) and bottom (Tb) uniform and non-uniform temperature of the plate is assumed to be 300°K, 600°K respectively.22,27

  • Stress strain relation
  • The constitutive relationship between stress and strain vectors in plane stress state for an isotropic layer accounting thermal effect can be written as47

{ σ }=[ Q ¯ ]{ ε } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=daGadaqaaK qzGeGaeq4WdmhakiaawUhacaGL9baajugibiabg2da9OWaamWaaeaa jugibiqadgfagaqeaaGccaGLBbGaayzxaaWaaiWaaeaajugibiabew 7aLbGccaGL7bGaayzFaaaaaa@4541@ or { σ x σ y σ xy σ yz σ xz }=[ Q ¯ 11 Q ¯ 12 Q ¯ 16 0 0 Q ¯ 12 Q ¯ 22 Q ¯ 26 0 0 Q ¯ 16 Q ¯ 26 Q ¯ 66 0 0 0 0 0 Q ¯ 44 0 0 0 0 0 Q ¯ 55 ]{ { ε x ε y ε xy ε yz ε zx }{ λ 1 λ 2 λ 12 0 0 }ΔT } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=daGadaqaaK qzGeqbaeqabuqaaaaakeaajugibiabeo8aZPWaaSbaaSqaaKqzadGa amiEaaWcbeaaaOqaaKqzGeGaeq4WdmNcdaWgaaWcbaqcLbmacaWG5b aaleqaaaGcbaqcLbsacqaHdpWCjuaGdaWgaaWcbaqcLbmacaWG4bGa amyEaaWcbeaaaOqaaKqzGeGaeq4WdmNcdaWgaaWcbaqcLbmacaWG5b GaamOEaaWcbeaaaOqaaKqzGeGaeq4Wdmxcfa4aaSbaaSqaaKqzadGa amiEaiaadQhaaSqabaaaaaGccaGL7bGaayzFaaqcLbsacqGH9aqpkm aadmaabaqcLbsafaqabeqbfaaaaaGcbaWaa0aaaeaajugibiaadgfa aaqcfa4aaSbaaSqaaKqzadGaaGymaiaaigdaaSqabaaakeaadaqdaa qaaKqzGeGaamyuaaaajuaGdaWgaaWcbaqcLbmacaaIXaGaaGOmaaWc beaaaOqaamaanaaabaqcLbsacaWGrbaaaKqbaoaaBaaaleaajugWai aaigdacaaI2aaaleqaaaGcbaqcLbsacaaIWaaakeaajugibiaaicda aOqaamaanaaabaqcLbsacaWGrbaaaKqbaoaaBaaaleaajugWaiaaig dacaaIYaaaleqaaaGcbaWaa0aaaeaajugibiaadgfaaaqcfa4aaSba aSqaaKqzadGaaGOmaiaaikdaaSqabaaakeaadaqdaaqaaKqzGeGaam yuaaaajuaGdaWgaaWcbaqcLbmacaaIYaGaaGOnaaWcbeaaaOqaaKqz GeGaaGimaaGcbaqcLbsacaaIWaaakeaadaqdaaqaaKqzGeGaamyuaa aajuaGdaWgaaWcbaqcLbmacaaIXaGaaGOnaaWcbeaaaOqaamaanaaa baqcLbsacaWGrbaaaKqbaoaaBaaaleaajugWaiaaikdacaaI2aaale qaaaGcbaWaa0aaaeaajugibiaadgfaaaGcdaWgaaWcbaqcLbmacaaI 2aGaaGOnaaWcbeaaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaa daqdaaqaaKqzGeGaamyuaaaajuaGdaWgaaWcbaqcLbmacaaI0aGaaG inaaWcbeaaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugi biaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaadaqdaa qaaKqzGeGaamyuaaaakmaaBaaaleaajugWaiaaiwdacaaI1aaaleqa aaaaaOGaay5waiaaw2faamaacmaabaWaaiWaaeaajugibuaabeqafe aaaaGcbaqcLbsacqaH1oqzkmaaBaaaleaajugWaiaadIhaaSqabaaa keaajugibiabew7aLPWaaSbaaSqaaKqzadGaamyEaaWcbeaaaOqaaK qzGeGaeqyTduMcdaWgaaWcbaqcLbmacaWG4bGaamyEaaWcbeaaaOqa aKqzGeGaeqyTduMcdaWgaaWcbaqcLbmacaWG5bGaamOEaaWcbeaaaO qaaKqzGeGaeqyTduMcdaWgaaWcbaqcLbmacaWG6bGaamiEaaWcbeaa aaaakiaawUhacaGL9baajugibiabgkHiTOWaaiWaaeaajugibuaabe qafeaaaaGcbaqcLbsacqaH7oaBjuaGdaWgaaWcbaqcLbmacaaIXaaa leqaaaGcbaqcLbsacqaH7oaBjuaGdaWgaaWcbaqcLbmacaaIYaaale qaaaGcbaqcLbsacqaH7oaBjuaGdaWgaaWcbaqcLbmacaaIXaGaaGOm aaWcbeaaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaaaaGccaGL7b GaayzFaaqcLbsacqqHuoarcaWGubaakiaawUhacaGL9baaaaa@DD2B@ (11)

Where λ 1 = Q ¯ 11 α 1 + Q ¯ 12 α 2 + Q ¯ 16 α 12 ; λ 2 = Q ¯ 12 α 1 + Q ¯ 22 α 2 + Q ¯ 26 α 12 ; λ 12 = Q ¯ 16 α 1 + Q ¯ 26 α 2 + Q ¯ 66 α 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeU 7aSLqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqpceWG rbGbaebakmaaBaaaleaajugWaiaaigdacaaIXaaaleqaaKqzGeGaeq ySdewcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgUcaRiqa dgfagaqeaOWaaSbaaSqaaKqzadGaaGymaiaaikdaaSqabaqcLbsacq aHXoqykmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRaWkceWG rbGbaebakmaaBaaaleaajugWaiaaigdacaaI2aaaleqaaKqzGeGaeq ySdeMcdaWgaaWcbaqcLbmacaaIXaGaaGOmaaWcbeaajugibiaacUda caaMe8UaaGjbVlaaysW7cqaH7oaBjuaGdaWgaaWcbaqcLbmacaaIYa aaleqaaKqzGeGaeyypa0Jabmyuayaaraqcfa4aaSbaaSqaaKqzadGa aGymaiaaikdaaSqabaqcLbsacqaHXoqyjuaGdaWgaaWcbaqcLbmaca aIXaaaleqaaKqzGeGaey4kaSIabmyuayaaraGcdaWgaaWcbaqcLbma caaIYaGaaGOmaaWcbeaajugibiabeg7aHLqbaoaaBaaaleaajugWai aaikdaaSqabaqcLbsacqGHRaWkceWGrbGbaebajuaGdaWgaaWcbaqc LbmacaaIYaGaaGOnaaWcbeaajugibiabeg7aHLqbaoaaBaaaleaaju gWaiaaigdacaaIYaaaleqaaKqzGeGaai4oaiaaysW7caaMe8UaaGjb VlabeU7aSLqbaoaaBaaaleaajugWaiaaigdacaaIYaaaleqaaKqzGe Gaeyypa0Jabmyuayaaraqcfa4aaSbaaKGbagaajugWaiaaigdacaaI 2aaajyaGbeaajugibiabeg7aHLqbaoaaBaaaleaajugWaiaaigdaaS qabaqcLbsacqGHRaWkceWGrbGbaebakmaaBaaaleaajugWaiaaikda caaI2aaaleqaaKqzGeGaeqySdewcfa4aaSbaaSqaaKqzadGaaGOmaa WcbeaajugibiabgUcaRiqadgfagaqeaOWaaSbaaSqaaKqzadGaaGOn aiaaiAdaaSqabaqcLbsacqaHXoqyjuaGdaWgaaWcbaqcLbmacaaIXa GaaGOmaaWcbeaajugibiaaysW7caaMe8oaaa@B6D5@

where { Q ¯ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paacmaaba qcLbsaceWGrbGbaebakmaaBaaaleaajugWaiaadMgacaWGQbaaleqa aaGccaGL7bGaayzFaaaaaa@3E1D@ , { σ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paacmaaba qcLbsacqaHdpWCaOGaay5Eaiaaw2haaaaa@3BA6@ and { ε } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paacmaaba qcLbsacqaH1oqzaOGaay5Eaiaaw2haaaaa@3B8A@ are transformed stiffness matrix, stress and strain vectors of the isotropic lamina, respectively, α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaaIXaaabeaaaaa@3AA0@ , α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaaIYaaabeaaaaa@3AA1@ , α 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaaIXaGaaGOmaaqabaaaaa@3B5C@ are the thermal expansion coefficients along x,y,z directions, which are obtained from the thermal coefficients in the longitudinal( α 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeg 7aHLqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacaGGSaaaaa@3F78@ ) and transverse ( α t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaiGc9yrVq0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeg 7aHLqbaoaaBaaaleaajugWaiaadshaaSqabaaaaa@3E77@ ) directions of the FGMs using transformation matrix. ∆T is the temperature rise. The constitutive relationship between stress resultants per unit length and mid-plain strains and curvatures is written in matrix form

{ N i M i P i }=[ A ij B ij E ij B ij D ij F ij E ij F ij H ij ]{ ε j 0 K j 0 K j 2 }{ N i T M i T P i T }(i,j=1,2,6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbqaq=paacmaaju gibqaabeGcbaqcLbsacaWGobWcdaWgaaqaaKqzadGaamyAaaWcbeaa aOqaaKqzGeGaamytaSWaaSbaaeaajugWaiaadMgaaSqabaaakeaaju gibiaadcfalmaaBaaabaqcLbmacaWGPbaaleqaaaaakiaawUhacaGL 9baajugibiabg2da9OWaamWaaeaajugibuaabeqabqaaaaGcbaqcLb safaqabeabbaaaaOqaaKqzGeGaamyqaOWaaSbaaSqaaKqzGeGaamyA aKqzadGaamOAaaWcbeaaaOqaaKqzGeGaamOqaSWaaSbaaeaajugWai aadMgacaWGQbaaleqaaaGcbaqcLbsacaWGfbWcdaWgaaqaaKqzadGa amyAaiaadQgaaSqabaaakeaaaaaabaqcLbsafaqabeabbaaaaOqaaK qzGeGaamOqaSWaaSbaaeaajugWaiaadMgacaWGQbaaleqaaaGcbaqc LbsacaWGebWcdaWgaaqaaKqzadGaamyAaiaadQgaaSqabaaakeaaju gibiaadAealmaaBaaabaqcLbmacaWGPbGaamOAaaWcbeaaaOqaaaaa aeaajugibuaabeqaeeaaaaGcbaqcLbsacaWGfbWcdaWgaaqaaKqzad GaamyAaiaadQgaaSqabaaakeaajugibiaadAealmaaBaaabaqcLbma caWGPbGaamOAaaWcbeaaaOqaaKqzGeGaamisaSWaaSbaaeaajugWai aadMgacaWGQbaaleqaaaGcbaaaaaqaaaaaaiaawUfacaGLDbaadaGa daqcLbsaeaqabOqaaKqzGeGaeqyTdu2cdaqhaaqcgayaaKqzadGaam OAaaqcgayaaKqzadGaaGimaaaaaOqaaKqzGeGaam4saSWaa0baaeaa jugWaiaadQgaaSqaaKqzadGaaGimaaaaaOqaaKqzGeGaam4saSWaa0 baaeaajugWaiaadQgaaSqaaKqzadGaaGOmaaaaaaGccaGL7bGaayzF aaqcLbsacqGHsislkmaacmaajugibqaabeGcbaqcLbsacaWGobWcda qhaaqaaKqzadGaamyAaaWcbaqcLbmacaWGubaaaaGcbaqcLbsacaWG nbWcdaqhaaqaaKqzadGaamyAaaWcbaqcLbmacaWGubaaaaGcbaqcLb sacaWGqbWcdaqhaaqaaKqzadGaamyAaaWcbaqcLbmacaWGubaaaaaa kiaawUhacaGL9baajugibiaacIcacaWGPbGaaiilaiaadQgacqGH9a qpcaaIXaGaaiilaiaaikdacaGGSaGaaGOnaiaacMcaaaa@AA2E@    (12)

{ Q 2 Q 1 }=[ A 4j D 4j A 5j D 5j ]{ ε j 0 K j 2 };{ R 2 R 1 }=[ D 4j F 4j D 5j F 5j ]{ ε j 0 K j 2 }(j=4,5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbqaq=paacmaaju gibqaabeGcbaqcLbsacaWGrbWcdaWgaaqaaKqzadGaaGOmaaWcbeaa aOqaaKqzGeGaamyuaSWaaSbaaeaajugWaiaaigdaaSqabaaaaOGaay 5Eaiaaw2haaKqzGeGaeyypa0JcdaWadaqcLbsaeaqabOqaaKqzGeGa amyqaSWaaSbaaeaajugWaiaaisdacaWGQbaaleqaaKqzGeGaamiraS WaaSbaaeaajugWaiaaisdacaWGQbaaleqaaaGcbaqcLbsacaWGbbWc daWgaaqaaKqzadGaaGynaiaadQgaaSqabaqcLbsacaWGebGcdaWgaa WcbaqcLbmacaaI1aGaamOAaaWcbeaaaaGccaGLBbGaayzxaaWaaiWa aKqzGeabaeqakeaajugibiabew7aLTWaa0baaeaajugWaiaadQgaaS qaaKqzadGaaGimaaaaaOqaaKqzGeGaam4saSWaa0baaeaajugWaiaa dQgaaSqaaKqzadGaaGOmaaaaaaGccaGL7bGaayzFaaqcLbsacaGG7a GcdaGadaqcLbsaeaqabOqaaKqzGeGaamOuaSWaaSbaaeaajugWaiaa ikdaaSqabaaakeaajugibiaadkfalmaaBaaabaqcLbmacaaIXaaale qaaaaakiaawUhacaGL9baajugibiabg2da9OWaamWaaKqzGeabaeqa keaajugibiaadsealmaaBaaabaqcLbmacaaI0aGaamOAaaWcbeaaju gibiaadAealmaaBaaabaqcLbmacaaI0aGaamOAaaWcbeaaaOqaaKqz GeGaamiraSWaaSbaaeaajugWaiaaiwdacaWGQbaaleqaaKqzGeGaam OraSWaaSbaaeaajugWaiaaiwdacaWGQbaaleqaaaaakiaawUfacaGL DbaadaGadaqcLbsaeaqabOqaaKqzGeGaeqyTdu2cdaqhaaqaaKqzad GaamOAaaWcbaqcLbmacaaIWaaaaaGcbaqcLbsacaWGlbWcdaqhaaqa aKqzadGaamOAaaWcbaqcLbmacaaIYaaaaaaakiaawUhacaGL9baaju gibiaacIcacaWGQbGaeyypa0JaaGinaiaacYcacaaI1aGaaiykaaaa @9C2B@     (13)

with

N i = [ N ix N iy N ixy ] T , M i = [ M ix M iy M ixy ] T , P i = [ P ix P iy P ixy ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbqaq=NqzGeGaam OtaSWaaSbaaeaajugWaiaadMgaaSqabaqcLbsacqGH9aqpcaGGBbGa amOtaSWaaSbaaeaajugWaiaadMgacaWG4baaleqaaKqzGeGaamOtaS WaaSbaaeaajugWaiaadMgacaWG5baaleqaaKqzGeGaamOtaSWaaSba aeaajugWaiaadMgacaWG4bGaamyEaaWcbeaajugibiaac2falmaaCa aabeqaaKqzadGaamivaaaajugibiaacYcacaWGnbGcdaWgaaWcbaqc LbmacaWGPbaaleqaaKqzGeGaeyypa0Jaai4waiaad2ealmaaBaaaba qcLbmacaWGPbGaamiEaaWcbeaajugibiaad2ealmaaBaaabaqcLbma caWGPbGaamyEaaWcbeaajugibiaad2eakmaaBaaaleaajugWaiaadM gacaWG4bGaamyEaaWcbeaajugibiaac2falmaaCaaabeqaaKqzadGa amivaaaajugibiaacYcacaWGqbWcdaWgaaqaaKqzadGaamyAaaWcbe aajugibiabg2da9iaacUfacaWGqbWcdaWgaaqaaKqzadGaamyAaiaa dIhaaSqabaqcLbsacaWGqbWcdaWgaaqcgayaaKqzadGaamyAaiaadM haaKGbagqaaKqzGeGaamiuaOWaaSbaaSqaaKqzadGaamyAaiaadIha caWG5baaleqaaKqzGeGaaiyxaOWaaWbaaSqabeaajugWaiaadsfaaa aaaa@82AD@

where Aij, Bij etc., are the plate stiffness defined in Appendix. Thermal stress resultants

N T i = [ N tx N ty N txy ] T , M i T = [ M tx M ty M txy ] T , P i T = [ P tx P ty P txy ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbqaq=NqzGeGaam OtaSWaaSaaaOqaaKqzadGaamivaaGcbaqcLbmacaWGPbaaaKqzGeGa eyypa0Jaai4waiaad6ealmaaBaaabaqcLbmacaWG0bGaamiEaaWcbe aajugibiaad6ealmaaBaaabaqcLbmacaWG0bGaamyEaaWcbeaajugi biaad6ealmaaBaaabaqcLbmacaWG0bGaamiEaiaadMhaaSqabaqcLb sacaGGDbWcdaahaaqabeaajugWaiaadsfaaaqcLbsacaGGSaGaamyt aSWaa0baaeaajugWaiaadMgaaSqaaKqzadGaamivaaaajugibiabg2 da9iaacUfacaWGnbGcdaWgaaWcbaqcLbmacaWG0bGaamiEaaWcbeaa jugibiaad2eakmaaBaaaleaajugWaiaadshacaWG5baaleqaaKqzGe GaamytaOWaaSbaaSqaaKqzadGaamiDaiaadIhacaWG5baaleqaaKqz GeGaaiyxaOWaaWbaaSqabeaajugWaiaadsfaaaqcLbsacaGGSaGaam iuaSWaa0baaeaajugWaiaadMgaaSqaaKqzadGaamivaaaajugibiab g2da9iaacUfacaWGqbWcdaWgaaqaaKqzadGaamiDaiaadIhaaSqaba qcLbsacaWGqbGcdaWgaaWcbaqcLbmacaWG0bGaamyEaaWcbeaajugi biaadcfalmaaBaaabaqcLbmacaWG0bGaamiEaiaadMhaaSqabaqcLb sacaGGDbWcdaahaaqabeaajugWaiaadsfaaaaaaa@8816@ are calculated by

[ N i T M i T P i T ]= m=1 N h/2 h/2 { Q 11 ¯ α 1 + Q 12 ¯ α 2 + Q 16 ¯ α 12 Q 12 ¯ α 1 + Q 22 ¯ α 2 + Q 26 ¯ α 12 Q 16 ¯ α 1 + Q 26 ¯ α 2 + Q 66 ¯ α 12 } ( 1,z, z 3 )ΔTdz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbqaq=paadmaaba qcLbsafaqabeqadaaakeaajugibiaad6ealmaaDaaabaqcLbmacaWG PbaaleaajugWaiaadsfaaaaakeaajugibiaad2ealmaaDaaabaqcLb macaWGPbaaleaajugWaiaadsfaaaaakeaajugibiaadcfalmaaDaaa baqcLbmacaWGPbaaleaajugWaiaadsfaaaaaaaGccaGLBbGaayzxaa qcLbsacqGH9aqpkmaaqahabaWaa8qmaeaadaGadaqcLbsaeaqabOqa amaanaaabaqcLbsacaWGrbWcdaWgaaqaaKqzadGaaGymaiaaigdaaS qabaaaaKqzGeGaeqySde2cdaWgaaqaaKqzadGaaGymaaWcbeaajugi biabgUcaROWaa0aaaeaajugibiaadgfalmaaBaaabaqcLbmacaaIXa GaaGOmaaWcbeaaaaqcLbsacqaHXoqykmaaBaaaleaajugWaiaaikda aSqabaqcLbsacqGHRaWkkmaanaaabaqcLbsacaWGrbWcdaWgaaqaaK qzadGaaGymaiaaiAdaaSqabaaaaKqzGeGaeqySde2cdaWgaaqaaKqz adGaaGymaiaaikdaaSqabaaakeaadaqdaaqaaKqzGeGaamyuaSWaaS baaeaajugWaiaaigdacaaIYaaaleqaaaaajugibiabeg7aHTWaaSba aeaajugWaiaaigdaaSqabaqcLbsacqGHRaWkkmaanaaabaqcLbsaca WGrbWcdaWgaaqaaKqzadGaaGOmaiaaikdaaSqabaaaaKqzGeGaeqyS deMcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey4kaSIcdaqdaa qaaKqzGeGaamyuaSWaaSbaaeaajugWaiaaikdacaaI2aaaleqaaaaa jugibiabeg7aHTWaaSbaaeaajugWaiaaigdacaaIYaaaleqaaaGcba Waa0aaaeaajugibiaadgfalmaaBaaabaqcLbmacaaIXaGaaGOnaaWc beaaaaqcLbsacqaHXoqylmaaBaaabaqcLbmacaaIXaaaleqaaKqzGe Gaey4kaSIcdaqdaaqaaKqzGeGaamyuaSWaaSbaaeaajugWaiaaikda caaI2aaaleqaaaaajugibiabeg7aHTWaaSbaaeaajugWaiaaikdaaS qabaqcLbsacqGHRaWkkmaanaaabaqcLbsacaWGrbGcdaWgaaWcbaqc LbmacaaI2aGaaGOnaaWcbeaaaaqcLbsacqaHXoqylmaaBaaabaqcLb macaaIXaGaaGOmaaWcbeaaaaGccaGL7bGaayzFaaaaleaajugWaiab gkHiTiaadIgacaGGVaGaaGOmaaWcbaqcLbmacaWGObGaai4laiaaik daaKqzGeGaey4kIipaaSqaaKqzadGaamyBaiabg2da9iaaigdaaSqa aKqzadGaamOtaaqcLbsacqGHris5aOWaaeWaaeaajugibiaaigdaca GGSaGaaGjbVlaadQhacaGGSaGaaGjbVlaadQhalmaaCaaabeqaaKqz adGaaG4maaaaaOGaayjkaiaawMcaaKqzGeGaeyiLdqKaamivaiaads gacaWG6baaaa@CC1C@

For FGM material the elastic constant are defined as.47

Q ¯ 11 = Q ¯ 22 = E(z,T) 1 ν 2 , Q ¯ 12 = νE(z,T) 1 ν 2 , Q ¯ 44 = Q ¯ 55 = Q ¯ 66 = E(z,T) 2(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb saceWGrbGbaebajuaGdaWgaaWcbaqcLbmacaaIXaGaaGymaaWcbeaa jugibiabg2da9iqadgfagaqeaKqbaoaaBaaaleaajugWaiaaikdaca aIYaaaleqaaKqzGeGaeyypa0JcdaWcaaqaaKqzGeGaamyraiaacIca caWG6bGaaiilaiaadsfacaGGPaaakeaajugibiaaigdacqGHsislcq aH9oGBjuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaqcLbsacaGGSaGa aGjbVlaaysW7ceWGrbGbaebajuaGdaWgaaWcbaqcLbmacaaIXaGaaG OmaaWcbeaajugibiabg2da9OWaaSaaaeaajugibiabe27aUjaadwea caGGOaGaamOEaiaacYcacaWGubGaaiykaaGcbaqcLbsacaaIXaGaey OeI0IaeqyVd4wcfa4aaWbaaSqabeaajugWaiaaikdaaaaaaKqzGeGa aiilaiaaysW7caaMe8UaaGjbVlqadgfagaqeaKqbaoaaBaaaleaaju gWaiaaisdacaaI0aaaleqaaKqzGeGaeyypa0Jabmyuayaaraqcfa4a aSbaaSqaaKqzadGaaGynaiaaiwdaaSqabaqcLbsacqGH9aqpceWGrb GbaebajuaGdaWgaaWcbaqcLbmacaaI2aGaaGOnaaWcbeaajugibiab g2da9OWaaSaaaeaajugibiaadweacaGGOaGaamOEaiaacYcacaWGub GaaiykaaGcbaqcLbsacaaIYaGaaiikaiaaigdacqGHRaWkcqaH9oGB caGGPaaaaaaa@8CCD@     (14)

The value of E is taken from Eq. (1).

 

  • Strain energy of the plate

The strain energy of the plate is given by

Π 1 = 1 2 V { ε } T [ σ ]dv MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacqqHGoaulmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0Jc daWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaOWaa8quaeaada GadaqaaKqzGeGaeqyTdugakiaawUhacaGL9baaaSqaaKqzadGaamOv aaWcbeqcLbsacqGHRiI8aOWaaWbaaSqabeaajugWaiaadsfaaaGcda WadaqaaKqzGeGaeq4WdmhakiaawUfacaGLDbaajugibiaadsgacaWG 2baaaa@5212@     (15)

U= 1 2 R [ N ix ε xx + N iy ε yy + N ixy γ xy + Q 1 γ xz + Q 2 γ yz ] dxdy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacaWGvbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsa caaIYaaaaKqbaoaapefakeaajuaGdaWdXaGcbaqcfa4aamWaaOqaaK qzGeGaamOtaSWaaSbaaeaajugWaiaadMgacaWG4baaleqaaKqzGeGa eqyTduwcfa4aaSbaaSqaaKqzadGaamiEaiaadIhaaSqabaqcLbsacq GHRaWkcaWGobWcdaWgaaqaaKqzadGaamyAaiaadMhaaSqabaqcLbsa cqaH1oqzjuaGdaWgaaWcbaqcLbmacaWG5bGaamyEaaWcbeaajugibi abgUcaRiaad6ealmaaBaaabaqcLbmacaWGPbGaamiEaiaadMhaaSqa baqcLbsacqaHZoWzlmaaBaaabaqcLbmacaWG4bGaamyEaaWcbeaaju gibiabgUcaRiaadgfalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGa eq4SdCwcfa4aaSbaaSqaaKqzadGaamiEaiaadQhaaSqabaqcLbsacq GHRaWkcaWGrbWcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiabeo7a NTWaaSbaaeaajugWaiaadMhacaWG6baaleqaaaGccaGLBbGaayzxaa aaleaajugWaiaadkfaaSqaaaqcLbsacqGHRiI8aaWcbaaabeqcLbsa cqGHRiI8aiaadsgacaWG4bGaamizaiaadMhaaaa@8368@      (16)

Potential energy due to thermal stresses

The potential energy ( Π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacqqHGoaukmaaBaaaleaajugWaiaaikdaaSqabaaaaa@3B6B@ ) storage by thermal load (uniform and transverse change in temperature across the thickness) Due to change in temperature, pre buckling stresses, i.e. in plane thermal compressive stress resultants in the plate are generated. These resultants are the reason for the buckling. The potential energy due to the in plane thermal stress resultants is expressed as

2 = 1 2 A [ N xtm ( w , x ) 2 + N ytm ( w , y ) 2 +2 N xytm ( w , x )( w , y ) ] dA = 1 2 A { w , x w , y } T [ N xtm N xytm N xytm N ytm ]{ w , x w , y }dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceiqabeaam8qaaK qzGeGaey4dIu9cdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiabg2da 9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGda WdraGcbaqcfa4aamWaaOqaaKqzGeGaamOtaKqbaoaaBaaaleaajugW aiaadIhacaWG0bGaamyBaaWcbeaajuaGdaqadaGcbaqcLbsacaWG3b GaaiilaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakiaawIcacaGL PaaalmaaCaaabeqaaKqzadGaaGOmaaaajugibiabgUcaRiaad6eaju aGdaWgaaWcbaqcLbmacaWG5bGaamiDaiaad2gaaSqabaqcfa4aaeWa aOqaaKqzGeGaam4DaiaacYcalmaaBaaabaqcLbmacaWG5baaleqaaa GccaGLOaGaayzkaaWcdaahaaqabeaajugWaiaaikdaaaqcLbsacqGH RaWkcaaIYaGaamOtaKqbaoaaBaaaleaajugWaiaadIhacaWG5bGaam iDaiaad2gaaSqabaqcfa4aaeWaaOqaaKqzGeGaam4DaiaacYcalmaa BaaabaqcLbmacaWG4baaleqaaaGccaGLOaGaayzkaaqcfa4aaeWaaO qaaKqzGeGaam4DaiaacYcajuaGdaWgaaWcbaqcLbmacaWG5baaleqa aaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaaleaajugWaiaadgeaaS qabKqzGeGaey4kIipacaaMe8UaamizaiaadgeaaOqaaKqzGeGaaGjb VlaaysW7caaMe8UaaGjbVlabg2da9Kqbaoaalaaakeaajugibiaaig daaOqaaKqzGeGaaGOmaaaajuaGdaWdraGcbaqcfa4aaiWaaOqaaKqz GeqbaeqabiqaaaGcbaqcLbsacaWG3bGaaiilaSWaaSbaaeaajugWai aadIhaaSqabaaakeaajugibiaadEhacaGGSaWcdaWgaaqaaKqzadGa amyEaaWcbeaaaaaakiaawUhacaGL9baaaSqaaKqzadGaamyqaaWcbe qcLbsacqGHRiI8aSWaaWbaaeqabaqcLbmacaWGubaaaKqbaoaadmaa keaajugibuaabeqaciaaaOqaaKqzGeGaamOtaKqbaoaaBaaaleaaju gWaiaadIhacaWG0bGaamyBaaWcbeaaaOqaaKqzGeGaamOtaSWaaSba aeaajugWaiaadIhacaWG5bGaamiDaiaad2gaaSqabaaakeaajugibi aad6ealmaaBaaabaqcLbmacaWG4bGaamyEaiaadshacaWGTbaaleqa aaGcbaqcLbsacaWGobWcdaWgaaqaaKqzadGaamyEaiaadshacaWGTb aaleqaaaaaaOGaay5waiaaw2faaKqbaoaacmaakeaajugibuaabeqa ceaaaOqaaKqzGeGaam4DaiaacYcajuaGdaWgaaWcbaqcLbmacaWG4b aaleqaaaGcbaqcLbsacaWG3bGaaiilaSWaaSbaaeaajugWaiaadMha aSqabaaaaaGccaGL7bGaayzFaaqcLbsacaWGKbGaamyqaaaaaa@CC99@     (17)

Where, Nxtm, Nytm and Nxytm are Thermo mechanical in-plane compressive stress resultant per unit length. with Nxtm = Nxm - Nxt ,Nxytm = Nxym - Nxyt and Nytm = Nym - Nyt .

Using finite element method5 and summing over the entire element the Eq. (17) can be written as

W= e=1 NE W ( e ) = e=1 NE { Λ ( e ) } T λ T [ K g ( e ) ] { Λ } ( e ) = λ T { q } T [ K g ]{ q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacaWGxbGaeyypa0tcfa4aaabCaOqaaKqzGeGaam4vaSWaaWbaaeqa baWaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaeaajugWai aadwgacqGH9aqpcaaIXaaaleaajugWaiaad6eacaWGfbaajugibiab ggHiLdGaeyypa0tcfa4aaabCaOqaaKqbaoaacmaakeaajugibiabfU 5amLqbaoaaCaaaleqabaWaaeWaaeaajugWaiaadwgaaSGaayjkaiaa wMcaaaaaaOGaay5Eaiaaw2haaKqbaoaaCaaaleqabaqcLbmacaWGub aaaKqzGeGaeq4UdWwcfa4aaSbaaSqaaKqzadGaamivaaWcbeaajuaG daWadaGcbaqcLbsacaWGlbWcdaWgaaqaaKqzadGaam4zaaWcbeaaju aGdaahaaWcbeqaamaaCaaameqabaWcdaqadaadbaqcLbmacaWGLbaa miaawIcacaGLPaaaaaaaaaGccaGLBbGaayzxaaqcfa4aaiWaaOqaaK qzGeGaeu4MdWeakiaawUhacaGL9baalmaaCaaabeqaamaabmaabaqc LbmacaWGLbaaliaawIcacaGLPaaaaaaabaqcLbmacaWGLbGaeyypa0 JaaGymaaWcbaqcLbmacaWGobGaamyraaqcLbsacqGHris5aiabg2da 9iabeU7aSLqbaoaaBaaaleaajugWaiaadsfaaSqabaqcfa4aaiWaaO qaaKqzGeGaamyCaaGccaGL7bGaayzFaaWcdaahaaqabeaajugWaiaa dsfaaaqcfa4aamWaaOqaaKqzGeGaam4saSWaaSbaaeaajugWaiaadE gaaSqabaaakiaawUfacaGLDbaajuaGdaGadaGcbaqcLbsacaWGXbaa kiaawUhacaGL9baaaaa@8F70@

Where λ T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacqaH7oaBjuaGdaWgaaWcbaqcLbmacaWGubaaleqaaaaa@3C66@ , [ K g ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcfa 4aamWaaOqaaKqzGeGaam4saKqbaoaaBaaaleaajugWaiaadEgaaSqa baaakiaawUfacaGLDbaaaaa@3E29@ are defined as critical thermal buckling temperature and global geometric stiffness matrix defined in appendix respectively.

Using Eqs. (5), (6) and (7), Eq. (16) can be written in expanded form as

U= 1 2 A { ε ¯ l } T [ D ] { ε ¯ l }dA+ 1 2 A [ ε ¯ l ] T [ D 3 ]{ A }{ θ }dA + 1 2 A { A } T { θ } T [ D 4 ]{ ε ¯ l }dA+ 1 2 A { A } T { θ } T [ D 5 ]{ A }{ θ }dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceiqabeaam8qaaK qzGeGaamyvaiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqz GeGaaGOmaaaajuaGdaWdraGcbaqcfa4aaiWaaOqaaKqzGeGafqyTdu MbaebalmaaBaaabaqcLbmacaWGSbaaleqaaaGccaGL7bGaayzFaaqc fa4aaWbaaSqabeaajugWaiaadsfaaaqcfa4aamWaaOqaaKqzGeGaam iraaGccaGLBbGaayzxaaaaleaajugWaiaadgeaaSqabKqzGeGaey4k IipajuaGdaGadaGcbaqcLbsacuaH1oqzgaqeaSWaaSbaaeaajugWai aadYgaaSqabaaakiaawUhacaGL9baajugibiaadsgacaWGbbGaey4k aSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaKqbao aapebakeaajuaGdaWadaGcbaqcLbsacuaH1oqzgaqeaSWaaSbaaeaa jugWaiaadYgaaSqabaaakiaawUfacaGLDbaajuaGdaahaaWcbeqaaK qzadGaamivaaaaaSqaaKqzadGaamyqaaWcbeqcLbsacqGHRiI8aKqb aoaadmaakeaajugibiaadseajuaGdaWgaaWcbaqcLbmacaaIZaaale qaaaGccaGLBbGaayzxaaqcfa4aaiWaaOqaaKqzGeGaamyqaaGccaGL 7bGaayzFaaqcfa4aaiWaaOqaaKqzGeGaeqiUdehakiaawUhacaGL9b aajugibiaadsgacaWGbbaakeaajugibiaaysW7caaMe8UaaGjbVlaa ysW7caaMe8Uaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLb sacaaIYaaaaKqbaoaapebakeaajuaGdaGadaGcbaqcLbsacaWGbbaa kiaawUhacaGL9baajuaGdaahaaWcbeqaaKqzadGaamivaaaajuaGda GadaGcbaqcLbsacqaH4oqCaOGaay5Eaiaaw2haaaWcbaqcLbmacaWG bbaaleqajugibiabgUIiYdqcfa4aaWbaaSqabeaajugWaiaadsfaaa qcfa4aamWaaOqaaKqzGeGaamiraSWaaSbaaeaajugWaiaaisdaaSqa baaakiaawUfacaGLDbaajuaGdaGadaGcbaqcLbsacuaH1oqzgaqeaS WaaSbaaeaajugWaiaadYgaaSqabaaakiaawUhacaGL9baajugibiaa dsgacaWGbbGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLb sacaaIYaaaaKqbaoaapebakeaajuaGdaGadaGcbaqcLbsacaWGbbaa kiaawUhacaGL9baajuaGdaahaaWcbeqaaKqzadGaamivaaaajuaGda GadaGcbaqcLbsacqaH4oqCaOGaay5Eaiaaw2haaKqbaoaaCaaaleqa baqcLbmacaWGubaaaaWcbaqcLbmacaWGbbaaleqajugibiabgUIiYd qcfa4aamWaaOqaaKqzGeGaamiraSWaaSbaaeaajugWaiaaiwdaaSqa baaakiaawUfacaGLDbaajuaGdaGadaGcbaqcLbsacaWGbbaakiaawU hacaGL9baajuaGdaGadaGcbaqcLbsacqaH4oqCaOGaay5Eaiaaw2ha aKqzGeGaamizaiaadgeaaaaa@D7FA@     (18)

where [D], [D3], [D4] and [D5] are the laminate stiffness matrices defined in appendix and { ε ¯ l } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacu aH1oqzgaqeamaaBaaaleaacaWGSbaabeaaaOGaay5Eaiaaw2haaaaa @3AED@ is the linear mid-plane strain vector.25

The strain energy function is computed for each element and then summed over all the elements to get the total strain energy.33 Following this procedure, Eq. (18), can be written as

U= e=1 NE U ( e ) = e=1 NE { Λ } ( e ) T [ K l + K nl ( Δ ) ] ( e ) { Λ } ( e ) = { q } T [ K i + K nl ( q ) ]{ q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacaWGvbGaaGjbVlaaysW7cqGH9aqpcaaMe8UaaGjbVNqbaoaaqaha keaajugibiaadwfalmaaCaaabeqaamaabmaabaqcLbmacaWGLbaali aawIcacaGLPaaaaaaabaqcLbmacaWGLbGaeyypa0JaaGymaaWcbaqc LbmacaWGobGaamyraaqcLbsacqGHris5aiaaysW7caaMe8Uaeyypa0 JaaGjbVlaaysW7juaGdaaeWbGcbaqcfa4aaiWaaOqaaKqzGeGaeu4M dWeakiaawUhacaGL9baalmaaCaaabeqaamaaCaaameqabaWcdaqada adbaqcLbmacaWGLbaamiaawIcacaGLPaaaaaWcdaahaaadbeqaaKqz adGaamivaaaaaaqcfa4aamWaaOqaaKqzGeGaam4saSWaaSbaaeaaju gWaiaadYgaaSqabaqcLbsacqGHRaWkcaWGlbqcfa4aaSbaaSqaaKqz adGaamOBaiaadYgaaSqabaqcfa4aaeWaaOqaaKqzGeGaeuiLdqeaki aawIcacaGLPaaaaiaawUfacaGLDbaalmaaCaaabeqaamaabmaabaqc LbmacaWGLbaaliaawIcacaGLPaaaaaqcfa4aaiWaaOqaaKqzGeGaeu 4MdWeakiaawUhacaGL9baalmaaCaaabeqaamaabmaabaqcLbmacaWG LbaaliaawIcacaGLPaaaaaaabaqcLbmacaWGLbGaeyypa0JaaGymaa WcbaqcLbmacaWGobGaamyraaqcLbsacqGHris5aiaaysW7cqGH9aqp caaMe8UaaGjbVNqbaoaacmaakeaajugibiaadghaaOGaay5Eaiaaw2 haaSWaaWbaaeqabaqcLbmacaWGubaaaKqbaoaadmaakeaajugibiaa dUealmaaBaaabaqcLbmacaWGPbaaleqaaKqzGeGaey4kaSIaam4saS WaaSbaaeaajugWaiaad6gacaWGSbaaleqaaKqbaoaabmaakeaajugi biaadghaaOGaayjkaiaawMcaaaGaay5waiaaw2faaKqbaoaacmaake aajugibiaadghaaOGaay5Eaiaaw2haaaaa@A8B3@    (19)

Where [ K l ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpmaadm aabaqcLbsacaWGlbGcdaWgaaWcbaqcLbmacaWGSbaaleqaaaGccaGL BbGaayzxaaaaaa@3D04@ , [ K nl ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpmaadm aabaqcLbsacaWGlbWcdaWgaaqaaKqzadGaamOBaiaadYgaaSqabaaa kiaawUfacaGLDbaaaaa@3DED@ and { q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpmaacm aabaqcLbsacaWGXbaakiaawUhacaGL9baaaaa@3B09@ are defined as global linear, nonlinear stiffness matrix and displacement vector respectively and defined in appendix.

External work done

Due to uniform temperature change, pre-buckling stresses in the FGM plate are generated. The in-plane pre-buckling stress resultants per unit length are reason for buckling. The work done (W) by the in-plane stress resultants in producing out of plane displacement ‘w’ is expressed as

W= 1 2 A [ N x ( w , x ) 2 + N y ( w , y ) 2 +2 N xy ( w , x )( w , y ) ] dA = 1 2 A { w , x w , y } T [ N x N xy N xy N y ]{ w , x w , y }dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceiqabeaam8qaaK qzGeGaam4vaiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqz GeGaaGOmaaaajuaGdaWdraGcbaqcfa4aamWaaOqaaKqzGeGaamOtaK qbaoaaBaaaleaajugWaiaadIhaaSqabaqcfa4aaeWaaOqaaKqzGeGa am4DaiaacYcajuaGdaWgaaWcbaqcLbmacaWG4baaleqaaaGccaGLOa GaayzkaaWcdaahaaqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaWG obqcfa4aaSbaaSqaaKqzadGaamyEaaWcbeaajuaGdaqadaGcbaqcLb sacaWG3bGaaiilaSWaaSbaaeaajugWaiaadMhaaSqabaaakiaawIca caGLPaaalmaaCaaabeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaik dacaWGobqcfa4aaSbaaSqaaKqzadGaamiEaiaadMhaaSqabaqcfa4a aeWaaOqaaKqzGeGaam4DaiaacYcajuaGdaWgaaWcbaqcLbmacaWG4b aaleqaaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaam4Daiaa cYcalmaaBaaabaqcLbmacaWG5baaleqaaaGccaGLOaGaayzkaaaaca GLBbGaayzxaaaaleaajugWaiaadgeaaSqabKqzGeGaey4kIipacaaM e8UaamizaiaadgeaaOqaaKqzGeGaaGjbVlaaysW7caaMe8UaaGjbVl abg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaa juaGdaWdraGcbaqcfa4aaiWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLb sacaWG3bGaaiilaSWaaSbaaeaajugWaiaadIhaaSqabaaakeaajugi biaadEhacaGGSaWcdaWgaaqaaKqzadGaamyEaaWcbeaaaaaakiaawU hacaGL9baaaSqaaKqzadGaamyqaaWcbeqcLbsacqGHRiI8aSWaaWba aeqabaqcLbmacaWGubaaaKqbaoaadmaakeaajugibuaabeqaciaaaO qaaKqzGeGaamOtaSWaaSbaaeaajugWaiaadIhaaSqabaaakeaajugi biaad6eajuaGdaWgaaWcbaqcLbmacaWG4bGaamyEaaWcbeaaaOqaaK qzGeGaamOtaKqbaoaaBaaaleaajugWaiaadIhacaWG5baaleqaaaGc baqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaamyEaaWcbeaaaaaaki aawUfacaGLDbaajuaGdaGadaGcbaqcLbsafaqabeGabaaakeaajugi biaadEhacaGGSaqcfa4aaSbaaSqaaKqzadGaamiEaaWcbeaaaOqaaK qzGeGaam4DaiaacYcalmaaBaaabaqcLbmacaWG5baaleqaaaaaaOGa ay5Eaiaaw2haaKqzGeGaamizaiaadgeaaaaa@BCE3@  (20)

Where, Nx, Ny and Nxy are thermal in-plane thermal compressive stress resultant per unit length.

Using finite element method5 and summing over the entire element the Eq. (20) can be written as

W= e=1 NE W ( e ) = e=1 NE { Λ ( e ) } T λ T [ K g ( e ) ] { Λ } ( e ) = λ T { q } T [ K g ]{ q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacaWGxbGaeyypa0tcfa4aaabCaOqaaKqzGeGaam4vaSWaaWbaaeqa baWaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaeaajugWai aadwgacqGH9aqpcaaIXaaaleaajugWaiaad6eacaWGfbaajugibiab ggHiLdGaeyypa0tcfa4aaabCaOqaaKqbaoaacmaakeaajugibiabfU 5amTWaaWbaaeqabaWaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMca aaaaaOGaay5Eaiaaw2haaSWaaWbaaeqabaqcLbmacaWGubaaaKqzGe Gaeq4UdW2cdaWgaaqaaKqzadGaamivaaWcbeaajuaGdaWadaGcbaqc LbsacaWGlbWcdaWgaaqaaKqzadGaam4zaaWcbeaajuaGdaahaaWcbe qaaKqbaoaaCaaameqabaWcdaqadaadbaqcLbmacaWGLbaamiaawIca caGLPaaaaaaaaaGccaGLBbGaayzxaaqcfa4aaiWaaOqaaKqzGeGaeu 4MdWeakiaawUhacaGL9baajuaGdaahaaWcbeqaamaabmaabaqcLbma caWGLbaaliaawIcacaGLPaaaaaaabaqcLbmacaWGLbGaeyypa0JaaG ymaaWcbaqcLbmacaWGobGaamyraaqcLbsacqGHris5aiabg2da9iab eU7aSLqbaoaaBaaaleaajugWaiaadsfaaSqabaqcfa4aaiWaaOqaaK qzGeGaamyCaaGccaGL7bGaayzFaaWcdaahaaqabeaajugWaiaadsfa aaqcfa4aamWaaOqaaKqzGeGaam4saKqbaoaaBaaaleaajugWaiaadE gaaSqabaaakiaawUfacaGLDbaajuaGdaGadaGcbaqcLbsacaWGXbaa kiaawUhacaGL9baaaaa@8F70@      (21)

 Where λ T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcLb sacqaH7oaBlmaaBaaabaqcLbmacaWGubaaleqaaaaa@3BD8@ , [ K g ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbyaq=haam8qcfa 4aamWaaOqaaKqzGeGaam4saKqbaoaaBaaaleaajugWaiaadEgaaSqa baaakiaawUfacaGLDbaaaaa@3E29@ are defined as critical thermal buckling temperature and global geometric stiffness matrix defined in appendix respectively.

  • Strain energy due to elastic foundations
  • Strain energy due to elastic foundations can be expressed as:

π= 1 2 A { K 1 w 2 + K 2 [ ( w ,x ) 2 + ( w ,y ) 2 ] }dA , π= 1 2 A { w w ,x w ,y } T [ K 1 0 0 0 K 2 0 0 0 K 2 ]{ w w ,x w ,y }dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabceqabaadpeaaju gibiabec8aWjabg2da9OWaaSaaaeaajugibiaaigdaaOqaaKqzGeGa aGOmaaaakmaapebabaWaaiWaaeaajugibiaadUealmaaBaaabaqcLb macaaIXaaaleqaaKqzGeGaam4DaSWaaWbaaeqabaqcLbmacaaIYaaa aKqzGeGaey4kaSIaam4saOWaaSbaaSqaaKqzadGaaGOmaaWcbeaakm aadmaabaWaaeWaaeaajugibiaadEhakmaaBaaaleaajugibiaacYca jugWaiaadIhaaSqabaaakiaawIcacaGLPaaalmaaCaaabeqaaKqzad GaaGOmaaaajugibiabgUcaROWaaeWaaeaajugibiaadEhakmaaBaaa leaajugibiaacYcajugWaiaadMhaaSqabaaakiaawIcacaGLPaaalm aaCaaabeqaaKqzadGaaGOmaaaaaOGaay5waiaaw2faaaGaay5Eaiaa w2haaKqzGeGaaGPaVlaadsgacaWGbbaaleaajugWaiaadgeaaSqabK qzGeGaey4kIipacaGGSaaakeaajugibiabec8aWjabg2da9OWaaSaa aeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaakmaapebabaWaaiWaae aajugibuaabeqadeaaaOqaaKqzGeGaam4DaaGcbaqcLbsacaWG3bGc daWgaaWcbaqcLbsacaGGSaqcLbmacaWG4baaleqaaaGcbaqcLbsaca WG3bGcdaWgaaWcbaqcLbsacaGGSaqcLbmacaWG5baaleqaaaaaaOGa ay5Eaiaaw2haamaaCaaaleqabaqcLbmacaWGubaaaOWaamWaaeaaju gibuaabeqadmaaaOqaaKqzGeGaam4saSWaaSbaaeaajugWaiaaigda aSqabaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsaca aIWaaakeaajugibiaadUeakmaaBaaaleaajugWaiaaikdaaSqabaaa keaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaake aajugibiaadUealmaaBaaabaqcLbmacaaIYaaaleqaaaaaaOGaay5w aiaaw2faamaacmaabaqcLbsafaqabeWabaaakeaajugibiaadEhaaO qaaKqzGeGaam4DaOWaaSbaaSqaaKqzGeGaaiilaKqzadGaamiEaaWc beaaaOqaaKqzGeGaam4DaOWaaSbaaSqaaKqzGeGaaiilaKqzadGaam yEaaWcbeaaaaaakiaawUhacaGL9baajugibiaaykW7caWGKbGaamyq aaWcbaqcLbmacaWGbbaaleqajugibiabgUIiYdaaaaa@ABF0@     (22)

Finite element models finite element models

In the present study a C° nine-noded isoparametric finite element with 7 DOFs per node is employed i.e., (u,v,w,. ψ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaea0=jugibiabeI 8a5TWaaSbaaeaajugWaiaadIhaaSqabaaaaa@3BEA@ , ψ y, θ x , θ y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK NcdaWgaaWcbaqcLbmacaWG5bqcLbsacaGGSaaaleqaaKqzGeGaeqiU deNcdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaaiilaiabeI7aXP WaaSbaaSqaaKqzadGaamyEaaWcbeaajugibiaacMcaaaa@4740@ is employed. For this type of element, the displacement vector and the element geometry are expressed as

{ Λ }= i=1 NN φ i { Λ } i ;x= i=1 NN φ i x i ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaqcLb sacqqHBoataOGaay5Eaiaaw2haaKqzGeGaeyypa0JcdaaeWbqaaKqz GeGaeqOXdOMcdaWgaaWcbaqcLbmacaWGPbaaleqaaaqaaKqzadGaam yAaiabg2da9iaaigdaaSqaaKqzadGaamOtaiaad6eaaKqzGeGaeyye IuoakmaacmaabaqcLbsacqqHBoataOGaay5Eaiaaw2haamaaBaaale aajugWaiaadMgaaSqabaqcLbsacaGG7aGaaGjbVlaaysW7caaMe8Ua aGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaadIhacq GH9aqpkmaaqahabaqcLbsacqaHgpGAlmaaBaaabaqcLbmacaWGPbaa leqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzadGaamOtai aad6eaaKqzGeGaeyyeIuoacaWG4bWcdaWgaaqaaKqzadGaamyAaaWc beaajugibiaacUdaaaa@760F@ and y= i=1 NN φ i y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyEai abg2da9OWaaabCaeaajugibiabeA8aQTWaaSbaaeaajugWaiaadMga aSqabaaabaqcLbmacaWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWGob GaamOtaaqcLbsacqGHris5aiaadMhakmaaBaaaleaajugWaiaadMga aSqabaaaaa@49F4@        (23)

where φ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpKqzGe GaeqOXdO2cdaWgaaqaaKqzadGaamyAaaWcbeaaaaa@3C09@ is the interpolation function for the ith node, { Λ } i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpmaacm aabaqcLbsacqqHBoataOGaay5Eaiaaw2haaSWaaSbaaeaajugWaiaa dMgaaSqabaaaaa@3DDC@ is the vector of unknown displacements for the ith node, NN is the number of nodes per element and xi and yi are Cartesian Coordinate of the ith node. The linear mid plane strain vector can be expressed in terms of mid plane displacement field and then the energy is computed for each element and then summed over all the elements to get the total strain energy.33 Following this, and using Eq. (23), Eq. (15) can be written as

Π 1 = e=1 NE Π 1 ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaga0=baadpKqzGe GaeuiOda1cdaWgaaqaaKqzadGaaGymaaWcbeaajugibiabg2da9OWa aabCaeaajugibiabfc6aqPWaaSbaaSqaaKqzadGaaGymaaWcbeaada ahaaqabeaadaqadaqaaKqzadGaamyzaaWccaGLOaGaayzkaaaaaaqa aKqzadGaamyzaiabg2da9iaaigdaaSqaaKqzadGaamOtaiaadweaaK qzGeGaeyyeIuoaaaa@4E82@     (24)

where, NE is the number of elements and U(e) is the elemental total potential energy which can be expressed

Π 1 = e=1 NE [ 1 2 { Λ ( e ) } T [ K l * ( e ) + K nl ( e ) { Λ } ( e ) ] { Λ } ( e ) { Λ ( e ) } T [ F t ( e ) ] ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiOda LcdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0JcdaaeWbqa amaadmaabaWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaakm aacmaabaqcLbsacqqHBoatlmaaCaaabeqaamaabmaabaqcLbmacaWG LbaaliaawIcacaGLPaaaaaaakiaawUhacaGL9baalmaaCaaabeqaaK qzadGaamivaaaakmaadmaabaqcLbsacaWGlbGcdaWgaaWcbaqcLbma caWGSbaaleqaaOWaaWbaaSqabeaajugibiaacQcaaaGcdaahaaWcbe qaaOWaaWbaaWqabeaalmaabmaameaajugWaiaadwgaaWGaayjkaiaa wMcaaaaaaaqcLbsacqGHRaWkcaWGlbGcdaWgaaWcbaqcLbmacaWGUb GaamiBaaWcbeaakmaaCaaaleqabaGcdaahaaadbeqaaSWaaeWaaWqa aKqzadGaamyzaaadcaGLOaGaayzkaaaaaaaakmaacmaabaqcLbsacq qHBoataOGaay5Eaiaaw2haamaaCaaaleqabaWaaeWaaeaajugWaiaa dwgaaSGaayjkaiaawMcaaaaaaOGaay5waiaaw2faamaacmaabaqcLb sacqqHBoataOGaay5Eaiaaw2haamaaCaaaleqabaWaaeWaaeaajugW aiaadwgaaSGaayjkaiaawMcaaaaajugibiabgkHiTOWaaiWaaeaaju gibiabfU5amPWaaWbaaSqabeaadaqadaqaaKqzadGaamyzaaWccaGL OaGaayzkaaaaaaGccaGL7bGaayzFaaWaaWbaaSqabeaajugWaiaads faaaGcdaWadaqaaKqzGeGaamOraOWaa0baaSqaaKqzadGaamiDaaWc baGcdaahaaadbeqaaSWaaeWaaWqaaKqzadGaamyzaaadcaGLOaGaay zkaaaaaaaaaOGaay5waiaaw2faaaGaay5waiaaw2faaaWcbaqcLbma caWGLbGaeyypa0JaaGymaaWcbaqcLbmacaWGobGaamyraaqcLbsacq GHris5aaaa@8F36@ = = 1 2 { q } T [ K l + K nl { q } ]{ q } { q } T [ F T ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaOWaaiWaaeaa jugibiaadghaaOGaay5Eaiaaw2haaSWaaWbaaeqabaqcLbmacaWGub aaaOWaamWaaeaajugibiaadUeakmaaBaaaleaajugWaiaadYgaaSqa baqcLbsacqGHRaWkcaWGlbGcdaWgaaWcbaqcLbmacaWGUbGaamiBaa WcbeaakmaacmaabaqcLbsacaWGXbaakiaawUhacaGL9baaaiaawUfa caGLDbaadaGadaqaaKqzGeGaamyCaaGccaGL7bGaayzFaaqcLbsacq GHsislkmaacmaabaqcLbsacaWGXbaakiaawUhacaGL9baadaahaaWc beqaaKqzadGaamivaaaakmaadmaabaqcLbsacaWGgbGcdaahaaWcbe qaaKqzadGaamivaaaaaOGaay5waiaaw2faaaaa@6025@    (25)

With

[ K l ]=[ K b ]+[ K s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb sacaWGlbGcdaWgaaWcbaqcLbmacaWGSbaaleqaaaGccaGLBbGaayzx aaqcLbsacqGH9aqpkmaadmaabaqcLbsacaWGlbGcdaWgaaWcbaqcLb macaWGIbaaleqaaaGccaGLBbGaayzxaaqcLbsacqGHRaWkkmaadmaa baqcLbsacaWGlbGcdaWgaaWcbaqcLbmacaWGZbaaleqaaaGccaGLBb Gaayzxaaaaaa@4A30@

where global bending stiffness matrix [Kb], shear stiffness matrix[Ks], global nonlinear stiffness matrix [Knl], global displacement vector {q}and thermal load vector [F] are defined as appendix.

  • Thermal post buckling anal thermal post buckling analysis

Using finite element model (Eq. (23), Eq. (15) can also be written as

Π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHGo aulmaaBaaabaqcLbmacaaIYaaaleqaaaaa@3A7F@ = 1 2 e=1 NE { Λ } T( e ) λ [ K g ] ( e ) { Λ } ( e ) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpkmaalaaabaqcLbsacaaIXaaakeaajugibiaaikdaaaGcdaaeWbqa amaacmaabaqcLbsacqqHBoataOGaay5Eaiaaw2haamaaCaaaleqaba qcLbmacaWGubWcdaqadaqaaKqzadGaamyzaaWccaGLOaGaayzkaaaa aKqzGeGaeq4UdWMcdaWadaqaaKqzGeGaam4saOWaaSbaaSqaaKqzad Gaam4zaaWcbeaaaOGaay5waiaaw2faaSWaaWbaaeqabaWaaeWaaeaa jugWaiaadwgaaSGaayjkaiaawMcaaaaakmaacmaabaqcLbsacqqHBo ataOGaay5Eaiaaw2haamaaCaaaleqabaWaaeWaaeaajugWaiaadwga aSGaayjkaiaawMcaaaaajugibiaadsgacaWGbbaaleaajugWaiaadw gacqGH9aqpcaaIXaaaleaajugWaiaad6eacaWGfbaajugibiabggHi Ldaaaa@6482@ = 1 2 λ { q } T [ K g ]{ q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpkmaalaaabaqcLbsacaaIXaaakeaajugibiaaikdaaaGaeq4UdWMc daGadaqaaKqzGeGaamyCaaGccaGL7bGaayzFaaWcdaahaaqabeaaju gWaiaadsfaaaGcdaWadaqaaKqzGeGaam4saSWaaSbaaeaajugWaiaa dEgaaSqabaaakiaawUfacaGLDbaadaGadaqaaKqzGeGaamyCaaGcca GL7bGaayzFaaaaaa@4BC6@    (26)

Were, λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3894@ and [ K g ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaju gibiaadUeakmaaBaaaleaajugWaiaadEgaaSqabaaakiaawUfacaGL Dbaaaaa@3C07@ are defined as the thermal post buckling load parameters and the global geometric stiffness matrix, respectively.

Π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHGo aulmaaBaaabaqcLbmacaaIYaaaleqaaaaa@3A7F@  

  • Strain energy due to elastic foundation

Using finite element notation after summed over all the element can be written as:

Π 3 =( e=1 NE Π fl ( e ) + e=1 NE Π fnl ( e ) ) , Π fl ( e ) = 1 2 A { Λ (e) } T [ K fl ] ( e ) { Λ (e) }dA , Π fnl ( e ) = 1 2 A { Λ (e) } T [ K fnl ] ( e ) { Λ (e) }dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiOda 1cdaWgaaqaaKqzadGaaG4maaWcbeaajugibiabg2da9iaacIcakmaa qahabaqcLbsacqqHGoaulmaaDaaabaWaaWbaaWqabeaajugWaiaadA gacaWGSbaaaaWcbaWaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMca aaaajugibiabgUcaROWaaabCaeaajugibiabfc6aqTWaa0baaeaada ahaaadbeqaaKqzadGaamOzaiaad6gacaWGSbaaaaWcbaWaaeWaaeaa jugWaiaadwgaaSGaayjkaiaawMcaaaaajugibiaacMcaaSqaaKqzad Gaamyzaiabg2da9iaaigdaaSqaaKqzadGaamOtaiaadweaaKqzGeGa eyyeIuoaaSqaaKqzadGaamyzaiabg2da9iaaigdaaSqaaKqzadGaam OtaiaadweaaKqzGeGaeyyeIuoacaGGSaGaeuiOda1cdaqhaaqaamaa CaaameqabaqcLbmacaWGMbGaamiBaaaaaSqaamaabmaabaqcLbmaca WGLbaaliaawIcacaGLPaaaaaqcLbsacqGH9aqpkmaalaaabaqcLbsa caaIXaaakeaajugibiaaikdaaaGcdaWdrbqaamaacmaabaqcLbsacq qHBoatlmaaCaaabeqaaKqzadGaaiikaiaadwgacaGGPaaaaaGccaGL 7bGaayzFaaWaaWbaaSqabeaajugWaiaadsfaaaGcdaWadaqaaKqzGe Gaam4saOWaaSbaaSqaaKqzadGaamOzaKqzGeGaamiBaaWcbeaaaOGa ay5waiaaw2faaSWaaWbaaeqabaWaaeWaaeaajugWaiaadwgaaSGaay jkaiaawMcaaaaakmaacmaabaqcLbsacqqHBoatlmaaCaaabeqaaKqz adGaaiikaiaadwgacaGGPaaaaaGccaGL7bGaayzFaaqcLbsacaWGKb GaamyqaaWcbaqcLbmacaWGbbaaleqajugibiabgUIiYdGaaiilaiab fc6aqTWaa0baaeaadaahaaadbeqaaKqzadGaamOzaiaad6gacaWGSb aaaaWcbaWaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaajugi biabg2da9OWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaakm aapefabaWaaiWaaeaajugibiabfU5amTWaaWbaaeqabaqcLbmacaGG OaGaamyzaiaacMcaaaaakiaawUhacaGL9baalmaaCaaabeqaaKqzad GaamivaaaakmaadmaabaqcLbsacaWGlbGcdaWgaaWcbaqcLbmacaWG MbGaamOBaiaadYgaaSqabaaakiaawUfacaGLDbaalmaaCaaabeqaam aabmaabaqcLbmacaWGLbaaliaawIcacaGLPaaaaaGcdaGadaqaaKqz GeGaeu4MdW0cdaahaaqabeaajugWaiaacIcacaWGLbGaaiykaaaaaO Gaay5Eaiaaw2haaKqzGeGaamizaiaadgeaaSqaaKqzadGaamyqaaWc beqcLbsacqGHRiI8aiaaysW7aaa@CB9F@  (27)

here [ K fl ] ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb sacaWGlbWcdaWgaaqaaKqzadGaamOzaiaadYgaaSqabaaakiaawUfa caGLDbaalmaaCaaabeqaamaabmaabaqcLbmacaWGLbaaliaawIcaca GLPaaaaaaaaa@405E@ ,and [ K fnl ] ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb sacaWGlbGcdaWgaaWcbaqcLbmacaWGMbGaamOBaiaadYgaaSqabaaa kiaawUfacaGLDbaalmaaCaaabeqaamaabmaabaqcLbmacaWGLbaali aawIcacaGLPaaaaaaaaa@415B@ are the elemental linear and nonlinear foundation stiffness matrices respectively.

Adopting Gauss quadrature integration numerical rule, the element stiffness and geometric stiffness respectively can be obtained by transforming expression in x, y coordinate system to natural coordinate system ᶓ, η.

Governing equation

The governing equation for thermally induced post buckling load of the plate analysis can be derived using the Lagrange’s equation of motion.

δ t 1 t 2 ( UWπ ) *TdT=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq McdaWdXaqaamaabmaabaqcLbsacaWGvbGaeyOeI0Iaam4vaiabgkHi Tiabec8aWbGccaGLOaGaayzkaaaaleaajugWaiaadshalmaaBaaame aajugWaiaaigdaaWqabaaaleaajugWaiaadshalmaaBaaameaajugW aiaaikdaaWqabaaajugibiabgUIiYdqcLbmacaGGQaqcLbsacaWGub GaamizaiaadsfacqGH9aqpcaaIWaaaaa@51BA@    (28)

Substituting Equations (25), (26) and (27) into Equation (28), ones obtain as in the form of non linear generalized eigen-value problem as

[K]{ q }+[ M ]{ q ¨ }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGBb Gaam4saiaac2fajuaGdaGadaGcbaqcLbsacaWGXbaakiaawUhacaGL 9baajugibiabgUcaRKqbaoaadmaakeaajugibiaad2eaaOGaay5wai aaw2faaKqbaoaacmaakeaajugibiqadghagaWaaaGccaGL7bGaayzF aaqcLbsacqGH9aqpcaaIWaaaaa@4984@    (29)

where,

The above Eq. (24) is the post buckling equation which can be solved literately as a linear eigen value problem assuming that the plate is vibrating in its principal mode in each iteration. For each iteration, the Eq. 19 can be expressed as generalized eigen-value problem as:

[ [K]λ[ M ] ]{ q }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqzGeGaai4waiaadUeacaGGDbGaeyOeI0Iaeq4UdWwcfa4aamWa aOqaaKqzGeGaamytaaGccaGLBbGaayzxaaaacaGLBbGaayzxaaqcfa 4aaiWaaOqaaKqzGeGaamyCaaGccaGL7bGaayzFaaqcLbsacqGH9aqp caaIWaaaaa@48DC@     (30)

where λ=ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBcqGH9aqpcqaHjpWDaaa@3B0C@ with ω is the natural frequency of the plate.

Equation (29) is the post buckling problem which is random in nature, being dependent on the system properties. Consequently, the natural frequency and mode shapes are random in nature. In deterministic environment, the Eq. (30) is evaluated using eigen-value formulation and solved employing a direct iterative methods, Newton-Rapsion methods, incremental methods etc. However, in random environment, it is not possible to solve the problem using the above mentioned methods without changing the nature of the equation. Based on this, the direct iterative method combined with nonlinear finite element method, i.e., (DISFEM) with a reasonable accuracy is used to determine the second-order statistics (mean and standard deviation) of post buckling response of the FGM plates. The nonlinear eigen-value problem as given in Eq. 30 is solved by employing a direct iterative method in conjunction with perturbation technique assuming that the random changes in eigenvector during iterations does not affect much the nonlinear stiffness matrix with the steps24‒46

Solution approach-disfem

We consider a class of problems where the zero mean random variation is very small when compared to the mean part of random system properties. In general, without any loss of generality any arbitrary random variable can be represented as the sum of its mean and a zero mean random variable, expressed by superscripts ‘d’ and ‘r’, respectively,25,26

K= K d + K r , K g = K g d + K g r , λ i = λ i d + λ i r , q i = q i d + q i r , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb Gaeyypa0Jaam4saSWaaWbaaKqbagqabaqcLbmacaWGKbaaaKqzGeGa ey4kaSIaam4saSWaaWbaaKqbagqabaqcLbmacaWGYbaaaKqzGeGaai ilaiaaysW7caWGlbWcdaWgaaqaaKqzadGaam4zaaWcbeaajugibiab g2da9iaaysW7caWGlbWcdaqhaaqaaKqzadGaam4zaaWcbaqcLbmaca WGKbaaaKqzGeGaey4kaSIaam4saSWaa0baaeaajugWaiaadEgaaSqa aKqzadGaamOCaaaajugibiaacYcacqaH7oaBlmaaBaaabaqcLbmaca WGPbaaleqaaKqzGeGaeyypa0Jaeq4UdW2cdaWgaaqaaKqzadGaamyA aaWcbeaadaahaaqabeaajugWaiaadsgaaaqcLbsacqGHRaWkcqaH7o aBlmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqaaKqzadGaamOC aaaajugibiaacYcacaaMe8UaaGjbVlaadghalmaaBaaabaqcLbmaca WGPbaaleqaaKqzGeGaeyypa0JaamyCaSWaaSbaaeaajugWaiaadMga aSqabaWaaWbaaeqabaqcLbmacaWGKbaaaKqzGeGaey4kaSIaamyCaS WaaSbaaeaajugWaiaadMgaaSqabaWaaWbaaeqabaqcLbmacaWGYbaa aKqzGeGaaiilaaaa@81F1@  (32)

The parameter p indicates the size of eigen problem.

Consider a class of problems where the random variation is very small as compared to the mean part of material properties. Further it is quite logical to assume that the dispersions in the derived quantities like andare also small as compared to their mean values. By substituting Eq. (32) in Eq. (30) and collecting same order of the magnitude term and keeping only up to first order terms, one obtains as,24

Zeroth order: [ K d ]{ q i d }= λ i d [ K g ]{ q i d } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajugibiaadUealmaaDaaabaaabaqcLbmacaWGKbaaaaGccaGLBbGa ayzxaaqcfa4aaiWaaOqaaKqzGeGaamyCaSWaa0baaeaajugWaiaadM gaaSqaaKqzadGaamizaaaaaOGaay5Eaiaaw2haaKqzGeGaeyypa0Ja eq4UdW2cdaqhaaqaaKqzadGaamyAaaWcbaqcLbmacaWGKbaaaKqbao aadmaakeaajugibiaadUealmaaDaaabaqcLbmacaWGNbaaleaaaaaa kiaawUfacaGLDbaajuaGdaGadaGcbaqcLbsacaWGXbWcdaqhaaqaaK qzadGaamyAaaWcbaqcLbmacaWGKbaaaaGccaGL7bGaayzFaaaaaa@5BBE@        (33)

First order: ( K d ){ q i r }+( K r ) q i d = λ i d [ K g r ]( q i d )+ λ i d [ K g d ]{ q i r }+ λ i r [ K g d ]( q i d ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG9ajuaGda qadaGcbaqcLbsacaWGlbqcfa4aaWbaaSqabeaajugWaiaadsgaaaaa kiaawIcacaGLPaaajuaGdaGadaGcbaqcLbsacaWGXbWcdaWgaaqaaK qzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadkhaaaaakiaawUha caGL9baajugibiabgUcaRKqbaoaabmaakeaajugibiaadUealmaaCa aabeqaaKqzadGaamOCaaaaaOGaayjkaiaawMcaaKqzGeGaamyCaSWa aSbaaeaajugWaiaadMgaaSqabaWaaWbaaeqabaqcLbmacaWGKbaaaK qzGeGaeyypa0Jaeq4UdW2cdaWgaaqaaKqzadGaamyAaaWcbeaadaah aaqabeaajugWaiaadsgaaaqcfa4aamWaaOqaaKqzGeGaam4saSWaa0 baaeaajugWaiaadEgaaSqaaKqzadGaamOCaaaaaOGaay5waiaaw2fa aKqbaoaabmaakeaajugibiaadghalmaaBaaabaqcLbmacaWGPbaale qaamaaCaaabeqaaKqzadGaamizaaaaaOGaayjkaiaawMcaaKqzGeGa ey4kaSIaeq4UdW2cdaWgaaqaaKqzadGaamyAaaWcbeaadaahaaqabe aajugWaiaadsgaaaqcfa4aamWaaOqaaKqzGeGaam4saSWaa0baaeaa jugWaiaadEgaaSqaaKqzadGaamizaaaaaOGaay5waiaaw2faaKqbao aacmaakeaajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaa CaaabeqaaKqzadGaamOCaaaaaOGaay5Eaiaaw2haaKqzGeGaey4kaS Iaeq4UdW2cdaWgaaqaaKqzadGaamyAaaWcbeaadaahaaqabeaajugW aiaadkhaaaqcfa4aamWaaOqaaKqzGeGaam4saSWaa0baaeaajugWai aadEgaaSqaaKqzadGaamizaaaaaOGaay5waiaaw2faaKqbaoaabmaa keaajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabe qaaKqzadGaamizaaaaaOGaayjkaiaawMcaaKqzGeGaaiilaaaa@9E90@        (34)

Eq. (33) is the deterministic equations relating to the mean eigen-values and corresponding mean eigenvectors, which can be determined by conventional eigen solution algorithms. For solving first order Eq. (34) the first order perturbation approach is employed.

According to the orthogonality properties, the normalized eigenvector meet the followi ng conditions.

{ q i d } T [ K g d ]{ q i d }= δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaiWaae aajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqa aKqzadGaamizaaaaaOGaay5Eaiaaw2haaSWaaWbaaeqabaqcLbmaca WGubaaaOWaamWaaeaajugibiaadUealmaaBaaabaqcLbmacaWGNbaa leqaamaaCaaabeqaaKqzadGaamizaaaaaOGaay5waiaaw2faamaacm aabaqcLbsacaWGXbWcdaWgaaqaaKqzadGaamyAaaWcbeaadaahaaqa beaajugWaiaadsgaaaaakiaawUhacaGL9baajugibiabg2da9iabes 7aKTWaaSbaaeaajugWaiaadMgacaWGQbaaleqaaaaa@57FD@    (35)

{ q i d } T [ K d ]{ q i d }= δ ij λ i d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaiWaae aajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqa aKqzadGaamizaaaaaOGaay5Eaiaaw2haaSWaaWbaaeqabaqcLbmaca WGubaaaOWaamWaaeaajugibiaadUealmaaCaaabeqaaKqzadGaamiz aaaaaOGaay5waiaaw2faamaacmaabaqcLbsacaWGXbWcdaWgaaqaaK qzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaaakiaawUha caGL9baajugibiabg2da9iabes7aKTWaaSbaaeaajugWaiaadMgaca WGQbaaleqaaKqzGeGaeq4UdW2cdaWgaaqaaKqzadGaamyAaaWcbeaa daahaaqabeaajugWaiaadsgaaaaaaa@5C86@ . ( i,j )=1,2,...,p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaeWaae aajugibiaadMgacaGGSaGaamOAaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaaiilaiaaysW7caaIYaGaaiilaiaaysW7caGGUaGaaG jbVlaac6cacaaMe8UaaiOlaiaacYcacaWGWbaaaa@49D0@     (36)

where δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG9ajugibi abes7aKTWaaSbaaeaajugWaiaadMgacaWGQbaaleqaaaaa@3C4A@ is the Kronecker delta.

The eigenvectors, which meet orthogonality, conditions after being properly, normalized form a complete orthonormal set and any vector in the space can be expressed as their linear combination of these eigenvectors. Hence, the ith random part of the eigenvectors can be expressed as

C ii r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGdbWcdaWgaaqaaKqzadGaamyAaiaadMgaaSqabaWaaWbaaeqabaqc LbmacaWGYbaaaKqzGeGaeyypa0JaaGimaaaa@3FF4@ . ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGPbGaeyiyIKRaamOAaaaa@3AF9@ , C ii r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGdbWcdaWgaaqaaKqzadGaamyAaiaadMgaaSqabaWaaWbaaeqabaqc LbmacaWGYbaaaKqzGeGaeyypa0JaaGimaaaa@3FF4@ , i=1, 2, …. , p (37)

where C ij r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGdbWcdaWgaaqaaKqzadGaamyAaiaadQgaaSqabaWaaWbaaeqabaqc LbmacaWGYbaaaaaa@3DA6@ s are small random coefficients to be determined.

Substituting Eq. (37) in Eq. (34), premultiplying, the first by { q i d } T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcfa4aai WaaOqaaKqzGeGaamyCaSWaaSbaaeaajugWaiaadMgaaSqabaWaaWba aeqabaqcLbmacaWGKbaaaaGccaGL7bGaayzFaaWcdaahaaqabeaaju gWaiaadsfaaaaaaa@41FE@ and second by { q j d } T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaiWaae aajugibiaadghalmaaBaaabaqcLbmacaWGQbaaleqaamaaCaaabeqa aKqzadGaamizaaaaaOGaay5Eaiaaw2haaSWaaWbaaeqabaqcLbmaca WGubaaaaaa@4147@ , ( ji ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaeWaae aajugibiaadQgacqGHGjsUcaWGPbaakiaawIcacaGLPaaaaaa@3C8C@ respectively and making use of orthogonality Eq. (36), one obtains as

λ i r = { q i d } T [ K r ]{ q i d } λ i d { q i d } T [ K g r ]{ q i d } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsacq aH7oaBlmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqaaKqzadGa amOCaaaajugibiabg2da9OWaaiWaaeaajugibiaadghalmaaBaaaba qcLbmacaWGPbaaleqaamaaCaaabeqaaKqzadGaamizaaaaaOGaay5E aiaaw2haamaaCaaaleqabaqcLbmacaWGubaaaOWaamWaaeaajugibi aadUealmaaCaaabeqaaKqzadGaamOCaaaaaOGaay5waiaaw2faamaa cmaabaqcLbsacaWGXbWcdaWgaaqaaKqzadGaamyAaaWcbeaadaahaa qabeaajugWaiaadsgaaaaakiaawUhacaGL9baajugibiabgkHiTiab eU7aSTWaaSbaaeaajugWaiaadMgaaSqabaWaaWbaaeqabaqcLbmaca WGKbaaaOWaaiWaaeaajugibiaadghalmaaBaaabaqcLbmacaWGPbaa leqaamaaCaaabeqaaKqzadGaamizaaaaaOGaay5Eaiaaw2haaSWaaW baaeqabaqcLbmacaWGubaaaOWaamWaaeaajugibiaadUealmaaBaaa baqcLbmacaWGNbaaleqaamaaCaaabeqaaKqzadGaamOCaaaaaOGaay 5waiaaw2faamaacmaabaqcLbsacaWGXbWcdaWgaaqaaKqzadGaamyA aaWcbeaadaahaaqabeaajugWaiaadsgaaaaakiaawUhacaGL9baaaa a@7A54@  (38)

C ij r = { q j d }[ K r ]{ q i d } λ i d { q j d }[ K g r ]{ q i d } ( λ i d λ j d ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGdbWcdaWgaaqaaKqzadGaamyAaiaadQgaaSqabaWaaWbaaeqabaqc LbmacaWGYbaaaKqzGeGaeyypa0JcdaWcaaqaamaacmaabaqcLbsaca WGXbWcdaWgaaqaaKqzadGaamOAaaWcbeaadaahaaqabeaajugWaiaa dsgaaaaakiaawUhacaGL9baadaWadaqaaKqzGeGaam4saSWaaWbaae qabaqcLbmacaWGYbaaaaGccaGLBbGaayzxaaWaaiWaaeaajugibiaa dghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqaaKqzadGaam izaaaaaOGaay5Eaiaaw2haaKqzGeGaeyOeI0Iaeq4UdW2cdaWgaaqa aKqzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaGcdaGada qaaKqzGeGaamyCaSWaaSbaaeaajugWaiaadQgaaSqabaWaaWbaaeqa baqcLbmacaWGKbaaaaGccaGL7bGaayzFaaWaamWaaeaajugibiaadU ealmaaBaaabaqcLbmacaWGNbaaleqaamaaCaaabeqaaKqzadGaamOC aaaaaOGaay5waiaaw2faamaacmaabaqcLbsacaWGXbWcdaWgaaqaaK qzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaaakiaawUha caGL9baaaeaadaqadaqaaKqzGeGaeq4UdW2cdaWgaaqaaKqzadGaam yAaaWcbeaadaahaaqabeaajugWaiaadsgaaaqcLbsacqGHsislcqaH 7oaBlmaaBaaabaqcLbmacaWGQbaaleqaamaaCaaabeqaaKqzadGaam izaaaaaOGaayjkaiaawMcaaaaaaaa@860C@  (39)

Substituting Eq. (34) into Eq. (32), we obtain

{ q i r }= j=1 p { q i d } { q j d } T [ K r ]{ q i d } λ i d { q j d } T [ K g r ]{ q i d } λ i d λ j d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaiWaae aajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqa aKqzadGaamOCaaaaaOGaay5Eaiaaw2haaKqzGeGaeyypa0JcdaaeWb qaamaacmaabaqcLbsacaWGXbWcdaWgaaqaaKqzadGaamyAaaWcbeaa daahaaqabeaajugWaiaadsgaaaaakiaawUhacaGL9baaaSqaaKqzad GaamOAaiabg2da9iaaigdaaSqaaKqzadGaamiCaaqcLbsacqGHris5 aOWaaSaaaeaadaGadaqaaKqzGeGaamyCaSWaaSbaaeaajugWaiaadQ gaaSqabaWaaWbaaeqabaqcLbmacaWGKbaaaaGccaGL7bGaayzFaaWa aWbaaSqabeaajugWaiaadsfaaaGcdaWadaqaaKqzGeGaam4saSWaaW baaeqabaqcLbmacaWGYbaaaaGccaGLBbGaayzxaaWaaiWaaeaajugi biaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqaaKqzad GaamizaaaaaOGaay5Eaiaaw2haaKqzGeGaeyOeI0Iaeq4UdW2cdaWg aaqaaKqzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaGcda GadaqaaKqzGeGaamyCaSWaaSbaaeaajugWaiaadQgaaSqabaWaaWba aeqabaqcLbmacaWGKbaaaaGccaGL7bGaayzFaaWaaWbaaSqabeaaju gWaiaadsfaaaGcdaWadaqaaKqzGeGaam4saSWaaSbaaeaajugWaiaa dEgaaSqabaWaaWbaaeqabaqcLbmacaWGYbaaaaGccaGLBbGaayzxaa WaaiWaaeaajugibiaadghalmaaBaaabaqcLbmacaWGPbaaleqaamaa CaaabeqaaKqzadGaamizaaaaaOGaay5Eaiaaw2haaaqaaKqzGeGaeq 4UdW2cdaWgaaqaaKqzadGaamyAaaWcbeaadaahaaqabeaajugWaiaa dsgaaaqcLbsacqGHsislcqaH7oaBlmaaBaaabaqcLbmacaWGQbaale qaamaaCaaabeqaaKqzadGaamizaaaaaaaaaa@9B95@ . ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGPbGaeyiyIKRaamOAaaaa@3AF9@    (40)

For the present case λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsacq aH7oaBaaa@3909@ , { q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaaiWaae aajugibiaadghaaOGaay5Eaiaaw2haaaaa@3A86@ , [ K ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaamWaae aajugibiaadUeaaOGaay5waiaaw2faaaaa@3A21@ , [ K f ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaamWaae aajugibiaadUealmaaBaaabaqcLbmacaWGMbaaleqaaaGccaGLBbGa ayzxaaaaaa@3C71@ , [ K f ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypWaamWaae aajugibiaadUealmaaBaaabaqcLbmacaWGMbaaleqaaaGccaGLBbGa ayzxaaaaaa@3C71@ and h are random because of random geometric and material properties. Let b 1 R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaqhaaqaaKqzadGaaGymaaWcbaqcLbmacaWGsbaaaaaa@3C62@ , b 2 R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaqhaaqaaKqzadGaaGOmaaWcbaqcLbmacaWGsbaaaaaa@3C63@ , b 3 R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaqhaaqaaKqzadGaaG4maaWcbaqcLbmacaWGsbaaaaaa@3C64@ , b q R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaqhaaqaaKqzadGaamyCaaWcbaqcLbmacaWGsbaaaaaa@3C9D@ …, denote random variables (system properties). The random variables (RVs) can be expressed as

b j R = b j d + b j r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaWgaaqaaKqzadGaamOAaaWcbeaadaahaaqabeaajugWaiaa dkfaaaqcLbsacqGH9aqpcaWGIbWcdaWgaaqaaKqzadGaamOAaaWcbe aadaahaaqabeaajugWaiaadsgaaaqcLbsacqGHRaWkcaWGIbWcdaWg aaqaaKqzadGaamOAaaWcbeaadaahaaqabeaajugWaiaadkhaaaaaaa@4AB3@    (41)

The FEM in conjunction with FOPT has been found to be accurate and efficient. According to this method, the random variables are expressed by Taylor’s series expansion. The expression only up to the first-order terms and neglecting the second- and higher-order terms are

λ i r = j=1 p λ i , d j h j r ;{ q i r }= j=1 p { q i d , j } h j r ; [ K r ]= j=1 p [ K d , j ] h j r ;[ K g r ]= j=1 p [ K g d , j ] h j r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabceqabaa2deaaju gibiabeU7aSTWaaSbaaeaajugWaiaadMgaaSqabaWaaWbaaeqabaqc LbmacaWGYbaaaKqzGeGaeyypa0JcdaaeWbqaaKqzGeGaeq4UdW2cda WgaaqaaKqzadGaamyAaaWcbeaakmaaDaaaleaajugibiaacYcaaSqa aKqzadGaamizaaaakmaaBaaaleaajugibiaadQgaaSqabaaabaqcLb macaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGWbaajugibiabggHi LdqcLbmacaWGObGcdaWgaaWcbaqcLbsacaWGQbaaleqaamaaCaaabe qaaKqzadGaamOCaaaajugibiaacUdacaaMe8UaaGjbVlaaysW7kmaa cmaabaqcLbsacaaMe8UaamyCaSWaaSbaaeaajugWaiaadMgaaSqaba WaaWbaaeqabaqcLbmacaWGYbaaaaGccaGL7bGaayzFaaqcLbsacqGH 9aqpkmaaqahabaWaaiWaaeaajugibiaadghalmaaBaaabaqcLbmaca WGPbaaleqaamaaCaaabeqaaKqzadGaamizaaaakmaaBaaaleaajugi biaacYcaaSqabaGcdaWgaaWcbaqcLbsacaWGQbaaleqaaaGccaGL7b GaayzFaaaaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaa dchaaKqzGeGaeyyeIuoacaWGObWcdaWgaaqaaKqzadGaamOAaaWcbe aadaahaaqabeaajugWaiaadkhaaaqcLbsacaGG7aGaaGjbVlaaysW7 caaMe8oakeaajugibiaaysW7kmaadmaabaqcLbsacaWGlbWcdaahaa qabeaajugWaiaadkhaaaaakiaawUfacaGLDbaajugibiabg2da9OWa aabCaeaadaWadaqaaKqzGeGaam4saOWaaWbaaSqabeaajugWaiaads gaaaGcdaWgaaWcbaqcLbsacaGGSaaaleqaaOWaaSbaaSqaaKqzGeGa amOAaaWcbeaaaOGaay5waiaaw2faaaWcbaqcLbmacaWGQbGaeyypa0 JaaGymaaWcbaqcLbmacaWGWbaajugibiabggHiLdGaamiAaSWaaSba aeaajugWaiaadQgaaSqabaWaaWbaaeqabaqcLbmacaWGYbaaaKqzGe Gaai4oaiaaysW7caaMe8UaaGjbVlaaysW7caaMe8UcdaWadaqaaKqz GeGaam4saSWaaSbaaeaajugWaiaadEgaaSqabaWaaWbaaeqabaqcLb macaWGYbaaaaGccaGLBbGaayzxaaqcLbsacqGH9aqpkmaaqahabaWa amWaaeaajugibiaadUealmaaBaaabaqcLbmacaWGNbaaleqaamaaCa aabeqaaKqzadGaamizaaaakmaaBaaaleaajugibiaacYcaaSqabaGc daWgaaWcbaqcLbsacaWGQbaaleqaaaGccaGLBbGaayzxaaaaleaaju gWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaadchaaKqzGeGaeyye IuoacaWGObWcdaWgaaqaaKqzadGaamOAaaWcbeaadaahaaqabeaaju gWaiaadkhaaaaaaaa@D65C@     (42)

Where (, j) denotes the partial differentiation with respect to b j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb WcdaWgaaqaaKqzadGaamOAaaWcbeaaaaa@39C0@ .

On substitution of Eq. (42) into Eq. (38), one obtain as

λ i , d j = { q i d } T [ K d , j ]{ q i d } λ i d { q i d } T [ K g d , j ]{ q i d } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsacq aH7oaBlmaaBaaabaqcLbmacaWGPbaaleqaaOWaa0baaSqaaKqzGeGa aiilaaWcbaqcLbmacaWGKbaaaSWaaSbaaeaajugWaiaadQgaaSqaba qcLbsacqGH9aqpkmaacmaabaqcLbsacaWGXbWcdaWgaaqaaKqzadGa amyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaaakiaawUhacaGL9b aadaahaaWcbeqaaKqzadGaamivaaaakmaadmaajuaGbaqcLbsacaWG lbGcdaahaaqcfayabeaajugWaiaadsgaaaGcdaWgaaqcfayaaKqzGe GaaiilaaqcfayabaGcdaWgaaqcfayaaKqzadGaamOAaaqcfayabaaa caGLBbGaayzxaaGcdaGadaqaaKqzGeGaamyCaSWaaSbaaeaajugWai aadMgaaSqabaWaaWbaaeqabaqcLbmacaWGKbaaaaGccaGL7bGaayzF aaqcLbsacqGHsislcqaH7oaBlmaaBaaabaqcLbmacaWGPbaaleqaam aaCaaabeqaaKqzadGaamizaaaakmaacmaabaqcLbsacaWGXbWcdaWg aaqaaKqzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadsgaaaaaki aawUhacaGL9baalmaaCaaabeqaaKqzadGaamivaaaakmaadmaajuaG baqcLbsacaWGlbWcdaWgaaqcgayaaKqzadGaam4zaaqcgayabaWcda ahaaqcgayabeaajugWaiaadsgaaaGcdaWgaaqcfayaaKqzGeGaaiil aaqcfayabaWcdaWgaaqcfayaaKqzadGaamOAaaqcfayabaaacaGLBb GaayzxaaGcdaGadaqaaKqzGeGaamyCaSWaaSbaaeaajugWaiaadMga aSqabaWaaWbaaeqabaqcLbmacaWGKbaaaaGccaGL7bGaayzFaaaaaa@8D08@    (43)

The variance of the eigen-values can now be expressed as

Var( λ i )= j=1 q k=1 q λ i,j d λ i,k d Cov( b j r , b k r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGwbGaamyyaiaadkhakmaabmaabaqcLbsacqaH7oaBkmaaBaaaleaa jugWaiaadMgaaSqabaaakiaawIcacaGLPaaajugibiabg2da9OWaaa bCaeaadaaeWbqaaKqzGeGaeq4UdWMcdaqhaaWcbaqcLbmacaWGPbqc LbsacaGGSaqcLbmacaWGQbaaleaajugWaiaadsgaaaaaleaajugWai aadUgacqGH9aqpcaaIXaaaleaajugWaiaadghaaKqzGeGaeyyeIuoa aSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzadGaamyCaaqcLb sacqGHris5aiabeU7aSPWaa0baaSqaaKqzadGaamyAaKqzGeGaaiil aKqzadGaam4AaaWcbaqcLbmacaWGKbaaaKqzGeGaam4qaiaad+gaca WG2bGcdaqadaqaaKqzGeGaamOyaSWaaSbaaeaajugWaiaadQgaaSqa baWaaWbaaeqabaqcLbmacaWGYbaaaKqzGeGaaiilaiaadkgalmaaBa aabaqcLbmacaWGRbaaleqaamaaCaaabeqaaKqzadGaamOCaaaaaOGa ayjkaiaawMcaaaaa@78DC@     (44)

where Cov( b j r , b k r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGdbGaam4BaiaadAhakmaabmaabaqcLbsacaWGIbWcdaWgaaqaaKqz adGaamOAaaWcbeaakmaaCaaaleqabaqcLbmacaWGYbaaaKqzGeGaai ilaiaadkgalmaaBaaabaqcLbmacaWGRbaaleqaamaaCaaabeqaaKqz adGaamOCaaaaaOGaayjkaiaawMcaaaaa@4891@ is the covariance between b j r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaypqcLbsaca WGIbWcdaWgaaqaaKqzadGaamOAaaWcbeaadaahaaqabeaajugWaiaa dkhaaaaaaa@3CD7@ and b k r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaciaaW9paypqcLb sacaWGIbWcdaWgaaqaaKqzadGaam4AaaWcbeaadaahaaqabeaajugW aiaadkhaaaaaaa@3E48@ . The standard deviation (SD) is obtained by the square root of the variance.

Results and discussion

A DISFEM probabilistic approach outlined in previous subsection for post buckling response of the elastically supported FGM plate in thermal environments with random material properties has been presented through numerical examples. The present probabilistic approach has been validated by comparing the results with those available in literature and independent MCS. Typical results for mean and coefficient of variance {COV (=SD/mean)} of the post buckling response for FGM square plate in the given following text are presented using MATLAB 7.10.0 (R2010a) software. A nine nodded Lagrangian isoparametric element with 63 degree of freedom per node for the present HSDT model has been used throughout the study. Based on the convergence study, a (4 x 4) mesh size has been used for numerical computation. Due to the linear nature of variation as mentioned earlier and passing through the origin, the results are only presented by COV of the system properties equal to 0.1. However, the obtained results revealed that the stochastic approach would be valid up to COV=0.2.44,45 Moreover, the presented results would be sufficient to extrapolate the results for other COV values keeping in mind the limitation of the FOPT. The assumed basic random input variables (bi) are sequenced and written as

b 1 = E c , b 2 = E m , b 3 = υ c , b 4 = υ m , b 5 =n, b 6 = k 1 , b 7 = k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGacaaC=da2dK qzGeGaamOyaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacqGH9aqp caWGfbWcdaWgaaqaaKqzadGaam4yaaWcbeaajugibiaacYcacaaMe8 UaaGjbVlaaysW7caWGIbWcdaWgaaqaaKqzadGaaGOmaaWcbeaajugi biabg2da9iaadwealmaaBaaabaqcLbmacaWGTbaaleqaaKqzGeGaai ilaiaaysW7caaMe8UaaGjbVlaadkgalmaaBaaabaqcLbmacaaIZaaa leqaaKqzGeGaeyypa0JaeqyXdu3cdaWgaaqaaKqzadGaam4yaaWcbe aajugibiaacYcacaaMe8UaaGjbVlaaysW7caWGIbWcdaWgaaqaaKqz adGaaGinaaWcbeaajugibiabg2da9iabew8a1LqbaoaaBaaaleaaju gWaiaad2gaaSqabaqcLbsacaGGSaGaaGjbVlaaysW7caaMe8UaamOy aKqbaoaaBaaaleaajugWaiaaiwdaaSqabaqcLbsacqGH9aqpcaWGUb GaaiilaiaaysW7caaMe8UaaGjbVlaadkgalmaaBaaabaqcLbmacaaI 2aaaleqaaKqzGeGaeyypa0Jaam4AaSWaaSbaaeaajugWaiaaigdaaS qabaqcLbsacaaMe8UaaiilaiaaysW7caaMe8UaaGjbVlaadkgalmaa BaaabaqcLbmacaaI3aaaleqaaKqzGeGaeyypa0Jaam4AaSWaaSbaae aajugWaiaaikdaaSqabaqcLbsacaaMe8oaaa@96AC@

where E c , E m , υ c , υ m , n, k 1 k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGacaaC=da2dK qzGeGaamyraSWaaSbaaeaajugWaiaadogaaSqabaqcLbsacaGGSaGa aGjbVlaadwealmaaBaaabaqcLbmacaWGTbaaleqaaKqzGeGaaiilai aaysW7cqaHfpqDlmaaBaaabaqcLbmacaWGJbaaleqaaKqzGeGaaiil aiaaysW7cqaHfpqDjuaGdaWgaaWcbaqcLbmacaWGTbaaleqaaKqzGe GaaiilaiaabccacaWGUbGaaiilaiaadUgalmaaBaaabaqcLbmacaaI XaaaleqaaKqzGeGaaeilaiaabccacaWGRbWcdaWgaaqaaKqzadGaaG OmaaWcbeaaaaa@5C1E@ are Young’s moduli, Poisson’s ratios, volume fraction index and Winkler, Pasternak elastic foundations. Foundation Parameters taken as:

K 1 = ( k 1 *D D 1 ) ( a 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabg2da9Kqbaoaa liaakeaajuaGdaqadaGcbaqcLbsacaWGRbqcfa4aaSbaaSqaaKqzad GaaGymaaWcbeaajugibiaacQcacaWGebGaamiraKqbaoaaBaaaleaa jugWaiaaigdaaSqabaaakiaawIcacaGLPaaaaeaajuaGdaqadaGcba qcLbsacaWGHbGcdaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPaaa aaaaaa@4CAC@ , K 2 = ( k 2 *D D 1 ) ( a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqbaoaa liaakeaajuaGdaqadaGcbaqcLbsacaWGRbqcfa4aaSbaaSqaaKqzad GaaGOmaaWcbeaajugibiaacQcacaWGebGaamiraKqbaoaaBaaaleaa jugWaiaaigdaaSqabaaakiaawIcacaGLPaaaaeaajuaGdaqadaGcba qcLbsacaWGHbGcdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaa aaaaaa@4CAC@ , where D D 1 = ( Ec* h 3 ) ( 12*( 1V c 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb GaamiraKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqp juaGdaWccaGcbaqcfa4aaeWaaOqaaKqzGeGaamyraiaadogacaGGQa GaamiAaKqbaoaaBaaaleaajugWaiaaiodaaSqabaaakiaawIcacaGL PaaaaeaajuaGdaqadaGcbaqcLbsacaaIXaGaaGOmaiaacQcajuaGda qadaGcbaqcLbsacaaIXaGaeyOeI0IaamOvaiaadogajuaGdaWgaaWc baqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaa aaaaaa@5364@ .

In the present study, the various combination of boundary edge support conditions namely, simply supported, clamped and two opposite edges are clamped and simply supported, the temperature of metal and ceramic are taken as Tc=600°K,Tm=300°K, with uniform temperature distribution. The boundary conditions can be written as

All edges simply supported (SSSS)

v=w= θ y = ψ y =0,atx=0,a;u=w= θ x = ψ x =0aty=0,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiGaaa3=aG9aju gibiaadAhacqGH9aqpcaWG3bGaeyypa0JaeqiUdexcfa4aaSbaaSqa aKqzadGaamyEaaWcbeaajugibiabg2da9iabeI8a5TWaaSbaaeaaju gWaiaadMhaaSqabaqcLbsacqGH9aqpcaaIWaGaaiilaiaaysW7caWG HbGaamiDaiaaysW7caWG4bGaeyypa0JaaGimaiaacYcacaaMe8Uaam yyaiaacUdacaaMf8UaamyDaiabg2da9iaadEhacqGH9aqpcqaH4oqC juaGdaWgaaWcbaqcLbmacaWG4baaleqaaKqzGeGaeyypa0JaeqiYdK 3cdaWgaaqaaKqzadGaamiEaaWcbeaajugibiabg2da9iaaicdacaaM e8UaamyyaiaadshacaaMe8UaamyEaiabg2da9iaaicdacaGGSaGaaG jbVlaadkgaaaa@72FF@ ;

All edges clamped (CCCC):

u=v=w= ψ x = ψ y = θ x = θ y =0,atx=0,aandy=0,b; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiGaaa3=aG9aju gibiaadwhacqGH9aqpcaWG2bGaeyypa0Jaam4Daiabg2da9iabeI8a 5LqbaoaaBaaaleaajugWaiaadIhaaSqabaqcLbsacqGH9aqpcqaHip qEjuaGdaWgaaWcbaqcLbmacaWG5baaleqaaKqzGeGaeyypa0JaeqiU de3cdaWgaaqaaKqzadGaamiEaaWcbeaajugibiabg2da9iabeI7aXL qbaoaaBaaaleaajugWaiaadMhaaSqabaqcLbsacqGH9aqpcaaIWaGa aiilaiaaysW7caWGHbGaamiDaiaaysW7caWG4bGaeyypa0JaaGimai aacYcacaaMe8UaamyyaiaaywW7caWGHbGaamOBaiaadsgacaaMe8Ua amyEaiabg2da9iaaicdacaGGSaGaaGjbVlaadkgacaGG7aaaaa@7028@

Two opposite edges clamped and other two simply supported (CSCS):

u=v=w= ψ x = ψ y = θ x = θ y =0,atx=0andy=0; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiGaaa3=aG9aju gibiaadwhacqGH9aqpcaWG2bGaeyypa0Jaam4Daiabg2da9iabeI8a 5TWaaSbaaeaajugWaiaadIhaaSqabaqcLbsacqGH9aqpcqaHipqElm aaBaaabaqcLbmacaWG5baaleqaaKqzGeGaeyypa0JaeqiUde3cdaWg aaqaaKqzadGaamiEaaWcbeaajugibiabg2da9iabeI7aXTWaaSbaae aajugWaiaadMhaaSqabaqcLbsacqGH9aqpcaaIWaGaaiilaiaaysW7 caWGHbGaamiDaiaaysW7caWG4bGaeyypa0JaaGimaiaaywW7caWGHb GaamOBaiaadsgacaaMe8UaamyEaiabg2da9iaaicdacaGG7aaaaa@6837@ ;

v=w= θ y = ψ y =0,atx=au=w= θ x = ψ x =0,aty=b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiGaaa3=aG9aju gibiaadAhacqGH9aqpcaWG3bGaeyypa0JaeqiUdexcfa4aaSbaaSqa aKqzadGaamyEaaWcbeaajugibiabg2da9iabeI8a5LqbaoaaBaaale aajugWaiaadMhaaSqabaqcLbsacqGH9aqpcaaIWaGaaiilaiaaysW7 caWGHbGaamiDaiaaysW7caWG4bGaeyypa0JaamyyaiaaysW7caaMe8 UaamyDaiabg2da9iaadEhacqGH9aqpcqaH4oqClmaaBaaabaqcLbma caWG4baaleqaaKqzGeGaeyypa0JaeqiYdK3cdaWgaaqaaKqzadGaam iEaaWcbeaajugibiabg2da9iaaicdacaGGSaGaaGjbVlaadggacaWG 0bGaaGjbVlaadMhacqGH9aqpcaWGIbaaaa@6E8F@

The dimensionless mean post buckling load of the FGM plate subjected to mechanical ( λ cr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX giofMCY92DaeXatLxBI9gBaerbuLwBLnhiov2DGi1BTfMBaerbd9wD YLwzYbItLDhariqtHjhBLrhDaibaieYdf9irVeeu0dXdh9vqqj=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiqaceaaceqaamWaeaWaau aaaOqaaKqzGeGaaq4UdSWaaSbaaeaajugWaiaadogacaWGYbaaleqa aaaa@3F9E@ ) and thermal ( λ Tcr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX giofMCY92DaeXatLxBI9gBaerbuLwBLnhiov2DGi1BTfMBaerbd9wD YLwzYbItLDhariqtHjhBLrhDaibaieYdf9irVeeu0dXdh9vqqj=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiqaceaaceqaamWaeaWaau aaaOqaaKqzGeGaaq4UdOWaaSbaaSqaaKqzadGaamivaiaadogacaWG Ybaaleqaaaaa@4081@ ) loadings is defined as respectively,22

λ cr = ( N b x ) cr b 2 E c h 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX giofMCY92DaeXatLxBI9gBaerbuLwBLnhiov2DGi1BTfMBaerbd9wD YLwzYbItLDhariqtHjhBLrhDaibaieYdf9irVeeu0dXdh9vqqj=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiqaceaaceqaamWaeaWaau aaaOqaciaaW9paypqcLbsacaaH7oWcdaWgaaqaaKqzadGaam4yaiaa dkhaaSqabaqcLbsacqGH9aqpkmaalaaabaqcLbsacaGGOaGaamOtaS WaaWbaaeqabaqcLbmacaWGIbaaaOWaaSbaaSqaaKqzGeGaamiEaaWc beaajugibiaacMcalmaaBaaabaqcLbmacaWGJbGaamOCaaWcbeaaju gibiaadkgalmaaCaaabeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaamyr aSWaaSbaaeaajugWaiaadogaaSqabaqcLbsacaWGObGcdaahaaWcbe qaaKqzadGaaG4maaaaaaaaaa@5913@ and λ Tcr =λα (ΔT) cr × 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX giofMCY92DaeXatLxBI9gBaerbuLwBLnhiov2DGi1BTfMBaerbd9wD YLwzYbItLDhariqtHjhBLrhDaibaieYdf9irVeeu0dXdh9vqqj=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiqaceaaceqaamWaeaWaau aaaOqaciaaW9paypqcLbsacaaH7oGcdaWgaaWcbaqcLbmacaWGubGa am4yaiaadkhaaSqabaqcLbsacqGH9aqpcaaH7oGaaqySdiaacIcacq GHuoarcaWGubGaaiykaSWaaSbaaeaajugWaiaadogacaWGYbaaleqa aKqzGeGaey41aqRaaGymaiaaicdalmaaCaaabeqaaKqzadGaaG4maa aaaaa@53F2@

Where, ( N b x ) cr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX giofMCY92DaeXatLxBI9gBaerbuLwBLnhiov2DGi1BTfMBaerbd9wD YLwzYbItLDhariqtHjhBLrhDaibaieYdf9irVeeu0dXdh9vqqj=hEe eu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=J b9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiqaceaaceqaamWaeaWaau aaaOqaciaaW9paypqcLbsacaGGOaGaamOtaOWaaWbaaSqabeaajugW aiaadkgaaaGcdaWgaaWcbaqcLbsacaWG4baaleqaaKqzGeGaaiykaS WaaSbaaeaajugWaiaadogacaWGYbaaleqaaaaa@477C@ is dimensionalized critical post buckling mechanical and thermal of the FGM plate. The following temperature dependent [TID] material properties of functionally graded materials are given in Table 1.36,38 The value of temperature T is taken as 300K for the analysis.

Validation study for mean buckling response

The comparisons of non-dimensional critical buckling loads obtained by the present theory and those given by Thai and Choi38 & Reddy et al.40 presented in Table 1 and observed the close agreement between the results. It can be seen that, in this loading condition also, the non-dimensional critical buckling load decreases with the increase of power-law index. Table 2 shows the comparison of critical buckling load ( N cr = N cr a 2 π 2 D 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaqcfa 4aaubeaOqabSqaaKqzadGaam4yaiaadkhaaSqab0qaaKqzGeGaamOt aaaacqGH9aqpjuaGdaWcaaGcbaqcfa4aaubeaOqabSqaaKqzadGaam 4yaiaadkhaaSqab0qaaKqzGeGaamOtaaaacaWGHbqcfa4aaWbaaSqa beaajugWaiaaikdaaaaakeaajugibiabec8aWLqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqbaoaavabakeqaleaajugWaiaaicdaaSqab0qa aKqzGeGaamiraaaaaaaakiaawIcacaGLPaaaaaa@50A8@ for a simply supported FGM plate with a/h = 100 and r/a = 0.2. The effective material properties are computed by rule of mixtures. In order to be consistent with the literature, the properties of the metallic phase is used for normalization Tc=400, Tm=300. a=50, b=50. S1 Quarter considering 9th & 81st nodes. It is clearly seen present [HSDT] results are in good agreement with Natarajan et al.37 & Zhao.43 Validation study of critical buckling temperature rise for square aluminum /alumina plate with uniform temperature rise and plate thickness ratio (a/h) = 20, a=50, b=50, a/b=1. Tc =600, Tm=300 is presented in Table 3. Again it can be seen that present HSDT results are in good agreement with the results of Wu et al.22 & Lal et al.42 The validation study of results using present outlined approach is compared with those available in literature through numerical examples. Figure 2 examines the comparison of dimensionless mean post buckling load, TID material properties of clamped supported square FGM plate subjected to mechanical loading with those of Wu et al.22 & Lal et al.42 various volume fraction index (n) and amplitude ratios (Wmax/h). Clearly, it is seen that the present DISFEM results using HSDT are in good agreement with the available analytical results.

Validation study for stochastic post buckling response

The validation study for the COV of the initial buckling load due to randomness in material properties (bi, (i=1,..., 4) for CCCC bi-axially compressed square Al/ZrO2 plate with volume fraction index (n = 0.0 and 0.2) and a/h = 10, is shown in Figure 3 and compared with those published results of Yang et al.28 & Lal et al.42 It can be seen that the present outlined approach [DISFEM] for different volume fraction index are in very good agreement with the results of Lal et al.42 & Yang et al.28 using FSDT with semi-analytical approach in conjunction with FOPT. The comparison of probabilistic results obtained by DISFEM and independent MCS technique of square FGM plate without and with Winkler and Pasternak elastic foundations with simply supported boundary conditions in thermal environments and amplitude ratios, b/h=10 having random change in Ec material property keeping other as deterministic to their mean position is shown in Figure 4. For the MCS approach 10,000 random number based on convergence is used to simulate the results of desired mean and standard deviation (SD) using normal distribution function. From the figure, it is seen that present results obtained by DISFEM approach are very good agreements with the independent MCS approach.

Parametric study for second order statistics of post buckling response

Table 4 shows the effects of volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plate with TID material properties, Plate thickness ratio (a/h) =10. It is observed that expected mean post buckling load increases on increasing volume fraction index and simultaneously on increasing amplitude ratio. COV is varying different for different volume fraction index and amplitude ratio with combined effect of all input random variables. Effects of elastic foundations, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties, Plate thickness ratio (a/h) =10 is presented in Table 5. It is seen that expected mean post buckling load increases on increasing volume fraction index and amplitude ratios with combined input random variables for both Winkler and Pasternak elastic foundations and more especially for Pasternak elastic foundations. However COV decreases much for Pasternak elastic foundations.

Table 6 shows the effects of plate thickness ratios (a/h) with types of loadings, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plate with TID material properties. It is noticed that expected mean post buckling load decreases on increasing plate thickness ratio, volume fraction index and amplitude ratios with combined input random variables, whereas the COV decreases in same conditions. Amplitude ratios are noticeable for thin plates. Effects of elastic foundations, plate thickness ratios (a/h) with types of loadings, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties is presented in Table 7. It is observed that the plate thickness ratios, combined effects of input random variables and amplitude ratios with volume fraction index significantly affect the expected mean post buckling load for both Winkler and Pasternak elastic foundations. For same conditions the COV is much affected for Pasternak elastic foundations compared to Winkler elastic foundations. Table 8 shows the effects of plate aspect ratios (a/b), volume fraction (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean post buckling load and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plate with TID material properties. a/h=20. It is noticed that for square plates when there is increase of volume fraction index, combined input random variables and amplitude ratios, the post buckling load decreases and COV increases. For rectangular plates the post buckling load further increases and COV decreases on increasing volume fraction index. Effects of elastic foundations, plate aspect ratios (b/a), volume fraction (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties. a/h=20 is presented in Table 9. It is noticed that for given volume fraction index, amplitude ratios (Wmax/h) and input random variables the mean post buckling load increase and COV decrease for square plates, however the post buckling load and COV is higher when plate is resting on Pasternak elastic foundations compared to Winkler elastic foundations. On changing aspect ratios for rectangular plates post buckling load and COV further increase for given conditions. Table 10 shows the effects of various support conditions, volume fraction (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plates with TID material properties, a/h=30. It is observed that for SSSS support plates increase in volume fraction index, amplitude ratios and input random variables the post buckling load decreases and COV increases. But when plates are CSCS and CCCC support the mean post buckling load further increase; however there is marginally decrease in COV. SSSS support is superior to CCCC and CSCS support. Effects of elastic foundations, various support conditions, volume fraction (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plate resting on Winkler and Pasternak elastic foundations with TID material properties, a/h=30 is presented in Table 11. It is noticed that for given SSSS support conditions, volume fraction index, amplitude ratios and combined input random variables the mean post buckling load e decreases and COV increase for both Winkler and Pasternak elastic foundations support conditions; however the values are more significant for Pasternak elastic foundations support conditions. But when same plates are CSCS and CCCC the mean post buckling load further increases and COV decreases. Post buckling load and COV are higher for Pasternak elastic foundations. Table 12 shows the effects of material properties with volume fraction index (n), amplitude ratios (Wmax/h) and input random variables bi, [(i =1,…, 7)=0.10] on the dimensionless mean and COV, of post buckling load and temperature square geometrically nonlinear simple support FGM plates with TD material properties., a/h=20. It is noticed that on increasing volume fraction index and amplitude ratios with input random variables the mean post buckling load decreases and there is marginal increase in COV for given conditions. But on changing plate material there is significant decrease in mean post buckling load and increase in COV at similar conditions. The effects of material properties with volume fraction index (n), amplitude ratios (Wmax/h) and input random variables bi, [(i =1,…, 7) =0.10] on the dimensionless mean and COV of post buckling load and temperature square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TD material properties., a/h=20 is presented in Table 13. It is noticed that on increasing volume fraction index, amplitude ratios with input random variables the mean post buckling load change for Winkler foundations and they further increase for Pasternak elastic foundations; however the COV decreases and in more for Pasternak foundations. On changing plate material for same conditions the mean post buckling load further decrease and COV is higher for plates resting on Pasternak elastic foundations.

Figure 1 Geometry of Functionally graded material plate.
Figure 2 The validation study for the mean of the initial buckling load.
Figure 3 The validation study for the COV of the initial buckling load due to randomness in material properties.
Figure 4 Validation study for COV, random material property Ec.

Power law index, n

 

 

 

 

 

 

 

 

 

a/b

a/h

Source

0

0.5

1

2

5

10

20

100

0.5

5

Thai & Choi36

5.376

3.539

2.733

2.116

1.719

1.537

1.369

1.099

Reddy et al38

5.371

3.527

2.715

2.092

1.7

1.527

1.364

1.097

Present [HSDT]

5.3668

0.6083

2.8064

2.1522

1.6598

1.452

1.3044

1.0817

Table 1 Comparison of non-dimensional critical buckling load (N ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaqcLb sacaGGOaGaamOtaaaacaGGPaaaaa@38B4@ of simply supported Al/Al2O3 plate subjected to biaxial compression, where a=1, b=2. (N ¯ )= Ncr* a 2 E m * h 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaqcLb sacaGGOaGaamOtaaaacaGGPaGccqGH9aqpdaWcaaqaaKqzGeGaamOt aiaadogacaWGYbqcLbmacaGGQaqcLbsacaWGHbWcdaahaaqabeaaju gWaiaaikdaaaaakeaajugibiaadweakmaaBaaaleaajugWaiaad2ga aSqabaqcLbmacaGGQaqcLbsacaWGObWcdaahaaqabeaajugWaiaaio daaaaaaaaa@4BB1@

Power index, n

Natarajan et al.37

Zhao et al.43

Present [HSDT]

0

5.2831

5.2611

5.2575

0.2

4.6919

4.6564

4.7165

1

3.663

3.6761

3.7855

2

3.3961

3.3672

3.4882

5

3.1073

3.1238

3.1898

10

2.8947

2.9366

3.1267

Table 2 Comparison of critical buckling load ( N cr = N cr a 2 π 2 D 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaavabakeqaleaajugibiaadogacaWGYbaaleqaneaajugi biaad6eaaaGaeyypa0tcfa4aaSaaaOqaaKqbaoaavabakeqaleaaju gWaiaadogacaWGYbaaleqaneaajugibiaad6eaaaqcfa4aaubiaOqa bSqabeaajugWaiaaikdaa0qaaKqzGeGaamyyaaaaaOqaaKqbaoaava cakeqaleqabaqcLbmacaaIYaaaneaajugibiabec8aWbaajuaGdaqf qaGcbeWcbaqcLbmacaaIWaaaleqaneaajugibiaadseaaaaaaaGcca GLOaGaayzkaaaaaa@5184@ for a simply supported FGM plate with a/h = 100 and r/a = 0.2. The effective material properties are computed by rule of mixtures. In order to be consistent with the literature, the properties of the metallic phase is used for normalization Tc=400, Tm=300. a=50, b=50. S1 Quarter considering 9th & 81st nodes

Boundary

 

 < λ Tcr ( αΔT× 10 3 )      MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaadsfacaWGJbGaamOCaaqabaGccaqG9aGaaeiiamaabmaa baGaeqySdeMaeuiLdqKaamivaiabgEna0kaaigdacaaIWaWaaWbaaS qabeaacaaIZaaaaaGccaGLOaGaayzkaaGaaeiiaiaabccacaqGGaGa aeiiaiaabccaaaa@48F1@

 

Conditions

n = 0

n = 1

SSSS

Present [HSDT]

0.4574

0.2314

Lal et al.42

0.5801

0.2737

Wu et al.22

0.43

0.226

CCCC

Present [HSDT]

1.2176

0.6242

Lal et al.42

1.0539

0.5024

Wu et al.22

1.192

0.944

CSCS

Present [HSDT]

1.0338

0.5299

Lal et al.42

1.0527

0.5017

Wu et al.22

1.01

0.38

Table 3 Validation study of critical buckling temperature rise for square aluminum /alumina plate with uniform temperature rise and plate thickness ratio (a/h) = 20, a=50, b=50, a/b=1. Tc =600, Tm=300

n

Wmax/h

Uniform temperature distribution

Without foundation (k1=0, k2=0)

Mean

COV

0

0.5

3.3767

0.1107

1

4.1679

0.1391

0.5

0.5

2.253

0.096

1

2.743

0.119

1

0.5

1.7521

0.0972

1

1.904

0.0885

2

0.5

1.3542

0.1148

1

1.7013

0.1242

5

0.5

1.0747

0.1087

1

1.1571

0.0955

10

0.5

0.9503

0.0946

1

1.1538

0.1173

Table 4 Effects of volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i=1,…,7)=0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plate with TID material properties, Plate thickness ratio (a/h) =10

n

Wmax/h

Uniform temperature distribution

 

Winkler foundation

 

Pasternak foundation

(k1=100, k2=0)

 

(k1=100, k2=10)

 

Mean

COV

Mean

COV

0

0.5

3.7179

0.1114

4.5512

0.4912

1

4.4789

0.14

5.3048

0.6746

0.5

0.5

2.588

0.098

3.388

0.296

1

3.1178

0.124

3.943

0.285

1

0.5

2.0774

0.0982

2.8526

1.0265

1

2.5357

0.1223

3.3447

0.9916

2

0.5

1.6652

0.1078

2.9764

0.701

1

3.0866

0.4965

1.7013

0.1242

5

0.5

1.3618

0.099

2.0663

1.2001

1

1.6196

0.1201

2.6598

0.9576

10

0.5

1.2232

0.0883

1.9041

0.589

1

1.3741

0.1068

2.097

0.6405

Table 5 Effects of elastic foundations, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i=1,…, 7) =0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties, Plate thickness ratio (a/h)=10

a/h

n

Wmax/h

Uniform temperature distribution

Without foundation (k1=0, k2=0)

Mean

COV

5

0.5

0.5

13.2938

0.1055

1

12.4514

0.2798

1

0.5

10.4303

0.1

1

11.8297

0.1451

2

0.5

7.9563

0.1027

1

9.0007

0.1381

5

0.5

5.8913

0.1004

1

6.1306

0.098

10

0.5

5.0413

0.1066

1

7.2093

0.0237

15

0.5

0.5

0.7336

0.095

1

0.9163

0.1191

1

0.5

0.5693

0.098

1

0.7179

0.1178

2

0.5

0.4413

0.1199

1

0.5559

0.1311

5

0.5

0.3567

0.1143

1

0.4369

0.1286

10

0.5

0.3178

0.0958

1

0.3791

0.1139

Table 6 Effects of plate thickness ratios (a/h) with types of loadings, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plate with TID material properties

a/h

n

Wmax/h

Uniform temperature distribution

 

Winkler foundation

Pasternak foundation

(k1=100, k2=0)

(k1=100, k2=10)

Mean

COV

Mean

COV

5

0.5

0.5

15.0098

0.114

19.1154

0.5609

1

15.8935

0.1282

19.2017

0.3122

1

0.5

12.1285

0.1083

17.2785

0.354

1

13.7303

0.1541

16.6147

1.4049

2

0.5

9.4429

0.0971

18.3407

0.2308

1

10.9041

0.3797

20.4814

0.165

5

0.5

9.0992

0.1053

14.9518

0.106

1

7.143

0.1651

14.219

0.7909

10

0.5

8.2368

0.1897

13.5099

0.1196

1

5.9524

0.355

10.2663

1.2258

15

0.5

0.5

0.8383

0.0973

1.0931

0.3443

1

1.021

0.1213

1.2706

0.3633

1

0.5

0.6724

0.0985

0.9197

1.1641

1

0.8195

0.1193

1.0492

1.2846

2

0.5

0.542

0.1123

0.7794

1.9097

1

0.6512

0.1285

0.8968

2.0143

5

0.5

0.4533

0.1044

0.6823

1.1346

1

0.5322

0.1212

0.761

1.6633

10

0.5

0.4113

0.0895

0.6322

0.8045

1

0.4731

0.1076

0.6998

0.945

Table 7 Effects of elastic foundations, plate thickness ratios (a/h) with types of loadings, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties

a/b

n

Wmax/h

Uniform temperature distribution

Without foundation (k1=0, k2=0)

Mean

COV

1

0.5

0.5

0.3259

0.0951

1

0.4037

0.1176

1

0.5

0.2527

0.0992

1

0.316

0.1178

2

0.5

0.1962

0.123

1

0.245

0.1341

5

0.5

0.1597

0.1175

1

0.1939

0.1315

10

0.5

0.1427

0.0968

1

0.1703

0.1163

2

0.5

0.5

1.2475

0.0853

1

1.8

0.1078

1

0.5

0.9707

0.0883

1

1.0947

0.0996

2

0.5

0.7359

0.098

1

1.0749

0.0946

5

0.5

0.5838

0.1151

1

0.7083

0.1199

10

0.5

0.5056

0.0945

1

0.6111

0.1149

Table 8 Effects of plate aspect ratios (a/b), volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of geometrically nonlinear simple support FGM plate with TID material properties. a/h=20

a/b

n

Wmax/h

Uniform temperature distribution

 

Winkler foundation

Pasternak foundation

(k1=100, k2=0)

(k1=100, k2=10)

Mean

COV

Mean

COV

1

0.5

0.5

0.3707

0.0972

0.4188

0.3381

1

0.4496

0.1195

0.5588

0.4078

1

0.5

0.2969

0.0992

0.4047

0.2275

1

0.3608

0.1187

0.4659

0.366

2

0.5

0.2397

0.1147

0.3278

0.3583

1

0.2891

0.1301

0.3929

0.1913

5

0.5

0.2022

0.1069

0.3021

0.6507

1

0.2365

0.1235

0.3376

0.8417

10

0.5

0.1843

0.0903

0.2818

0.8936

1

0.2119

0.1095

0.274

0.3863

2

0.5

0.5

1.2639

0.0855

1.4149

0.5259

1

1.8078

0. 0953

1.6686

0.5819

1

0.5

0.9872

0.0884

1.138

0.4088

1

1.4283

0.1055

1.4296

0.6815

2

0.5

0.7525

0.0979

0.9028

0.0844

1

1.088

0.0963

1.0163

0.1198

5

0.5

0.5999

0.1126

0.7377

0.0749

1

0.7118

0.1195

0.7884

0.1118

10

0.5

0.4513

0.0923

0.6583

0.1995

1

0.6196

0.1129

0.6914

0.2868

Table 9 Effects of elastic foundations, plate aspect ratios (a/b), volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature of geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties. a/h=20

Support conditions

n

Wmax/h

Uniform temperature distribution

Without foundation (k10, k2=0)

Mean

COV

SSSS

0.5

0.5

0.1023

0.0948

1

0.1255

0.1164

2

0.5

0.0616

0.1259

1

0.0762

0.1372

5

0.5

0.0505

0.1205

1

0.0607

0.135

10

0.5

0.0453

0.0981

1

0.0535

0.1173

CSCS

0.5

0.5

0.2102

0.0821

1

0.2261

0.0877

2

0.5

0.1271

0.0928

1

0.1435

0.0942

5

0.5

0.1023

0.0894

1

0.1112

0.0909

10

0.5

0.0906

0.0767

1

0.1078

0.076

CCCC

0.5

0.5

0.211

0.0819

1

0.2347

0.0884

2

0.5

0.1276

0.0925

1

0.1608

0.0883

5

0.5

0.1026

0.0891

1

0.1151

0.0892

10

0.5

0.0909

0.0765

1

0.1077

0.0762

Table 10 Effects of various support conditions, volume fraction (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plate with TID material properties, a/h=30

Support conditions

n

Wmax/h

Uniform temperature distribution

 

Winkler foundation

 

Pasternak foundation

(k1=100, k2=0)

 

(k1=100, k2=10)

Mean

COV

Mean

COV

SSSS

0.5

0.5

0.1156

0.0966

0.1491

0.3936

1

0.139

0.1176

0.1732

0.4528

2

0.5

0.0746

0.1168

0.1069

0.1148

1

0.0896

0.1315

3.9701

0.2063

5

0.5

0.0634

0.1091

0.0945

0.7642

1

0.0739

0.1258

0.105

0.0113

10

0.5

0.058

0.091

0.0885

0.9727

1

0.0665

0.1099

0.0972

0.1838

CSCS

0.5

0.5

0.2243

0.0834

0.2621

0.1452

1

0.2727

0.0979

0.2803

0.1189

2

0.5

0.1411

0.09

0.1787

0.5367

1

0.1635

0.0943

0.1928

0.0848

5

0.5

0.1162

0.085

0.1536

0.2724

1

0.1319

0.095

0.1546

0.0934

10

0.5

0.1045

0.0742

0.1413

0.7181

1

0.1164

0.0848

0.1415

0.4438

CCCC

0.5

0.5

0.2251

0.0832

0.2629

0.1452

1

0.268

0.1025

0.2879

0.1708

2

0.5

0.1417

0.0897

0.1794

0.5179

1

0.1672

0.101

0.1826

0.2046

5

0.5

0.1166

0.0848

0.154

0.2679

1

0.1256

0.0848

0.155

0.0924

10

0.5

0.1048

0.0741

0.1414

0.7231

1

0.1127

0.0785

0.1419

0.4431

Table 11 Effects of elastic foundations, various support conditions, volume fraction index (n), amplitude ratios (Wmax/h) and input random variables [bi, (i =1,…, 7) = 0.10] on the dimensionless mean and COV, of thermally induced post buckling load and temperature square geometrically nonlinear simple support FGM plates resting on Winkler and Pasternak elastic foundations with TID material properties, a/h=30

Types of
Material Properties

n

Wmax/h

Non-uniform temperature distribution

Without foundation (k1=0, k2=0)

Mean

COV

ZrO2/ Ti-6Al-4V

0.5

0.5

8.1124

0.0621

1

11.0616

0.1029

1

0.5

7.7498

0.0565

1

10.2511

0.0951

2

0.5

7.3481

0.0543

1

8.4

0.0883

5

0.5

6.758

0.0565

1

9.1007

0.1016

10

0.5

6.3506

0.0616

1

9.41

0.4281

Al/Al2O3

0.5

0.5

0.3259

0.0951

1

0.4037

0.1176

1

0.5

0.2527

0.0992

1

0.316

0.1178

2

0.5

0.1962

0.123

1

0.245

0.1341

5

0.5

0.1597

0.1175

1

0.1939

0.1315

10

0.5

0.1427

0.0968

1

0.1703

0.1163

Table 12 Effects of material properties with types of loadings, volume fraction index (n), with amplitude ratios (Wmax/h) and input random variables bi, [(i =1,…, 7) = 0.10] on the dimensionless mean and COV, of post buckling load and temperature square geometrically nonlinear simple support FGM plates with TD material properties., a/h=20

Conclusion

The DISFEM stochastic procedure outlined for present study to obtain the mean and COV of the post-buckling response of square FGM plates resting on Winkler and Pasternak elastic foundations, subjected to uniform change in temperature distribution with TID material properties. The characteristics of the post buckling load of plates are significantly influenced by the foundation stiffness parameters, amplitude ratios, material properties, support conditions, the plate thickness ratios, the aspect ratios and volume fraction index.48‒51 The mean and the dispersion of the post buckling load of plates are higher when the plates are subjected to ZrO2/Ti-6Al-4V material properties as compared to Al/Al2O3 case. The COV of post buckling response of the plate increases as the change in elastic foundations. This brings out the importance of considering thermal loading as one of the essential parameter from the design point of view especially in aerospace applications where reliability of the components resting on Pasternak elastic foundations in the presence of thermal loading is of significance. Increase in the volume fraction index, the post buckling load of the plate decreases especially it is less for plates resting on Pasternak elastic foundations. The effect of volume fraction index on the variation in post buckling response depends strongly on the material compositions and for reliability point of view; minimum volume fraction index should be taken as with Pasternak elastic foundations.

Acknowledgement

None.

Conflicts of Interest

Author declares that there is no conflict of interest.

Nomenclature

Aij, Bij, etc

:

Laminate stiffnesses

a, b                     

:

Plate length and breadth

h

:

Thickness of the plate

Ef, Em      

:

Elastic moduli of fiber and matrix, respectively.

Gf, Gm

:

Shear moduli of fiber and matrix, respectively.

vf, vm          

:

Poisson’s ratio of fiber and matrix, respectively.

Vm, Vf

:

Volume fraction of fiber and matrix, respectively.

αf, αm         

:

Coefficient of thermal expansion of fiber and matrix, respectively.

bi

:

Basic random material properties

E11, E22

:

Longitudinal and Transverse elastic moduli

G12, G13, G23

:

Shear moduli

Kl,

:

Linear bending stiffness matrix

Kg

:

Thermal geometric stiffness matrix

D

 

Elastic stiffness matrices

M αβ , m αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9yqqrpepu0dbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiabeg7aHjabek7aIbqabaGccaGGSaGaamyBamaaBaaaleaa cqaHXoqycqaHYoGyaeqaaaaa@3F52@

:

Mass and inertia matrices

ne, n

:

Number of elements, number of layers in the laminated plate

Nx, Ny, Nxy

 

In-plane thermal buckling loads

nn

:

Number of nodes per element

 Ni

:

Shape function of ith node

C ¯ p ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9yqqrpepu0dbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaai aadoeaaaWaaSbaaSqaamaaCaaameqabaWaaWbaaeqabaWaaWbaaeqa baGaamiCaaaaaaaaaSGaamyAaiaadQgacaWGRbGaamiBaaqabaaaaa@3C31@ /p>

:

Reduced elastic material constants

f, {f}(e)

:

Vector of unknown displacements, displacement vector of eth element

u, v, w

:

Displacements of a point on the mid plane of plate

u ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeea0dXdh9vqai=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzagGabmyDay aaraGcdaWgaaWcbaGaaGymaaqabaaaaa@385A@ , u ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeea0dXdh9vqai=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzagGabmyDay aaraGcdaWgaaWcbaGaaGOmaaqabaaaaa@385B@ , u ¯ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYlf9irVeea0dXdh9vqai=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzagGabmyDay aaraGcdaWgaaWcbaGaaG4maaqabaaaaa@385C@

:

Displacement of a point (x, y, z)

σ ¯ ij , ε ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9yqqrpepu0dbbG8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaai abeo8aZbaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiilamaanaaa baGaeqyTdugaamaaBaaaleaacaWGPbaabeaakmaaBaaaleaacaWGQb aabeaaaaa@3E8A@

:

Stress vector, Strain vector

ψy, ψx

:

Rotations of normal to mid plane about the x and y axis respectively

θx, θy, θk

:

Two slopes and angle of fiber orientation wrt x-axis for kth layer

x, y, z

:

Cartesian coordinates

ρ, λ, Var(.)

:

Mass density, eigenvalue, variance

ω , ϖ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqO1dyhaaa@37CF@

:

Fundamental frequency and its dimensionless form

RVs 

:

Random variables

DT, DC,

:

Difference in temperatures and moistures

α1, α2,  β1,  β2

:

Thermal expansion and hygroscopic coefficients along x and y direction, respectively.

Appendix

D=[ A 1ij B ij E ij 0 0 B ij D 1ij F 1ij 0 0 E ij F 1ij H ij 0 0 0 0 0 A 2ij D 2ij 0 0 0 D 2ij F 2ij ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiGaaa3=aG9aju gibiaadseacqGH9aqpkmaadmaabaqcLbsafaqabeqbfaaaaaGcbaqc LbsacaWGbbGcdaWgaaWcbaqcLbmacaaIXaGaamyAaiaadQgaaSqaba aakeaajugibiaadkealmaaBaaabaqcLbmacaWGPbGaamOAaaWcbeaa aOqaaKqzGeGaamyraSWaaSbaaeaajugWaiaadMgacaWGQbaaleqaaa GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaamOqaSWa aSbaaeaajugWaiaadMgacaWGQbaaleqaaaGcbaqcLbsacaWGebWcda WgaaqaaKqzadGaaGymaiaadMgacaWGQbaaleqaaaGcbaqcLbsacaWG gbWcdaWgaaqaaKqzadGaaGymaiaadMgacaWGQbaaleqaaaGcbaqcLb sacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaamyraSWaaSbaaeaa jugWaiaadMgacaWGQbaaleqaaaGcbaqcLbsacaWGgbWcdaWgaaqaaK qzadGaaGymaiaadMgacaWGQbaaleqaaaGcbaqcLbsacaWGibWcdaWg aaqaaKqzadGaamyAaiaadQgaaSqabaaakeaajugibiaaicdaaOqaaK qzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqz GeGaaGimaaGcbaqcLbsacaWGbbWcdaWgaaqaaKqzadGaaGOmaiaadM gacaWGQbaaleqaaaGcbaqcLbsacaWGebWcdaWgaaqaaKqzadGaaGOm aiaadMgacaWGQbaaleqaaaGcbaqcLbsacaaIWaaakeaajugibiaaic daaOqaaKqzGeGaaGimaaGcbaqcLbsacaWGebWcdaWgaaqaaKqzadGa aGOmaiaadMgacaWGQbaaleqaaaGcbaqcLbsacaWGgbWcdaWgaaqaaK qzadGaaGOmaiaadMgacaWGQbaaleqaaaaaaOGaay5waiaaw2faaaaa @8F62@

( A 1ij , B ij , D 1ij , E ij , F 1ij , H ij )= h/2 h/2 Q ij (1,z, z 2 , z 3 , z 4 , z 6 )dz (i,j=1,2,6) ( A 2ij , D 2ij , F 2ij )= h/2 h/2 Q ij (1, z 2 , z 4 )dz (i,j=4,5) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceiqabiaaW9payp qaaKqzGeGaaiikaiaadgealmaaBaaabaqcLbmacaaIXaGaamyAaiaa dQgaaSqabaqcLbsacaGGSaGaamOqaSWaaSbaaeaajugWaiaadMgaca WGQbaaleqaaKqzGeGaaiilaiaadseajuaGdaWgaaWcbaqcLbmacaaI XaGaamyAaiaadQgaaSqabaqcLbsacaGGSaGaamyraSWaaSbaaeaaju gWaiaadMgacaWGQbaaleqaaKqzGeGaaiilaiaadAealmaaBaaabaqc LbmacaaIXaGaamyAaiaadQgaaSqabaqcLbsacaGGSaGaamisaSWaaS baaeaajugWaiaadMgacaWGQbaaleqaaKqzGeGaaiykaiabg2da9Kqb aoaapedakeaajugibiaadgfalmaaBaaabaqcLbmacaWGPbGaamOAaa WcbeaajugibiaacIcacaaIXaGaaiilaiaadQhacaGGSaGaamOEaSWa aWbaaeqabaqcLbmacaaIYaaaaKqzGeGaaiilaiaadQhalmaaCaaabe qaaKqzadGaaG4maaaajugibiaacYcacaWG6bWcdaahaaqabeaajugW aiaaisdaaaqcLbsacaGGSaGaamOEaSWaaWbaaeqabaqcLbmacaaI2a aaaKqzGeGaaiykaiaadsgacaWG6baaleaajugWaiabgkHiTiaadIga caGGVaGaaGOmaaWcbaqcLbmacaWGObGaai4laiaaikdaaKqzGeGaey 4kIipacaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Ua aGjbVlaacIcacaWGPbGaaiilaiaadQgacqGH9aqpcaaIXaGaaiilai aaysW7caaIYaGaaiilaiaaysW7caaI2aGaaiykaaGcbaqcLbsacaaM e8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaays W7caaMe8UaaGjbVlaacIcacaWGbbqcfa4aaSbaaSqaaKqzadGaaGOm aiaadMgacaWGQbaaleqaaKqzGeGaaiilaiaadsealmaaBaaabaqcLb macaaIYaGaamyAaiaadQgaaSqabaqcLbsacaGGSaGaamOraSWaaSba aeaajugWaiaaikdacaWGPbGaamOAaaWcbeaajugibiaacMcacqGH9a qpjuaGdaWdXaGcbaqcLbsacaWGrbWcdaWgaaqaaKqzadGaamyAaiaa dQgaaSqabaqcLbsacaGGOaGaaGymaiaacYcacaWG6bWcdaahaaqabe aajugWaiaaikdaaaqcLbsacaGGSaGaamOEaSWaaWbaaeqabaqcLbma caaI0aaaaKqzGeGaaiykaiaadsgacaWG6baaleaajugWaiabgkHiTi aadIgacaGGVaGaaGOmaaWcbaqcLbmacaWGObGaai4laiaaikdaaKqz GeGaey4kIipacaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7ca aMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaa ysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaiikaiaadMgacaGGSaGaamOAaiabg2da9iaa isdacaGGSaGaaGjbVlaaiwdacaGGPaaaaaa@0F25@

[ K b ]= i=1 n A ( e ) [ B b ( e ) ] T [ D b ][ B b ( e ) ]dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaciaaW9paypWaam WaaeaajugibiaadUealmaaBaaabaqcLbmacaWGIbaaleqaaaGccaGL BbGaayzxaaqcLbsacqGH9aqpkmaaqahabaWaa8qeaeaadaWadaqaaK qzGeGaamOqaSWaaSbaaeaajugWaiaadkgaaSqabaWaaWbaaeqabaWa aeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaOGaay5waiaaw2 faamaaCaaaleqabaqcLbmacaWGubaaaaWcbaqcLbmacaWGbbWcdaah aaadbeqaaSWaaeWaaWqaaKqzadGaamyzaaadcaGLOaGaayzkaaaaaa WcbeqcLbsacqGHRiI8aaWcbaqcLbmacaWGPbGaeyypa0JaaGymaaWc baqcLbmacaWGUbaajugibiabggHiLdGcdaWadaqaaKqzGeGaamiraS WaaSbaaeaajugWaiaadkgaaSqabaaakiaawUfacaGLDbaadaWadaqa aKqzGeGaamOqaSWaaSbaaeaajugWaiaadkgaaSqabaWaaWbaaeqaba WaaeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaOGaay5waiaa w2faaKqzGeGaamizaiaadgeaaaa@6DC5@ , [ K s ]= i=1 n A ( e ) [ B s ( e ) ] T [ D s ][ B s ( e ) ]dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaciaaW9paypWaam WaaeaajugibiaadUealmaaBaaabaqcLbmacaWGZbaaleqaaaGccaGL BbGaayzxaaqcLbsacqGH9aqpkmaaqahabaWaa8qeaeaadaWadaqaaK qzGeGaamOqaSWaaSbaaeaajugWaiaadohaaSqabaWaaWbaaeqabaWa aeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaOGaay5waiaaw2 faaSWaaWbaaeqabaqcLbmacaWGubaaaaWcbaqcLbmacaWGbbWcdaah aaadbeqaaSWaaeWaaWqaaKqzadGaamyzaaadcaGLOaGaayzkaaaaaa WcbeqcLbsacqGHRiI8aaWcbaqcLbmacaWGPbGaeyypa0JaaGymaaWc baqcLbmacaWGUbaajugibiabggHiLdGcdaWadaqaaKqzGeGaamiraO WaaSbaaSqaaKqzadGaam4CaaWcbeaaaOGaay5waiaaw2faamaadmaa baqcLbsacaWGcbWcdaWgaaqaaKqzadGaam4CaaWcbeaadaahaaqabe aadaqadaqaaKqzadGaamyzaaWccaGLOaGaayzkaaaaaaGccaGLBbGa ayzxaaqcLbsacaWGKbGaamyqaaaa@6E13@

[ K g ]= i=1 n A ( e ) [ B g ( e ) ] T [ N 0 ][ B s ( e ) ]dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB LrhDaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaciaaW9paypWaam WaaeaajugibiaadUealmaaBaaabaqcLbmacaWGNbaaleqaaaGccaGL BbGaayzxaaqcLbsacqGH9aqpkmaaqahabaWaa8qeaeaadaWadaqaaK qzGeGaamOqaSWaaSbaaeaajugWaiaadEgaaSqabaWaaWbaaeqabaWa aeWaaeaajugWaiaadwgaaSGaayjkaiaawMcaaaaaaOGaay5waiaaw2 faamaaCaaaleqabaqcLbmacaWGubaaaaWcbaqcLbmacaWGbbWcdaah aaadbeqaaSWaaeWaaWqaaKqzadGaamyzaaadcaGLOaGaayzkaaaaaa WcbeqcLbsacqGHRiI8aaWcbaqcLbmacaWGPbGaeyypa0JaaGymaaWc baqcLbmacaWGUbaajugibiabggHiLdGcdaWadaqaaKqzGeGaamOtaO WaaSbaaSqaaKqzadGaaGimaaWcbeaaaOGaay5waiaaw2faamaadmaa baqcLbsacaWGcbWcdaWgaaqaaKqzadGaam4CaaWcbeaadaahaaqabe aadaqadaqaaKqzadGaamyzaaWccaGLOaGaayzkaaaaaaGccaGLBbGa ayzxaaqcLbsacaWGKbGaamyqaaaa@6DC7@ , { q }= e=1 NE { Λ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaju gibiaadghaaOGaay5Eaiaaw2haaKqzGeGaeyypa0JaaGjbVlaaysW7 kmaaqahabaWaaiWaaeaajugibiabfU5ambGccaGL7bGaayzFaaaale aajugWaiaadwgacqGH9aqpcaaIXaaaleaajugWaiaad6eacaWGfbaa jugibiabggHiLdGaaGjbVlaaysW7kiaaysW7aaa@50B2@

[ F T ]= i=1 n A ( e ) [ [ B 1i ] T [ N T ]+ [ B b1i ] T [ M T ]+ [ B b2i ] T [ P T ] ] dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaju gibiaadAealmaaCaaabeqaaKqzadGaamivaaaaaOGaay5waiaaw2fa aKqzGeGaeyypa0JcdaaeWbqaamaapebabaWaamWaaeaadaWadaqaaK qzGeGaamOqaSWaaSbaaeaajugWaiaaigdacaWGPbaaleqaaOWaaWba aSqabeaaaaaakiaawUfacaGLDbaalmaaCaaabeqaaKqzadGaamivaa aakmaadmaabaqcLbsacaWGobGcdaahaaWcbeqaaKqzadGaamivaaaa aOGaay5waiaaw2faaKqzGeGaey4kaSIcdaWadaqaaKqzGeGaamOqaS WaaSbaaeaajugWaiaadkgacaaIXaGaamyAaaWcbeaakmaaCaaaleqa baaaaaGccaGLBbGaayzxaaWcdaahaaqabeaajugWaiaadsfaaaGcda WadaqaaKqzGeGaamytaSWaaWbaaeqabaqcLbmacaWGubaaaaGccaGL BbGaayzxaaqcLbsacqGHRaWkkmaadmaabaqcLbsacaWGcbWcdaWgaa qaaKqzadGaamOyaiaaikdacaWGPbaaleqaaOWaaWbaaSqabeaaaaaa kiaawUfacaGLDbaalmaaCaaabeqaaKqzadGaamivaaaakmaadmaaba qcLbsacaWGqbWcdaahaaqabeaajugWaiaadsfaaaaakiaawUfacaGL DbaaaiaawUfacaGLDbaaaSqaaKqzadGaamyqaSWaaWbaaWqabeaalm aabmaameaajugWaiaadwgaaWGaayjkaiaawMcaaaaaaSqabKqzGeGa ey4kIipaaSqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzadGaam OBaaqcLbsacqGHris5aiaadsgacaWGbbGaaGjbVlaaysW7aaa@8514@

K l = A { B } T [ D ]{ B }dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb WcdaWgaaqaaKqzadGaamiBaaWcbeaajugibiabg2da9iaaysW7caaM e8UcdaWdrbqaamaacmaabaqcLbsacaWGcbaakiaawUhacaGL9baada ahaaWcbeqaaKqzadGaamivaaaaaSqaaKqzadGaamyqaaWcbeqcLbsa cqGHRiI8aOWaamWaaeaajugibiaadseaaOGaay5waiaaw2faamaacm aabaqcLbsacaWGcbaakiaawUhacaGL9baajugibiaadsgacaWGbbGa aGjbVdaa@538E@

Where

[ D b ]=[ φ i,x 0 0 0 0 0 0 φ i,y 0 0 0 0 0 0 0 φ i,x 0 0 0 0 0 0 φ i,y 0 0 0 0 0 0 0 φ i,x 0 0 0 0 0 0 φ i,y 0 0 0 0 0 0 0 C 1 φ i,x 0 0 0 0 0 0 0 C 1 φ i,y 0 0 0 0 0 C 1 φ i,y C 1 φ i,x 0 0 0 0 0 C 2 φ i,x 0 C 2 φ i,x 0 0 0 0 0 C 2 φ i,y 0 C 2 φ i,y 0 0 0 C 2 φ i,y C 2 φ i,x C 2 φ i,y C 2 φ i,x ]{ q } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaju gibiaadsealmaaBaaabaqcLbmacaWGIbaaleqaaaGccaGLBbGaayzx aaqcLbsacqGH9aqpkmaadmaabaqcLbsafaqabeadhaaaaaaaaOqaaK qzGeGaeqOXdOMcdaWgaaWcbaqcLbmacaWGPbGaaiilaiaadIhaaSqa baaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWa aakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaa keaajugibiabeA8aQTWaaSbaaeaajugWaiaadMgacaGGSaGaamyEaa WcbeaaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaa icdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaic daaOqaaKqzGeGaaGimaaGcbaqcLbsacqaHgpGAlmaaBaaabaqcLbma caWGPbGaaiilaiaadIhaaSqabaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGa aGimaaGcbaqcLbsacaaIWaaakeaajugibiabeA8aQPWaaSbaaSqaaK qzadGaamyAaiaacYcacaWG5baaleqaaaGcbaqcLbsacaaIWaaakeaa jugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaaju gibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugi biabeA8aQTWaaSbaaeaajugWaiaadMgacaGGSaGaamiEaaWcbeaaaO qaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqa aKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaK qzGeGaeqOXdO2cdaWgaaqaaKqzadGaamyAaiaacYcacaWG5baaleqa aaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaa GcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGc baqcLbsacaaIWaaakeaajugibiaadoealmaaBaaabaqcLbmacaaIXa aaleqaaKqzGeGaeqOXdO2cdaWgaaqaaKqzadGaamyAaiaacYcacaWG 4baaleqaaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGe GaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGa aGimaaGcbaqcLbsacaaIWaaakeaajugibiaadoealmaaBaaabaqcLb macaaIXaaaleqaaKqzGeGaeqOXdO2cdaWgaaqaaKqzadGaamyAaiaa cYcacaWG5baaleqaaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaO qaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqa aKqzGeGaam4qaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacqaHgp GAlmaaBaaabaqcLbmacaWGPbGaaiilaiaadMhaaSqabaaakeaajugi biaadoealmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaeqOXdO2cda WgaaqaaKqzadGaamyAaiaacYcacaWG4baaleqaaaGcbaqcLbsacaaI WaaakeaajugibiaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWa aakeaajugibiaaicdaaOqaaKqzGeGaeyOeI0Iaam4qaSWaaSbaaeaa jugWaiaaikdaaSqabaqcLbsacqaHgpGAlmaaBaaabaqcLbmacaWGPb GaaiilaiaadIhaaSqabaaakeaajugibiaaicdaaOqaaKqzGeGaeyOe I0Iaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacqaHgpGAkm aaBaaaleaajugWaiaadMgacaGGSaGaamiEaaWcbeaaaOqaaKqzGeGa aGimaaGcbaqcLbsacaaIWaaakeaajugibiaaicdaaOqaaKqzGeGaaG imaaGcbaqcLbsacaaIWaaakeaajugibiabgkHiTiaadoealmaaBaaa baqcLbmacaaIYaaaleqaaKqzGeGaeqOXdOMcdaWgaaWcbaqcLbmaca WGPbGaaiilaiaadMhaaSqabaaakeaajugibiaaicdaaOqaaKqzGeGa eyOeI0Iaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacqaHgp GAlmaaBaaabaqcLbmacaWGPbGaaiilaiaadMhaaSqabaaakeaajugi biaaicdaaOqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaakeaajugibi abgkHiTiaadoealmaaBaaabaqcLbmacaaIYaaaleqaaKqzGeGaeqOX dO2cdaWgaaqaaKqzadGaamyAaiaacYcacaWG5baaleqaaaGcbaqcLb sacqGHsislcaWGdbWcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiab eA8aQTWaaSbaaeaajugWaiaadMgacaGGSaGaamiEaaWcbeaaaOqaaK qzGeGaeyOeI0Iaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsa cqaHgpGAlmaaBaaabaqcLbmacaWGPbGaaiilaiaadMhaaSqabaaake aajugibiabgkHiTiaadoeakmaaBaaaleaajugWaiaaikdaaSqabaqc LbsacqaHgpGAlmaaBaaabaqcLbmacaWGPbGaaiilaiaadIhaaSqaba aaaaGccaGLBbGaayzxaaWaaiWaaeaajugibiaadghaaOGaay5Eaiaa w2haaKqzGeGaaGjbVlaaysW7caaMe8UccaaMe8UaaGjbVlaaysW7aa a@4867@

[ N 0 ]=[ N x N xy N xy N y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaju gibiaad6ealmaaBaaabaqcLbmacaaIWaaaleqaaaGccaGLBbGaayzx aaqcLbsacqGH9aqpkmaadmaabaqcLbsafaqabeGacaaakeaajugibi aad6ealmaaBaaabaqcLbmacaWG4baaleqaaaGcbaqcLbsacaWGobGc daWgaaWcbaqcLbmacaWG4bGaamyEaaWcbeaaaOqaaKqzGeGaamOtaO WaaSbaaSqaaKqzadGaamiEaiaadMhaaSqabaaakeaajugibiaad6ea lmaaBaaabaqcLbmacaWG5baaleqaaaaaaOGaay5waiaaw2faaKqzGe GaaGzaVlaaysW7caaMe8oaaa@5609@

References

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