Thermal Postbuckling Analysis Research Articles

Thermal post-buckling analysis of functionally graded beams with temperature-dependent physical properties

This paper focuses on thermal post-buckling analysis of functionally graded beams with temperature dependent physical properties by using the total Lagrangian Timoshenko beam element approximation. Material properties of the beam change in the thickness direction according to a power-law function. The beam is clamped at both ends. In the case of beams with immovable ends, temperature rise causes compressible forces and therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. Also, the material properties (Young's modulus, coefficient of thermal expansion, yield stress) are temperature dependent: That is the coefficients of the governing equations are not constant in this study. This situation suggests the physical nonlinearity of the problem. Hence, the considered problem is both geometrically and physically nonlinear. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. In this study, the differences between temperature dependent and independent physical properties are investigated for functionally graded beams in detail in post-buckling case. With the effects of material gradient property and thermal load, the relationships between deflections, critical buckling temperature and maximum stresses of the beams are illustrated in detail in post-buckling case.

Thermal postbuckling behavior of 3D braided beams with initial geometric imperfection under different type temperature distributions

Thermal postbuckling analysis for 3D (three-dimension) braided beams to initial geometrical imperfection in general modes subjected to uniform, linear and non-linear temperature distribution through the thickness are presented. The cross-section of 3D braided composite beam may be treated as a cell system and the geometry of each cell is deeply dependent on its position. A generic imperfection function for one-dimensional composite beam is introduced to model various possible initial geometrical imperfection including sine type, local type, and global type imperfections. Based on first-order shear deformation beam theory incorporating von Kármán nonlinear strain displacement relations, the governing equation is nonlinear integral–differential equations. An analytical solution for thermal postbuckling of 3D braided beams with and without imperfection obtained as a function of the applied thermal load is employed to determine buckling temperatures and postbuckling equilibrium paths of 3D braided beams. The results reveal that the temperature dependent properties, temperature distribution, geometric parameter, fiber volume fraction, initial geometrical imperfections and braiding angle have a significant effect on thermal postbuckling behavior of braided composite beams.

Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory

Thermal post-buckling and nonlinear vibration behaviors of FGM beams are analyzed by using concept of physical neutral surface, von Karman strain-displacement relationships and high order shear deformation theory. Material properties are assumed to be temperature dependent and vary along the thickness. The prominent character of physical neutral surface higher-order shear deformation beam theory is that stretching-bending couplings are eliminated in constitutive equations, and governing equations have the similar forms as homogeneous isotropic beams. Approximate solutions are given out by Ritz method, and influences played by different supported boundaries, thermal environmental conditions and volume fraction index are discussed in detail.

Thermal postbuckling analysis of fiber–metal laminated plates including interfacial damage

Considering the effects of interfacial damage, geometric nonlinearity and transverse shear deformation, thermal postbuckling of fiber–metal laminated plates including interfacial damage is analyzed in detail. Firstly, the Heaviside step function and higher order shear deformation functions are introduced into displacement field so that the damage degree can be characterized. Then, the shape functions can be determined by using the stress continuity conditions between interfaces and the stress boundaries on surfaces. By using the generalized variational principle, the thermal postbuckling equilibrium equations of fiber–metal laminated plates including interfacial damage are established. Finally, the thermal postbuckling problem is solved by adopting finite difference method and iteration method. In numerical examples, the effects of interfacial damage, width-to-thickness ratio and thermal load on the thermal postbuckling of fiber–metal laminated plates including interfacial damage are investigated.

An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams

Abstract In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.

Thermal post-buckling analysis of square plates resting on elastic foundation: A simple closed-form solutions

Thermal post-buckling paths of homogeneous, isotropic, square plate configurations resting on elastic foundation (Winkler type) subjected to biaxial compressive thermal loads are expressed as simple closed-form solutions by using the Rayleigh–Ritz method based on coupled displacement fields. Geometric non-linearity of von-Karman type is considered. The in-plane displacement field variations used in the formulation of Rayleigh–Ritz method are derived by using the governing in-plane static differential equations of the plate which in turn simplifies the difficulty of assuming an in-plane displacement field variations of the square plate. Accuracy and robustness of the proposed closed-form solutions are demonstrated by using the non-linear finite element formulation results which are obtained from an equilibrium path approach.

Thermal postbuckling of shear-deformable anisotropic laminated cylindrical shells with temperature-dependent properties

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell of finite length. The temperature field considered is assumed to be a uniform or non-uniform parabolic temperature distribution varying in the circumferential or axial direction. Temperature-dependent material properties are taken into account. The governing equations are based on Reddy's higher order shear deformation shell theory with von Karman-Donnell-type of kinematic non-linearity and including the extension/torsion, extension/flexure, flexure/torsion couplings and thermal effects. The non-linear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling temperature and postbuckling load-deflection curves. The numerical illustrations concern the postbuckling response of perfect and imperfect, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a compressive stress along with an associated shear stress and torsional effect when the anisotropic shell is subjected to thermal loads. They also confirm that the thermal postbuckling equilibrium path is stable or weakly unstable for the moderately long cylindrical shell and the shell structure is virtually imperfection- insensitive.

Thermal buckling and postbuckling of laminated composite beams with temperature-dependent properties

The thermal buckling and postbuckling analysis of laminated composite beams with temperature-dependent material properties is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. The differential quadrature method (DQM) as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. A direct iterative method is employed to obtain the critical temperature (bifurcation point) as well as the nonlinear equilibrium path (the postbuckling behavior) of symmetrically laminated beams. The applicability, rapid rate of convergence and high accuracy of the method are established via different examples and by comparing the results with those of existing in literature. Then, the effects of temperature dependence of the material properties, boundary conditions, length-to-thickness ratios, number of layers and ply angle on the thermal buckling and postbuckling characteristic of symmetrically laminated beams are investigated.

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells

Thermal postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to a uniform temperature rise. The SWCNTs are assumed to be aligned and straight with a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, FG- and UD-CNTRC shells under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the shell can be increased as a result of a functionally graded reinforcement. It is found that in most cases the CNTRC shell with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.

Thermal post-buckling analysis of uniform slender functionally graded material beams

Two or more distinct materials are combined into a single functionally graded material (FGM) where the microstructural composition and properties change gradually. Thermal post-buckling behavior of uniform slender FGM beams is investigated independently using the classical Rayleigh-Ritz (RR) formulation and the versatile Finite Element Analysis (FEA) formulation developed in this paper. The von-Karman strain-displacement relations are used to account for moderately large deflections of FGM beams. Bending-extension coupling arising due to heterogeneity of material through the thickness is included. Simply supported and clamped beams with axially immovable ends are considered in the present study. Post-buckling load versus deflection curves and buckled mode shapes obtained from both the RR and FEA formulations for different volume fraction exponents show an excellent agreement with the available literature results for simply supported ends. Response of the FGM beam with clamped ends is studied for the first time and the results from both the RR and FEA formulations show a very good agreement. Though the response of the FGM beam could have been studied more accurately by FEA formulation alone, the authors aim to apply the RR formulation is to find an approximate closed form post-buckling solutions for the FGM beams. Further, the use of the RR formulation clearly demonstrates the effect of bending-extension coupling on the post-buckling response of the FGM beams.

Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

Dynamic Thermal Postbuckling Analysis of Piezoelectric Functionally Graded Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is analyzed using an incremental numerical technique. The shell is graded across the thickness according to a power law form function. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the classical shell theory with Sanders' nonlinear kinematic relations. Then, using Hamilton's principle, equations of motion are derived for the piezoelectric FGM cylindrical shell. A finite difference based method combined with the Runge–Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent of FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known data in the literature.

Thermal post-buckling analysis of a laminated composite spherical shell panel embedded with shape memory alloy fibres using non-linear finite element method

In this article, the buckling and post-buckling behaviours of a laminated composite spherical shallow shell panel embedded with shape memory alloy (SMA) fibres are studied under a thermal environment. System equations for a laminated composite spherical shell panel embedded with SMA fibres are for the first time derived by modelling the geometric non-linearity in the Green—Lagrange sense and the material non-linearity in SMA fibres in the framework of the higher-order shear deformation theory. The shell panel model is discretized by using a non-linear finite-element approach. The governing algebraic equations are then derived by the variational approach and solved using a direct iterative technique. Influences of the thickness ratio, boundary condition, aspect ratio, curvature ratio, lamination scheme, SMA volume fraction, percentage of prestrain, and amplitude ratio on the buckling and post-buckling temperatures of a laminated composite shell panel with and without SMA have been examined in detail. The results are computed using the present model and compared with those available in the literature.

Thermal post-buckling analysis of slender columns using the concept of coupled displacement field

Thermal post-buckling analysis of columns with an axially immovable ends is studied using the Rayleigh–Ritz (R–R) method, where the admissible displacement functions are chosen based on the concept of coupled displacement field (CDF) criteria. Geometric non-linearity is considered using the von-Karman strain displacement relations of the beam. Furthermore, the displacement fields derived from CDF criteria are used in an intuitive formulation, where the thermal post-buckling behavior can be predicted by using two parameters namely tension developed in the column and linear buckling load. An exhaustive set of column boundary conditions are considered namely classical such as hinged–hinged, clamped–clamped, clamped–hinged and non-classical such as clamped–guided and hinged–guided. Post-buckling analysis results are presented in the form of closed form expressions, where the ratio of post-buckling load to linear buckling load parameter is expressed as a function of central amplitude of the column for all the boundary conditions considered. The amount of non-linearity predicted using the present formulations (R–R method and intuitive method) based on the concept of coupled displacement field (CDF) criteria shows an excellent agreement with the available literature results for both classical and non-classical boundary conditions.

Thermal Buckling and Post-Buckling of Laminated Composite Plates with SMA Fibers using Layerwise Theory

Thermal buckling and post-buckling analysis of laminated composite plates with SMA fibers subjected to uniform temperature distribution is presented using non-linear FEM. A 3-D Layerwise theory is employed in developing the system equations using variational approach. The Lagrangian interpolation functions are used to approximate the displacement components along the thickness as well as in the in-plane direction. The pre-buckling stresses are considered in the derivation of the geometric stiffness matrix. An incremental load technique and an incremental displacement technique using Newton-Raphson method are adopted to obtain thermal buckling temperature. The results are compared with the existing results in the literature.

Comparative study of thermal post-buckling analysis of columns using an equilibrium path method and an equivalent eigenvalue problem

Thermal post-buckling analysis of uniform slender and shear flexible columns with axially immovable ends is studied for various boundary conditions using a load-deflection curve (bifurcation diagram). Linear buckling load is computed as an eigenvalue problem and the location of initial perturbation in the non-linear analysis is chosen based on the linear eigenvector. Geometric non-linearity is considered using von-Karman strain displacement relations and load-displacement curves are established for the aforementioned boundary conditions. Elegant closed form solutions are obtained for the post-buckling load parameter to linear buckling load parameter as a function of amplitude ratio for the boundary conditions considered using least squares approximation. Numerical values obtained are in good agreement with available literature for the classical boundary conditions, and for the non-classical boundary conditions results are validated with the results obtained from the available FE formulation which is based...

Instability of Functionally Graded Panels with Structural Damping under Supersomic Flow

In this study, thermal post-buckling behaviors and linear flutter analysis of Functionally Graded(FG) panels with structural damping under a supersonic airflow are investigated. First-order shear deformation theory (FSDT) is applied to model the panel, and the von-Karman strain-displacement relations are adopted to consider the geometric nonlinearity. In addition, the damping is modeled as the Rayleigh damping, and the first-order piston theory is applied for the supersonic aerodynamic load.

Comparative study of thermal post-buckling analysis of uniform slender & shear flexible columns using rigorous finite element and intuitive formulations

Thermal post-buckling analysis of uniform, isotropic, slender and shear flexible columns is presented using a rigorous finite element formulation and a much simpler intuitive formulation. The ends of the columns are axially restrained to move and consequently any temperature rise above the stress free condition of the column produces an equivalent constant compressive mechanical load that causes the column to buckle at a critical temperature. Further increase in temperature beyond critical temperature results in the thermal post-buckling phenomenon. As a result of constraints imposed on the axial displacement at the ends of the column, the post-buckling phenomenon is governed by the von-Karman strain displacement relation applicable to one dimensional problems. Empirical formula for ratio of nonlinear axial load to critical load (equivalent constant mechanical load for a given temperature rise) as a function of the central deflection are obtained using both the rigorous finite element and intuitive formulations for various boundary conditions. The boundary conditions considered are the classical such as hinged–hinged, clamped–clamped and clamped–hinged conditions and nonclassical boundary conditions like the hinged–guided or the clamped–guided conditions. Post-buckling analysis results pertaining to nonclassical boundary conditions are meagre in the literature. It is observed that results obtained from both the formulations are in excellent agreement for all boundary conditions considered. Also the accuracy and simplicity of the intuitive formulation is aptly demonstrated to slender and shear flexible columns.

Thermal Post-Buckling Analysis of Functionally Graded Panels in Hypersonic Airflows

In this study, thermal post-buckling characteristics of functionally graded (FG) panels in hypersonic airflows are investigated. The volume fraction and the material properties of FGMs are continuously changed from ceramic to metal in the thickness direction agreeably to a simple power law distribution and a linear rule of mixture, respectively. Using the principle of virtual work, the governing equations are derived and the finite element method is applied to obtain the solutions. Based on the first-order shear deformation theory (FSDT), the FG panels are modeled by the von Karman strain-displacement relation for the structural non-linearity. Also, the third-order piston theory is employed to consider the aerodynamic non-linearity.