Thermal buckling analysis of annular FGM plate having variable thickness under thermal load of arbitrary distribution by finite element method

In this paper, a finite element formulation is developed for analyzing the axisymmetric thermal buckling of FGM annular plates of variable thickness subjected to thermal loads generally distributed nonuniformly along the plate radial coordinate. The FGM assumed to be isotropic with material properties graded in the thickness direction according to a simple power-law in terms of the plate thickness coordinate, and has symmetry with respect to the plate midplane. At first, the pre-buckling plane elasticity problem is developed and solved using the finite element method, to determine the distribution of the pre-buckling in-plane forces in terms of the temperature rise distribution. Subsequently, based on Kierchhoff plate theory and using the principle of minimum total potential energy, the weak form of the differential equation governing the plate thermal stability is derived, then by employing the finite element method, the stability equations are solved numerically to evaluate the thermal buckling load factor. Convergence and validation of the presented finite element model are investigated by comparing the numerical results with those available in the literature. Parametric studies are carried out to cover the effects of parameters including thickness-to-radius ratio, taper parameter and boundary conditions on the thermal buckling load factor of the plates.     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.     

Buckling of polar orthotropic annular plates under internal radial load and torsion

The buckling problem of clamped, polar orthotropic annular plates under internal radial load and torsion is studied theoretically with asymmetric modes taken into consideration. The problem is solved by means of the Galerkin method with the deflection function assumed in the form of cosine series in conjunction with a coordinate transformation. The critical combinations (λ_sc, λ_ic) of torsion and internal radial load are determined for a wide range of polar orthotropic material properties and various hole sizes of the orthotropic annulus. It is found that there are some ranges of λ_ic where λ_sc increases with λ_ic, principally when the internal radial load λ_ic > 0 (radial compression) acts on the annular plate.     

Nonlinear bending response of laminated composite spherical shell panel with system randomness subjected to hygro-thermo-mechanical loading

In the present paper, the effect of random system properties on transverse nonlinear central deflection of laminated composite spherical shell panel subjected to hygro-thermo-mechanical loading is investigated. System properties such as material properties, thermal expansion coefficients, hygro-contraction coefficients, load intensity and lamina plate thickness are taken as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic mathematical formulation. The elastic and hygrothermal properties of the composite material, which are considered to be dependent on temperature and moisture concentration, have been obtained based on micromechanical modeling. A direct iterative based C⁰ nonlinear finite element method combined with mean centered first-order perturbation technique (FOPT) proposed by present authors for the plate is extended for the spherical shell panel subjected to hygro-thermo-mechanical loading. The influences of random system properties with plate geometry, stacking sequences, support conditions, fiber volume fraction and temperature, and moisture distributions on the response of laminated spherical shell panel are examined in detail. The performance of the proposed approach is validated through comparison with those available in the literature and independent Monte Carlo simulation (MCS).     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties

Thermal postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate under thermal loading. Two cases of temperature field, i.e. in-plane non-uniform parabolic temperature distribution and heat conduction are considered. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents, and the material properties of FGM layers are assumed to be temperature-dependent. The governing equations are based on a higher-order shear deformation plate theory that includes thermal effects. The initial geometric imperfection of the plate is taken into account. A two-step perturbation technique is employed to determine buckling temperature and postbuckling equilibrium paths. The numerical illustrations concern the thermal postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates under different sets of loading conditions. The results reveal that the temperature dependency has a significant effect on the thermal postbuckling behavior of FGM plates. The results also confirm that for the case of heat conduction, the postbuckling path for geometrically perfect plates is no longer of the bifurcation type.     

The secondary buckling and post-secondary-buckling behaviours of rectangular plates

Secondary buckling and post-secondary-buckling behaviours are theoretically studied for simply-supported rectangular plates, whose primary buckling modes of deflection contain more than half-waves in the load acting direction. Modal coupling effects more complex than one two-term-coupling effects are incorporated into the secondary buckling and post-secondary-buckling analyses. Then, unstable or stable symmetric secondary branching points are found on the post-primary-buckling paths and “snap through” motions involving an abrupt change in wave-form are shown to be possible. Wave-form variations along post-secondary-buckling paths are also disclosed by means of a numerical analysis of equilibrium paths.     

Collapse of pressurized elastoplastic tubular members under lateral loads

The present work examines the collapse of tubular members subjected to lateral (transverse) quasi-static loading in the presence of uniform pressure. In particular, it investigates pressure effects on the ultimate lateral load of tubes and on their energy absorption capacity. External pressure is mainly considered, whereas internal pressure effects are also discussed. Tubes are modeled with shell finite elements, accounting for geometric and material nonlinearities. Relatively thick steel and aluminum tubes (D/t50), which exhibit significant inelastic deformations, are considered. Two-dimensional cases are examined first, where lateral loading is imposed by either two rigid plates or by two opposite radial loads. Three-dimensional cases are also analyzed, where the load is applied either through a pair of opposite wedge-shaped indenters or a single spherical indenter. The results are presented in terms of load–deflection curves for different levels of pressure, and indicate that the presence of pressure has significant effects on tube response. Deformed shapes of tubes are depicted and discussed, and comparison with test data from non-pressurized pipes is conducted. Finally, simplified analytical models are presented for two-dimensional and three-dimensional load configurations, which yield closed-form expressions, compare fairly well with the finite element results and illustrate some important features of tube response in an elegant manner.     

Collapse behaviour of aluminium plates

Theoretical predictions of the collapse behaviour of thin rectangular aluminium plates are presented and discussed. The plates are simply supported around all four edges, and are subjected to uniaxial in-plane compression. The method of analysis developed by the author is described elsewhere. The material behaviour of aluminium is characterised by the Ramberg-Osgood stress-strain formula. Two methods of using this formula are discussed—one based on the usual percentage proof stress, and the other on the stress at which the plastic strain is a stated proportion of the elastic strain. It is shown why the latter method is more suited to the normalisation of results for collapse analyses. The results presented here give an indication of the effects on plate strength of plate slenderness, initial out-of-flatness, and n, the “knee” factor of the material stress-strain curve.     

Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory

Second-order shear deformation theory (SSDT) is employed to analyze vibration of temperature-dependent solar functionally graded plates (SFGP’s). Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Two different types of SFGP’s such as ZrO₂/Ti-6Al-4V and Si₃N₄/SUS304 are considered. Uniform, linear, nonlinear, heat-flux and sinusoidal thermal conditions are imposed at the upper and lower surface for simply supported SFGPs. The energy method is applied to derive equilibrium equations, and solution is based on Fourier series that satisfy the boundary conditions (Navier’s method). Non-dimensional results are compared for temperature-dependent and temperature-independent SFGP’s and validated with known results in the literature. Numerical results indicate the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results obtained using the SSDT are very close to results from other shear deformation theories.     

加筋平托盘均布载荷作用下的变形研究

根据板的小变形理论,对平托盘次要结构进行简化,将加筋平托盘面板看成厚度变化的加筋平板,利用在均布载荷条件下板的变形公式,解出了此时的加筋板变形挠度解,并通过改变加筋板各几何参数,得到了加筋板不同的刚度分布,进而分析对最大变形的影响。研究表明,加筋板各几何参数在不同的取值范围对板的最大变形贡献有差异。为筋的厚度与面板厚度之间尺寸的协调设计提供了依据,对平托盘在结构尺寸的初期设计,如何分布配置材料具有一定的实际指导意义。     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

Nonlinear bending of Reissner–Mindlin plates with free edges under transverse and in-plane loads and resting on elastic foundations

A nonlinear bending analysis is presented for a rectangular Reissner–Mindlin plate with free edges subjected to combined transverse partially distributed load and compressive edge loading and resting on a two-parameter (Pasternak-type) elastic foundation. The formulations are based on the Reissner–Mindlin plate theory considering the first-order shear deformation effect, and including the plate-foundation interaction. The analysis uses a mixed Galerkin-perturbation technique to determine the load–deflection curves and load–bending moment curves. Numerical examples are presented that relate to the performances of moderately thick rectangular plates with free edges subjected to combined loading and resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, loaded area, the plate aspect ratio and initial compressive load are studied. Typical results are presented in dimensionless graphical form.     

Buckling analysis of FGM plates under uniform,linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.

     

A comparative computational investigation on the effects of randomly distributed general corrosion on the post-buckling behaviour of uniaxially loaded plates

Post-buckling behaviour and ultimate strength of imperfect corroded steel plates used in ships and other related marine structures are investigated. Nonlinear elastic-plastic large deflection finite element analyses are performed on corroded steel plates. General corrosion wastage is considered to be distributed randomly on either one or both surfaces of the analyzed plates. The effects of general corrosion are introduced into the finite element models using a random thickness surface model. The effects of corroded plate parameters on the plate post-buckling and ultimate strengths are evaluated in detail. It was realized that the aspect ratio and thickness (slenderness) of the corroded plates affects their strength characteristics. Age of the plate models affects mainly their post-buckling-strength regimes and degrades their buckling/ultimate strength. Also, nonlinear post-buckling characteristics of the plates suffering either one-side or both-side random corrosion exhibit some differences. Finally, simple empirical formulations are proposed in order to give rough estimations of the ultimate strength of randomly corroded plates.     

Postbuckling of shear deformable laminated plates under biaxial compression and lateral pressure and resting on elastic foundations

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to biaxial compression combined with uniform lateral pressure and resting on an elastic foundation. The lateral pressure is first converted into an initial deflection and the initial geometrical imperfection of the plate is also taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and including the plate-foundation interaction. The analysis uses a perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates under combined loading and resting on Pasternak-type or softening nonlinear elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, the biaxial load ratio and initial lateral pressure are studied.     

Reproducing hoop stress–strain behavior for tubular material using lateral compression test

Identification of material properties in the hoop direction, such as stress–strain behavior, is essential in tube hydroforming processes. Conventional tests such as uniaxial tension and compression tests have some drawbacks and limitations. In the current investigations a simple technique to identify the stress–strain behavior in the hoop direction for tubular material is introduced, based on the experimental data obtained from tube lateral compression test. In the proposed technique, an assumed stress–strain curve is used in finite element simulation to predict the load deflection curve of the tube lateral compression. An iterative algorithm is used to compare the calculated and experimental load deflection curves until a good agreement with a percentage deviation less than 4% is obtained. The suggested technique was used to obtain the material properties of Cu–40%Zn brass tube. The predicted stress–strain curve was compared with that obtained from uniaxial compression test. Comparison between the experimental and predicted stress–strain curve showed that the proposed technique is effective in the prediction of the material properties from the tube lateral compression test with percentage deviation less than 1%.     

A new approach for bending analysis of thin circular plates with large deflection

This paper is concerned with bending analysis of an axisymmetric simply supported circular plate with large deflection. Based on the linear theory of thin plates, the incremental load technique is developed for solving the bending problem of a thin circular plate with large deflection. In the proposed method, the total applied load is divided into various small load steps. In each load step, the plate stress behavior is simplified to be linear. The incremental formulations are presented for the deflection and stresses of the plate when external loads increase. A numerical example is given to show simplicity and accuracy of the present method. It is found that the proposed method can be an alternative useful tool for engineering applications.     

A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

Viscoelastic analysis of an exponentially graded sandwich plate

Bending response of an exponentially graded fiber-reinforced viscoelastic (EGFV) sandwich plate is investigated using various plate theories. The plate consists of viscoelastic material faces and an elastic material core. The effective moduli and Illyushin’s approximation methods are used to solve the equations governing the bending of simply-support EGFV sandwich plates. Numerical results for deflections and stresses are presented and some of them are compared. The effects due to aspect ratio, side-to-thickness ratio and constitutive parameter as well as time parameter are all investigated. Concluding remarks are made.