Post-buckling Paths Research Articles

Non-Linear Thermal Stability Analysis of SMA Wire-Embedded Hybrid Laminated Composite Timoshenko Beams on Non-Linear Hardening Elastic Foundation

In the present study, non-linear thermal post-buckling analysis of hybrid laminated composite Timoshenko beams embedded with the shape memory alloy (SMA) wires resting on a non-linear hardening elastic foundation were studied. Mechanical properties of composite media are considered temperature-dependent. The theory of Timoshenko beams and von Kármán's strain-displacement relations are applied simultaneously in virtual work principles to derive the system of non-linear equilibrium equations. Various types of boundary conditions such as clamped, simply supported, and rolled edges were studied for edge supports. Generalized Differential Quadrature Method (GDQM) was utilized to discrete the equilibrium equations in space domain. Different types of lay-ups, such as symmetric and asymmetric, were considered. Post-buckling paths are depicted for different values of non-linear elastic foundation parameters, volume fractions, pre-strains of the SMA fibers, and boundary conditions. The one-dimensional thermo-mechanical constitutive law suggested by Brinson [1] is applied to model the SMA wires. Numerical results make it possible to recognize that increases in the volume fraction and pre-strain of SMA lead to a dramatic enhancement in thermal buckling and post-buckling capacity of the beam. Pre-buckling, buckling, and post-buckling behavior of the beam are totally different and this is due to variations among critical buckling, austenite start and finish temperatures. Due to the recovery stress of SMA wires, particular consequences are shown.

Imperfection sensitivity analysis of laminated folded plates

A novel methodology for imperfection sensitivity analysis is presented. Koiter׳s perturbation method is used to calculate the imperfection paths emanating from mode interaction bifurcations, which occur on the post-buckling paths of the single modes. The Monte Carlo method is used to test a large number of modes and all possible interactions among them. The computational cost is low because of the efficiency of Koiter׳s method. The demands of Koiter׳s method for accurate evaluations of higher order derivatives of the potential energy are met by a mixed, corotational element.

Dynamic response of thin FG plates with a static unsymmetrical stable postbuckling path

The present paper deals with a dynamic interactive response of square FGM plates subjected to in-plane pulse loading. The structures are assumed to be simply supported along all edges. In order to obtain the equations of motion of individual plate, the classical laminate plate theory (CLPT) has been modified in such a way that it additionally accounts for all components of inertial forces. An effect of the imperfection sign (sense) influence on the dynamic response of plates made of functionally graded materials (FGMs) has been analyzed. In the case of the static problem, Koiter’s theory explains this sign (sense) sensitivity by an unsymmetrical stable equilibrium path of the FGM plate. The investigations deal with a comparison of the semi-analytical method (SAM) and the finite element method (FEM) applied to the dynamic postbuckling nonlinear analysis of thin-walled FG plated structures.

Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments

This paper presents a postbuckling analysis of carbon nanotube-reinforced composite (CNTRC) cylindrical panels resting on elastic foundations and subjected to axial compression in thermal environments. The cylindrical panels are reinforced by aligned single-walled carbon nanotubes (SWCNTs) which are assumed to be functionally graded (FG) through the thickness direction with different types of distributions. The material properties of FG-CNTRC panels are estimated through an extended rule of mixture micromechanical model. The governing equations are based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The panel–foundation interaction and thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique along with a two-step perturbation approach is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the CNT volume fraction, temperature rise, foundation stiffness, and the panel geometric parameters have a significant effect on the buckling load and postbuckling behavior of CNTRC cylindrical panels. The results for uniformly distributed (UD) CNTRC cylindrical panels are compared with those of FG-CNTRC cylindrical panels. The results also confirm that for an CNTRC cylindrical panel with immovable unloaded straight edges, the postbuckling path of the panel is no longer the bifurcation type.

Experimental buckling path of compressed bars under displacement control

Buckling and postcritical analysis of prismatic metal bars, including complex cycles of loading are considered in the paper. The postbuckling path analysis of bars loaded in the form of increasing displacements is discussed in detail. The results of experimental research are reported as a dependence of the force P versus lateral deflection f as well as total shortening of the rod u. It is stated that only the simultaneous analysis of both methods of presentation of results comprises a complete tool for developing the column response. Particularly, the response that appears in the first, linear elastic phase of loading can be observed on P-u diagrams. The increase of the compressing force causes visible shortening of the bar without its buckling. This important phenomenon is not usually included in stability considerations of prismatic rods.

The Instability Improvement of the Shape Memory Alloy Composite Plates Subjected to In-Plane Parabolic Temperature Distribution

The designs of thin structure components of aerospace vehicles require the consideration of thermal buckling and post-buckling problems. Thermal buckling of the structures in the aerospace environment may occur due to non-uniformly distributed temperature field. A finite element method study on the post-buckling of composite plates with embedded shape memory alloy wires was conducted. The plates were subjected to in-plane and through-thickness non-uniform thermal loadings where the non-uniform temperature distributions considered were parabolic in-plane and linearly varying through-thickness thermal loadings that may act separately or in combination. Recovery stress induced by the shape memory alloy was exploited to improve the thermal buckling behaviours of the composite plates. A non-linear finite element model along with its source codes that considered the recovery stress of the shape memory alloy, the non-uniform temperature field, the temperature dependent properties of the SMA and the composite matrix were developed. The post-buckling paths that showed the effect of the shape memory alloy on the thermal post-buckling behaviour of composite plates were generated using the source codes. It was found that the strain energy tuning method of the shape memory alloy greatly improved the post-buckling behaviour of composite plates subjected to the non-uniform temperature distributions.

Postbuckling of FGM rings

Buckling and post-buckling behaviors of a moderately thick closed ring made of a through-the-thickness functionally graded material (FGM) is investigated in this research. It is assumed that material properties of the ring are distributed across the thickness in terms of a power law model. The first order theory of shear deformable rings along with the von Kármán type of geometrical non-linearity is incorporated with the virtual displacements principle to establish the complete form of the coupled non-linear equilibrium conditions. The analysis is restricted to the case of hydrostatic pressure. An exact closed-form solution is developed to trace the pre-buckling deformations of the ring. Non-linear adjacent equilibrium criterion, which is an efficient criterion to deduce the post-buckling path, is implemented to establish the highly non-linear coupled stability equations. The generalized differential quadrature (GDQ) method is adopted to discrete the equilibrium equations and the associated boundary conditions. Numerical illustrations cover the post-buckled shape and deflection of the ring in the intermediate post-buckling regime. It is shown that, unlike the case of the flat beams, response of closed FGM rings is similar to the case of isotropic homogeneous ring. Consequently, hydrostatically loaded FGM rings exhibit the imperfection insensitivity feature through the loading process.

Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane

This paper investigates the axisymmetric postbuckling of functionally graded material (FGM) annular microplates based on the modified couple stress theory, Mindlin plate theory and von Kármán geometric nonlinearity. Material properties are assumed to be graded in the thickness direction according to Mori–Tanaka homogenization method. By using the physical neutral plane, the bending-extension coupling is eliminated in both nonlinear governing equations and boundary conditions of FGM microplates. The differential quadrature (DQ) method is employed to discretise the governing equations, which are then solved to obtain the postbuckling path of FGM microplates with different boundary conditions. Numerical results are presented to highlight the effects of length scale parameter, gradient index, inner-to-outer radius ratio and radius-to-thickness ratio on the postbuckling characteristics of FGM microplates. It is found that FGM microplates do exhibit the bifurcation-type buckling when the applied in-plane compressive load acts along the physical neutral plane.

Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments

Abstract This paper deals with the large amplitude vibration, nonlinear bending and thermal postbuckling of functionally graded material (FGM) beams resting on an elastic foundation in thermal environments. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation beam theory that includes beam–foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The numerical illustrations concern the nonlinear vibration, nonlinear bending and thermal postbuckling of FGM beams resting on Pasternak elastic foundations under different thermal environmental conditions. It is found that the FGM beam with intermediate material properties does not necessarily have intermediate nonlinear frequencies. The thermal postbuckling path of simply supported FGM beams is no longer of the bifurcation type for both uniform and non-uniform temperature fields.

Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage

Based on the elasto-plastic theory, considering the effect of spherical stress tensor on the elasto-plastic deformation and using the slicing treatment to deal with the plasticity of functionally graded coatings, the elasto-plastic increment constitutive equations of the sandwich plates with functionally graded metal-metal face sheets can be derived. Applying the weak bonded theory to the interfacial constitutive relation and taking into account the geometric nonlinearity, the nonlinear increment differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy principle. The finite difference method and the iterative method are used to obtain the post-buckling path. When the effect of geometrical nonlinearity of the plate is ignored, the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets can be solved by the Galerkin method and the iterative method. In the numerical examples, the effects of the interface damages, the induced load ratio, the functionally graded index, and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load are investigated.

Non-standard stabilization of the post-buckling path for elastic–plastic cylindrical shells under combined state of loadings

The post-buckling behavior of elastic–plastic cylindrical shells under external pressure and twisting moment is unstable. The effect of stabilization of the post-buckling behavior is in most cases obtained by changing geometry of the structure. In this paper an alternative concept of modification of the post-buckling behavior is applied, namely modification of post-buckling path is achieved by application of additional loadings acting on the structure without changing the shape or sizes of the optimized element. Calculations are performed using ANSYS code for elastic–plastic shells of different length and thickness for the whole range of load from a pure external pressure to a pure twisting moment. It turned out that the active stabilizing force can always stabilize the post-buckling paths whereas the passive and ‘mixed’ variant of stabilizing loads are able to stabilize the post-buckling paths only for thicker elastic–plastic shells and for limited states of loading.

Size-dependent vibrations of post-buckled functionally graded Mindlin rectangular microplates

In this paper, the free vibration behavior of post-buckled functionally graded (FG) Mindlin rectangular microplates are described based on the modified couple stress theory (MCST). This theory enables the consideration of the size-effect through introducing material length scale parameters. The FG microplates made of a mixture of metal and ceramic are considered whose volume fraction of components is expressed by a power law function. By means of Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro-plates in the postbuckling domain. The governing equations and boundary conditions are then discretized by using the generalized differential quadrature (GDQ) method before solving numerically by the pseudo-arclength continuation technique. In the solution procedure, the postbuckling problem of microplates is investigated first. Afterwards, the free vibration of microplates around the buckled configuration is discussed. The effects of dimensionless length scale parameter, material gradient index and aspect ratio on the on the postbuckling path and frequency of FG microplates subject to arbitrary edge supports are thoroughly discussed.

Postbuckling of laminated composite plates using NURBS-based isogeometric analysis

In this paper, postbuckling behavior of laminated composite plates is investigated using NURBS-based isogeometric analysis (IGA). A Green strain tensor with small rotations is proposed. Both von-Karman strain tensor and the proposed strain tensor are formulated. Governing equations are derived in the framework of first-order shear deformation theory (FSDT). Finite elements based on NUBRS show the robustness in performing a laminate with intentional imperfections as well as deformed mode shapes. Quadratic NURBS elements (Q9Nurbs) are used to construct physical meshes in C1 continuity. Newton–Raphson scheme is adjusted to geometric imperfections and employed for non-linear analysis procedure. In numerical examples, the postbuckling paths of laminated composite plates in sense of von-Karman strain theory and the proposed strain theory are presented. The obtained results of isotropic, symmetric, antisymmetric angle-ply and cross-ply laminates are verified with those available in literature.

The Post-Buckling Behavior of the Composite Plates with Embedded Shape Memory Alloy Subjected to Combined Loading Using Finite Element Method

Thin composite structures that are used in aerospace applications can be subjected to buckling failure due to combined mechanical and thermal loadings. This paper presents the work on the thermal post-buckling improvement of composite plates previously subjected to mechanical loading. Pre-strained shape memory alloy wires were embedded within laminated composite plate so that the recovery stress that can improve strain energy of the plate can be induced when the wires were heated. A geometric non-linear finite element formulation of the shape memory alloy composite plate and its source codes were developed. The formulation is based on total strain for the case of mechanical loading and incremental strain for the case of thermal loading. Using the codes, post-buckling paths were determined for quasi-isotropic and anti-symmetric cross-ply composite plates. It was found that by embedding shape memory alloy wires within composite plates, thermal post-buckling paths can be improved significantly even after the degradation of the thermal buckling resistance of composite plates due to the application of the mechanical loading.

The post-buckling behavior of the composite plates with embedded shape memory alloy subjected to combined loading using finite element method

Thin composite structures that are used in aerospace applications can be subjected to buckling failure due to combined mechanical and thermal loadings. This paper presents the work on the thermal post-buckling improvement of composite plates previously subjected to mechanical loading. Pre-strained shape memory alloy wires were embedded within laminated composite plate so that the recovery stress that can improve strain energy of the plate can be induced when the wires were heated. A geometric non-linear finite element formulation of the shape memory alloy composite plate and its source codes were developed. The formulation is based on total strain for the case of mechanical loading and incremental strain for the case of thermal loading. Using the codes, post-buckling paths were determined for quasi-isotropic and anti-symmetric cross-ply composite plates. It was found that by embedding shape memory alloy wires within composite plates, thermal post-buckling paths can be improved significantly even after the degradation of the thermal buckling resistance of composite plates due to the application of the mechanical loading.

Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach

This paper presents a postbuckling analysis of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under axial compression. Based on kernel particle approximations for the field variables, the Ritz method is employed to obtain the discretized governing equations. The cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) which are assumed to be graded through the thickness direction with different types of distributions. The effective material properties of CNTR-FG cylindrical panels are estimated through a micromechanical model based on the extended rule of mixture. To eliminate shear locking for a very thin cylindrical panel, the system’s bending stiffness is evaluated by a stabilized conforming nodal integration scheme and the membrane as well as shear terms are calculated by the direct nodal integration method. In the present study, the arc-length method combined with the modified Newton–Raphson method is used to trace the postbuckling path. Detailed parametric studies are carried out to investigate effects of various parameters on postbuckling behaviors of CNTR-FG cylindrical panels and results for uniformly distributed (UD) CNTR-FG cylindrical panel are provided for comparison.

Elasto-plastic pre- and post-buckling analysis of functionally graded beams under mechanical loading

In this paper, the elasto-plastic pre- and post-buckling behavior of beams made of functionally graded materials subjected to mechanical loading is investigated. A continuum-based finite element formulation is developed to determine the major characteristics of buckling. The arc-length algorithm is employed to analyze the stability problem. A plane stress von Mises model with isotropic hardening is utilized for the elasto-plastic nonlinear analysis of the beam. Basic idea in geometric and material nonlinear analysis of functionally graded material beams is to use the plasticity capacity of metal phase as a ductile material during loading. The influences of number of axial modes, material index, geometrical parameters and boundary conditions on the critical buckling point, pre- and post-buckling paths, plastic bifurcation point and stress distribution are fully studied. A good agreement between generated results and existing data in the literature is observed.

Development of a semi-analytical nonlinear finite element formulation for cylindrical shells

This article introduces a new semi-analytical nonlinear finite element formulation for thin cylinders according to a continuum-based approach. A comparison between the continuum-based approach and the classical approach for the buckling behavior of isotropic and orthotropic perfect cylinders validates the results. The classical approach is defined according to thin shell theories based on the von Karman approximation. A mathematical modeling for geometry imperfection of cylinders is derived according to a continuum-based approach whose results are compared with the results of the classical approach for imperfect cylinders. The influence of neglecting some nonlinear terms in the classical approach for perfect and imperfect cylinders on the buckling path is investigated. In the buckling analysis, two methods, i.e. the perturbation and load disturbance methods, which undertake to switch to the post-buckling path, are compared to each other.

Influence of symmetries and imperfections on the non-linear vibration modes of archetypal structural systems

The objective of the present work is to investigate how symmetries, initial geometric imperfections and energy level influence the number and stability of the non-linear normal modes and the existence of multimode solutions in structural systems liable to unstable buckling. To this aim, two archetypal models exhibiting interactive buckling phenomena, which lead to several unstable post-buckling paths and high imperfection sensitivity, are considered. As many structural elements, these models have several planes of symmetry. The inherent symmetries and post-buckling solutions have a marked influence on the underlying potential function and, consequently, on the non-linear dynamics of the system, whose stable solutions are limited by the energy level associated with the saddles lying on the boundary of the pre-buckling well. A detailed non-linear modal analysis is accomplished for increasing energy levels, by first considering the nominally perfect systems. Then, the symmetry breaking effect of initial geometric imperfections on the number and stability of the non-linear normal modes is investigated. Finally, some examples illustrate the influence of the superabundance of modes on the resonant forced behavior of the system.

Study on the Buckling Deformation of Orthotropic Strip

Because of the crystal preferential orientation, the rolled strip exhibits orthotropy. The mechanism and deformation behavior of the orthotropic strip buckling (included pre-buckling and post buckling) was studied by analytical and finite element method. Through the energy principle, the mathematical model of critical buckling was established, and the critical conditions were obtained. The generating path of post buckling was found with perturbation variational solution. At the same time, using software ANSYS to simulate the buckling deformation, then the buckling mode, the critical stress and the post-buckling path were obtained, which were in good accordance with the analytical results. It verified the accuracy of analytical results.