Post-buckling Problems Research Articles

Thermal post-buckling of slender composite and FGM columns through a simple and novel FE formulation

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.

Postbuckling analysis of microscale beams based on a strain gradient finite element approach

The postbuckling problem of Euler–Bernoulli (EB) microbeams under different boundary conditions is addressed in this paper using a non-classical finite element (FE) approach with a novel beam element. The proposed element is capable of considering the strain gradient effects and hence is appropriate to use at microscale. First, based on Mindlin’s strain gradient theory (SGT), a size-dependent EB beam model including strain gradient effects is developed. Then, by a FE approach, the non-classical beam element is constructed which needs two additional degrees of freedom per node as compared to the classical EB beam element. The new element is based upon Mindlin’s SGT, and can be easily reduced to that based on various higher-order elasticity theories such as the modified versions of strain gradient and couple stress theories. Selected numerical results are presented to show the reliability of the developed FE formulation, and also to study the small scale influences on the bifurcation diagrams of microbeams.

Thermomechanical initial post-buckling of Timoshenko's beam on elastic foundations

ABSTRACTThe objective of this work is to propose an analytical solution for the thermomechanical initial post-buckling response of a thick beam resting on a linear elastic foundation and subject to a uniform temperature rise throughout its cross-section. The thermal strain is assumed to follow a linear law with the temperature rise and the material properties are considered temperature independent. The beam cross-section geometrical properties are constant along the beam length, and the formulation is consistent with the small strain assumption. The beam ends are assumed pinned and immovable, thermal expansion is not allowed and as a consequence compressive forces arise and the beam may buckle. If the temperature is increased further, the beam continues to deflect laterally, hence the problem is geometrically non-linear. In addition, the model is appropriate to describe the behavior of short beams as it takes into account transverse shear deformations. The governing equations are derived and made non-dimensional, and it is seen that three non-dimensional parameters control the thermomechanical initial post-buckling problem: The elastic foundation stiffness, the beam slenderness ratio and the beam cross-section shear coefficient. A classical perturbation method is applied to the non-linear set of differential governing equations, therefore the critical buckling temperatures (loads) and modes and the initial post-buckling behavior may be analytically determined. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature, the elastic foundation.

Postbuckling Analysis of Functionally Graded Beams Using Nonlinear Model

The major novelty of the paper in the study, post-buckling of simply supported FGM beams using various theory, classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), parabolic shear deformation beam theory (PSDBT) and exponential shear deformation beam theory (ESDBT). Governing equations of FGM beam for post-buckling problem were found by applying Hamilton principle and Navier type solution method was used to solve post-buckling problem. It is assumed that elasticity modulus is changing in the thickness direction and all other material properties are taken to be constant. Variation of elasticity modulus in the thickness direction, are described by a simple power law distribution in terms of the volume fractions of constituents. The shear effect is shown to have a significant contribution to both the buckling and post-buckling behaviors. Results of this analysis show that classical and first-order theories underestimate the amplitude of buckling while all higher order theories, considered in this study, yield very close results for the static post-buckling response.

Large deflection and post-buckling analysis of non-linearly elastic rods by wavelet method

We propose a wavelet method in the present study to analyze the large deflection bending and post-buckling problems of rods composed of non-linearly elastic materials, which are governed by a class of strong non-linear differential equations. This wavelet method is established based on a modified wavelet approximation of an interval bounded L2-function, which provides a new method for the large deflection bending and post-buckling problems of engineering structures. As an example, in this study, we considered the rod structures of non-linear materials that obey the Ludwick and the modified Ludwick constitutive laws. The numerical results for both large deflection bending and post-buckling problems are presented, illustrating the convergence and accuracy of the wavelet method. For the former, the wavelet solutions are more accurate than the finite element method and the shooting method embedded with the Euler method. For the latter, both bifurcation and limit loads can be easily and directly obtained by solving the extended systems. On the other hand, for the shooting method embedded with Runge–Kutta method, to obtain these values usually needs to choose a good starting value and repeat trial solutions many times, which can be a tough task.

A direct minimization technique for finding minimum energy configurations for beam buckling and post-buckling problems with constraints

We present a novel technique to simulate the deformation of a cantilever elastic beam constrained in a curved solid channel subject to end forces. We pose this as the minimization of an energy functional and solve it by a variant of a dynamic programming approach called the Viterbi algorithm. The core idea of this approach is to discretize the variables describing the potential energy and to construct a set of admissible configurations of the beam. The Viterbi algorithm is then employed to search through the set of possible beam configurations and locate the one with the minimum potential energy in a very computationally efficient way. The new approach does not require any gradient computations and could be considered as a direct search method, and thus can be guaranteed to find the global minimum potential energy. Also the constraints can be automatically satisfied by constructing the proper set of all the possible configurations. The approach can also be used to find feasible starting configurations associated with conventional minimizing algorithms.

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.

Torsional buckling and post-buckling behavior of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium

The nonlinear buckling and post-buckling problems of functionally graded stiffened toroidal shell segments surrounded by an elastic medium under torsion based on an analytical approach are investigated. The rings and stringers are attached to the shell, and material properties of the shell are assumed to be continuously graded in the thickness direction. The classical shell theory with the geometrical nonlinearity in von Karman sense and the smeared stiffeners technique are applied to establish theoretical formulations. The three-term approximate solution of deflection is chosen more correctly, and the explicit expression to find critical load and post-buckling torsional load-deflection curves is given. The effects of geometrical parameters and the effectiveness of stiffeners on the stability of the shell are investigated.

Buckling and post-buckling behaviour of elastic seven-layered cylindrical shells – FEM study

The paper is devoted to buckling and post-buckling problems of an elastic seven-layered cylindrical shell under uniformly distributed pressure. The shell is an untypical sandwich structure composed of main corrugated core and two three-layer faces. Numerical FEM model for the shell has been elaborated. The calculations have been performed with the use of the ANSYS code for elastic shells of different dimensions. The linear and non-linear analyses of the shells have been performed with the use of the finite elements method. Critical pressure and equilibrium paths for the family of seven-layered shells subjected to uniformly distributed external pressure are calculated. The influence of corrugation pitch of main core and the length of the shell on the critical pressure has been determined. The results of these investigations are presented on the graphs.

Post-Buckling Analysis of Axially Functionally Graded Three-Dimensional Beams

Post-buckling analysis of an axially functionally graded (AFG) cantilever beam subjected to an axial nonfollower compression load is studied in this paper by using the total Lagrangian finite element model of three-dimensional continuum approximations. Material properties of the beam change in the axial direction according to a power-law function. In this study, finite element model of the beam is constructed by using total Lagrangian finite element model of three-dimensional continuum for an eight-node quadratic element. It is known that post-buckling problems are geometrically nonlinear problems. The considered highly nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton–Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations. The obtained results are compared with the published results. In this study, the effects of the material distribution on the post-buckling response of the AFG beam are investigated in detail. The differences between of material distributions are investigated in the post-buckling analysis. Numerical results show that the above-mentioned effects play a very important role on the post-buckling responses of the beam, and it is believed that new results are presented for post-buckling of AFG beams which are of interest to the scientific and engineering community in the area of FGM structures.

On Post-Buckling Behavior of Edge Cracked Functionally Graded Beams Under Axial Loads

This paper presents the post-buckling analysis of an edge cracked cantilever beam composed of functionally graded material (FGM) subjected to axial compressive loads by using the total Lagrangian Timoshenko beam element approximation. Material properties of the beam change in the height direction according to the exponential distribution. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. For beams subjected to compression loads, the load rise causes compressible forces and therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. The highly nonlinear problem considered herein is solved incrementally by using the finite element method in conjunction with the Newton–Raphson method, by which the full geometric nonlinearity is considered. There is no restriction on the magnitudes of deflections and rotations in contradistinction to the von Karman strain displacement relations of the beam. In the study, the effects of the location and depth of the crack, and different material distributions on the post-buckling behavior of the FGM beam are investigated in detail.

Postbuckling Analysis of Functionally Graded Beams Using Hyperbolic Shear Deformation Theory

In the present study, post-buckling of a thick FGM rectangular beam is carried out using the hyperbolic shear deformation theory (HYSDT). The theory accounts for parabolic distribution of transverse shear stresses across the thickness satisfying the stress free boundary conditions at top and bottom surfaces of the beam. It is assumed that elasticity modulus is changing in the thickness direction and all other material properties are taken to be constant. Variation of elasticity modulus in the thickness direction, are described by a simple power law distribution in terms of the volume fractions of constituents. Governing equations of FGM beam for post-buckling problem were found by applying Hamilton principle and Navier type solution method was used to solve post-buckling problem. The results obtained for post-buckling analysis of functionally graded beams are compared with those obtained by other theories, to validate the accuracy of the presented theory.

Free vibration and postbuckling of laminated composite Timoshenko beams

Abstract A numerical solution method was developed to investigate the postbuckling behavior and vibrations around the buckled configurations of symmetrically and unsymmetrically laminated composite Timoshenko beams subject to different boundary conditions. The Hamilton principle was employed to derive the governing equations and corresponding boundary conditions which are then discretized by introducing a set of matrix differential operators. The pseudo-arc-length continuation method was used to solve the postbuckling problem. To study the free vibration that takes place around the buckled configurations, the corresponding eigenvalue problem was solved by means of the postbuckling configuration modes obtained in the previous step. The static bifurcation diagrams for composite beams with different lay-up laminates are given, and it is shown that the lay-up configuration considerably affects the magnitude of critical buckling load and postbuckling behavior. The study of the vibrations of composite beams with different laminations around the buckled configurations indicates that the natural frequency in the prebuckling domain increases as the stiffness of a beam increases, while there is no specific relation between the lay-up lamination and natural frequency in the postbuckling domain which necessitates conducting an accurate analysis in this area.

Large post-buckling behavior of Timoshenko beams under axial compression loads

Large post-buckling behavior of Timoshenko beams subjected to non-follower axial compression loads are studied in this paper by using the total Lagrangian Timoshenko beam element approximation. Two types of support conditions for the beams are considered. In the case of beams subjected to compression loads, load rise causes compressible forces end therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. 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. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of lower-Carbon Steel. In the study, the relationships between deflections, rotational angles, critical buckling loads, post-buckling configuration, Cauchy stress of the beams and load rising are illustrated in detail in post-buckling case.

MACRO-MICRO ANALYSIS FOR ANTI-PENETRATING POSTBUCKLING OF FIBER REINFORCED, DEBONED LAMINATES WITH CONTACT EFFECTS

This paper presents a close-form solution for the anti-penetrating postbuckling problem of fiber reinforced, rectangular laminates. The laminate has a through-width debonded region buried at the center position and is simply supported at its four edges subjected to the in-plane compressive load. Based on the previous analysis of the contact kinematics from micro mechanics of composite material, the nonlinear contact effects at the interfaces of the debonded region are considered and characterized by a contact factor. Governing equations are established and are solved using the perturbation approach. Asymptotic solutions from perturbation analysis are presented. It is found that for fiber reinforced, debonded laminates the postbuckling deformation involves global deflection and local buckling with amplitude of the global one 104 times larger than the local one. The global deflection is induced by the postbuckling load and the local buckling is caused by the anti-penetrating contact effects. Due to the contact effects, the load carry capacity of the delaminated laminate is tremendously enhanced. Further examination indicates that both the buckling load and the contact performances are intrinsic properties of the laminates. Parametric analyses show that (i) the buckling load decreases with the increasing of the delamination length and the delamination depth and is very sensitive to the changing of the aspect ratio, (ii) the contact performances, including the location and size of the contact areas, rely only on the geometries of the debonded region and the material properties but are hardly affected by the aspect ratio. The analysis is in effect by comparing with the well established results.

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.

Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory

This study proposes a computationally efficient approach to nonlinear bending and thermal post-buckling problems in Euler–Bernoulli microbeams based on modified couple stress theory under geometrically accurate relationships. The governing equations, which consider the size effect and the axis extensibility, are formulated via the equilibrium of an infinitesimal element. The proposed model, which encompasses the size-independent and Von Kármán nonlinear theory, is solved using the shooting technique after transformation into a two-point boundary value problem. The proposed method was validated based on comparisons with several case studies using existing simulations. The influences of the length scale parameter and the Poisson ratio on the bending and thermal post-buckling behaviors of microbeams are discussed in detail. The numerical results show that the intrinsic size dependency of the material and the Poisson ratio make the microbeam behave in a relatively stiff manner, thereby leading to smaller deformations and greater increases in the buckling temperature.

Buckling and postbuckling of single-walled carbon nanotubes based on a nonlocal Timoshenko beam model

In this research, the axial buckling and postbuckling configurations of single-walled carbon nanotubes (SWCNTs) under different types of end conditions are investigated based on an efficient numerical approach. The effects of transverse shear deformation and rotary inertia are taken into account using the Timoshenko beam theory. The nonlinear governing equations and associated boundary conditions are derived by the virtual displacements principle and then discretized via the generalized differential quadrature method. The small scale effect is incorporated into the model through Eringen's nonlocal elasticity. To obtain the critical buckling loads, the set of linear discretized equations are solved as an eigenvalue problem. Also, to address the postbuckling problem, the pseudo arc-length continuation method is applied to the set of nonlinear parameterized equations. The effects of nonlocal parameter, boundary conditions, aspect ratio and buckling mode on the critical buckling load and postbuckling behavior are studied. Moreover, a comparison is made between the results of Timoshenko beam model and those of its Euler-Bernoulli counterpart for various magnitudes of nonlocal parameter.

Flexibility-Corotational Formulation of Space Frames with Large Elastic Deformations and Buckling

While the joints of a frame move in the space with large displacements and rotations, the deformations of the frame member with respect to the imaginary line connecting end joints, known as the chord, remain relatively small but still affect the stress field along the member length. A separation, known also as the corotational formulation, is simple for small displacements and rotations, and for small deformations, while in presence of large geometric changes the kinematic and dynamic interaction between the two systems (rigid and deformational) requires enhanced treatment. An enhanced corotational formulation, based on the flexibility modeling of elastic space frames with large deformations and rotations, expanding the prior developments by the authors, is presented in this article. The enhancements with respect to previous formulations cover: (1) rigid translations and expanded rotations of the chord; (2) a combination of axial and flexural deformations expanded to include torsional and shear influences; (3) a consistent formulation of geometric stiffness matrix; and (4) a rederivation of the formulation in time-independent incremental form. The entire reformulation enables the application of flexibility-based solutions to space frames addressing flexural and torsional buckling or postbuckling problems and allows for further integration of material nonlinearity.

Solution of Post-Buckling & Limit Load Problems,Without Inverting the Tangent Stiffness Matrix & WithoutUsing Arc-Length Methods

In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicit- ly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of (Kondoh and Atluri (1986)). By us- ing the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton's method (and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale prob- lems. On the contrary, by using the presently developed Scalar Homotopy Meth- ods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conduct- ing post-buckling and limit-load analyses of nonlinear structures. While the simple Williams' toggle is considered as an illustrative example in this paper, extension