Post-buckling Paths Research Articles

A nonlinear Chebyshev-based collocation technique to frequency analysis of thermally pre/post-buckled third-order circular sandwich plates

In the current research, a comprehensive frequency analysis is performed for the circular sandwich plates in the pre-and post-buckling ranges occurred because of an in-plane thermal field. A sandwich system is fabricated with an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets based on the functionally graded models (FG-GPLRC). The displacement field is counted using Reddy’s third-order shear deformation theory because the thick layers form the desired circular structure. Due to large deformations in the post-buckling situation, the geometrically nonlinear strain–displacement relations based on the von Kármán model are employed. The Chebyshev collocation solution is implemented to attain the discrete form of equilibrium and dynamic equations. A displacement control iterative procedure is also adopted to address the nonlinear equilibrium state of the system. Moreover, the adjacent-equilibrium criterion is considered to recognize the static paths from small-amplitude vibrations. After performing validation studies with available articles, novel results are displayed to verify the influence of geometrical and physical characteristics on the post-buckling path and fundamental natural frequency of circular sandwich plates.

Direct evaluation of fire resistance for restrained steel columns in frames using consistent rigid-plastic models and energy formulations

The paper develops a simple and expeditious numerical methodology to evaluate directly the resistance of restrained steel columns in fire conditions. During fire, the column expands due to heating and a compressive restraining force is applied by the surrounding frame, which makes the column prone to buckling. Besides, the column’s mechanical properties degrade with heating. The column’s resistance in fire is defined by its collapse temperature, at which the column is no longer capable to carry the serviceability load applied prior to the fire event. The collapse temperature occurs deeply in the plastic range, along the thermal post-buckling path. Based on the critical thermal buckling mode, a plastic mechanism is adopted to represent the column at collapse. This plastic mechanism is analysed by means of a consistent energy formulation. The methodology is able to represent a wide range of cases in practice, such as longitudinal gradients of temperature, general supporting conditions, any level of longitudinal restraining stiffness and a serviceability load with any adequate magnitude. The expeditious procedure computes the collapse temperature always within a small and acceptable error when compared to more sophisticated numerical tools – in some cases, it surpasses commercial Finite Element packages and it is always much more precise than the simplified method presented in the European code of practice. Predictably, it will become a quite useful tool at the engineering office.

Nonlinear thermo-elastic stability of variable stiffness curvilinear fibres based layered composite beams by shear deformable trigonometric beam model coupled with modified constitutive equations

Nonlinear thermo-elastic buckling characteristics of composite variable stiffness beam with layers making use of curvilinear fibres under thermal environment is attempted here. The model is based on a shear deformable theory introducing trigonometric function, and considering von Kármán’s assumptions based geometrical nonlinear effect. The beam constitutive equation is modified according to the stress-free situation in the width direction of beam-Poisson’s effect in the formulation for predicting the behaviour of general lay-up composite beams. By the principle of minimum total potential energy, the governing equations in terms of incremental stiffness matrices are formed introducing general beam finite element. The global equilibrium equations formulated are solved for envisaging the post-buckling path through eigenvalue analysis iteratively, thus establishing the relationship of thermal temperature against moderate amplitude level of beam deflection. A systematic parametric analysis considering different lamina properties such as curvilinear fibre path angles and including lay-up sequences, thermal expansion coefficient, mixed laminate combining straight and curvilinear fibres-based layers is carried out on thermo-structural stability of curvilinear fibre-based beams. Also, the influence of geometric factors, flexible beam end support, and variation in thermal profile, etc. over the stability behaviour of beam is examined.

Buckling and postbuckling behaviors of symmetric/asymmetric double-beam systems

Double-beam systems composed of two interconnected parallel beams have become basic mechanical components of modern engineering structures. Focusing on the asymmetric features in boundary conditions and other structural parameters, this paper studies the buckling and postbuckling behaviors of a double-beam system supported on an elastic foundation. The nonlinear governing differential equations of the system are derived by using the Hamilton's principle, and discretized via the differential quadrature method. Then the pseudo-arc length algorithm is adopted to overcome difficulties of numerical continuation around bifurcation points. Compared with single beam systems, double-beam systems exhibit some peculiarities like multivalued critical buckling loads, in-phase and out-of-phase postbuckling paths. Based on the proposed procedures, these buckling and postbuckling behaviors are investigated in detail. For symmetric boundary cases of the two sub-beams, the multivalued critical buckling loads of the system compose a cluster of critical curves that intersect at a point determined only by the critical loads of single beams. Hence a simple criterion based on single beam critical loads is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system. However for asymmetric boundary cases, the cluster of critical curves locate in a region that depends on the intrinsic double-beam characteristics, and the criterion is revised accordingly. Also for asymmetric systems, some unique features of the out-of-phase postbuckling paths are detected. It is found that their stabilities are quite different from those of symmetric systems, and that strengthening the system stiffness may even trigger the out-of-phase buckling more easily.

On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams

In this work, thermal buckling and postbuckling behaviors of graphene platelet (GPL) reinforced porous nanocomposite beams are studied with inclusion of temperature-dependent material properties for the first time. The Halpin-Tsai micromechanics model and extended rule of mixture as well as open-cell metal foam model are applied to determine the effective properties of nanocomposites. A high order shear deformation theory with the aid of von Kármán nonlinearity is implemented to yield governing equations corresponding to thermal buckling and postbuckling problems. The numerically stable admissible functions constructed through Gram-Schmidt procedure are developed to describe end restraints of beams. An eigenvalue equation without geometric nonlinearity is solved to obtain critical buckling temperatures, while an iterative methodology is implemented to find the temperature-dependent thermal postbuckling paths of beams. An extensive numerical study is performed to check the influences of internal pores and GPL reinforcements on the thermal buckling and postbuckling responses of GPL-reinforced metal foam beams. It is found that the temperature-dependency of material properties plays important roles on thermal buckling/postbuckling behaviors of GPL-reinforced porous beams, and dispersing more GPLs on the upper and lower surfaces and fabricating more internal pores near central portion of beams can improve thermal instability-resistance capability of composite structures.

Symmetric and asymmetric thermo-induced buckling and postbuckling of rotating GPLRC annular plates rested on elastic foundation

Investigating the instability of annular plate which revolves on its axis in thermal environment is a valuable subject in vehicle engineering and aerospace industry, for instance, extending the service life of engine bearing. The symmetric and asymmetric buckling and postbuckling induced by the temperature rise of a rotating graphene-platelets-reinforced composite annular plate rested on Pasternak foundation are studied in present paper. Asymmetric deformation forms are captured by adopting the first-order shear deformation theory. Introducing von Kármán’s nonlinearity ensures the accuracy of postbuckling equilibrium paths. Highly coupled governing equations are solved by using the differential quadrature method and postbuckling paths are traced by utilizing the pseudo-arclength continuation algorithm. Parametric studies are conducted to discuss the influences of the dispersion patterns, weight fraction, and dimensions of GPLs, annular plate sizes, elastic foundation, and rotation motion on buckling and postbuckling. Results indicate that the sequence in which symmetric and asymmetric buckling modes appear is inextricably linked to the radial wave number of buckling configuration. Both adding GPLs and adjusting the foundation parameters appropriately can improve the ability to resist symmetric and asymmetric deformation of annular plate. The effect of rotation motion shows a transitional change along with the temperature rise.

Post-buckling shear resistance of slender girder webs: Stiffener participation and flange contributions

A series of validated finite element models are used to parametrically evaluate the participation of flanges and transverse stiffeners in the post-buckling web shear mechanics of welded I-shaped steel plate girders. The models are validated against the results of six large-scale web shear-buckling tests from the existing literature on highly slender plate girders (with web slenderness ratios of 250–267) and varying transverse stiffener spacing (with web panel aspect ratios from 1 to 4). All girders exhibited a 3-stage web shear-buckling response: 1) an initial linear elastic stage, 2) a post-buckling stage I during which shear stiffness begins to progressively decrease, and 3) a post-buckling stage II with either significantly reduced or negative stiffness after the buckled web panel has formed a yield mechanism. Increasing the flange thickness in a given girder configuration can change the mode of the post-buckling stage II response from gradual unloading (with ultimate shear reached at the end of post-buckling stage I) to positive hardening (with ultimate shear instead reached during post-buckling stage II at much larger displacement and with a 5–15% increase in shear resistance from the end of post-buckling stage I). At ultimate shear, the transverse stiffeners develop axial forces that equal only 10–30% of the applied shear load depending on their sizing, the panel aspect ratio, and the web plate thickness. The stiffeners are primarily engaged in the web's out-of-plane direction to impose panelization and less as an axial strut for the post-buckling load path.

Probing the Stability of Ladder-Type Coilable Space Structures

This paper analyzes the buckling and postbuckling behavior of ultralight ladder-type coilable structures, called strips, composed of thin-shell longerons connected by thin rods. Based on recent research on the stability of cylindrical and spherical shells, the stability of strip structures loaded by normal pressure is studied by applying controlled perturbations through localized probing. A plot of these disturbances for increasing pressure is the stability landscape for the structure, which gives insight into the structure’s buckling, postbuckling, and sensitivity to disturbances. The probing technique is generalized to higher-order bifurcations along the postbuckling path, and low-energy escape paths into buckling that cannot be predicted by a classical eigenvalue formulation are identified. It is shown that the stability landscape for a pressure-loaded strip is similar to the landscape for classical shells, such as the axially loaded cylinder and the pressure-loaded sphere. Similarly to classical shells, the stability landscape for the strip shows that an early transition into buckling can be triggered by small disturbances; however, while classical shell structures buckle catastrophically, strip structures feature a large stable postbuckling range.

A couple-stress-based moving Kriging meshfree shell model for axial postbuckling analysis of random checkerboard composite cylindrical microshells

The current investigation intends to devise a couple stress continuum-based moving Kriging meshfree shell model for axial nonlinear buckling and postbuckling responses of random checkerboard composite cylindrical microshells. Accordingly, the graphene nanoplatelet reinforcements are supposed to be randomly dispersed based upon a checkerboard scheme within the resin matrix, the effective material properties of which are estimated based upon a probabilistic method. The modified couple stress continuum elasticity is implemented into the higher-order shear deformation shell theory incorporating geometrical nonlinearity. The established microstructural-dependent shell model is then numerically analyzed via the moving Kriging meshfree technique with the ability to take the essential boundary conditions into account accurately by employing proper moving Kriging shape function. By tracing the postbuckling paths, the snap-through phenomenon is observed. It is found that for a higher graphene nanoplatelet volume fraction, the significance of the stiffening character associated with the rotation gradient tensor increases. Furthermore, for a specific value of graphene nanoplatelet volume fraction, by reducing the length to width aspect ratio of graphene nanoplatelet reinforcements, the role of couple stress size dependency in the critical buckling load and critical shortening of an axially compressed cylindrical microshell becomes more important.

Thermally induced instability on asymmetric buckling analysis of pinned-fixed FG-GPLRC arches

In the present work, an analytical study on the asymmetric static and dynamic buckling of a pinned-fixed functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) arch under thermal conditions is presented. The reinforcement of graphene nanoplatelets (GPLs) is dispersed along arch thickness by following a power law distribution. Making use of the modified Halpin–Tsai micromechanical model and energy method, the static buckling load of the arch under an arbitrary radial point load and the dynamic buckling load of the arch under an arbitrary radial step point load can be derived, which could be applied to determine the existence of dynamic buckling and the phenomenon of multiple limit points under a static state. A numerical analysis is conducted to verify the accuracy of the analytical method, a good prediction on the static and dynamic buckling of the pinned-fixed FG-GPLRC arch is demonstrated. In this study, the influence of GPLs weight fraction, concentration and geometry on the static and dynamic buckling of the arch is comprehensively discussed. The dynamic and static buckling loads of the arch are found to be sensitive to applied load position under various elevated temperatures. Arch buckling load decreases as the power law index increases, but it increases as temperature rises. It is also found that the present approach is able to trace the postbuckling paths of the arch for stability analysis. Besides, accurate first-known thermal buckling solutions are also presented.

Thermal post-buckling analyses of magneto-electro-elastic laminated beams via generalized differential quadrature method

The present paper is concerned with the thermal post-buckling behaviors of magneto-electro-elastic laminated beams. Timoshenko beam theory accompanying the von Kármán geometric nonlinearity is considered to model the beams. Temperature and magneto-electric fields are obtained based on satisfying the interfacial continuity conditions and corresponding boundary conditions, and then introduced into the stress-strain constitutive relationships. Afterwards, three governing equations of motion are established with respect to the generalized kinematical variables according to Hamilton's principle, and solved by the generalized differential quadrature method and pseudo-arc length continuation technique. The numerical results aim at the studies on thermal post-buckling paths, post-buckling vibration characteristics and magneto-electric potential responses, the effects on which the lay-up modes, thermal load types and magneto-electric fields exert are discussed.

An analogy between analytical, approximate and numerical methods in nonlinear buckling of functionally graded columns

Purpose The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column. Design/methodology/approach The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation. Findings An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated. Originality/value The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

Analytical and mathematical simulation for nonlinear stability of scale-dependent magneto-electro-elastic system

By using differential quadrature method (DQM), a numerical investigation was provided for nonlinear stability behavior of magneto-electro-elastic (MEE) cylindrical shells at microscale. It is assumed that the cylindrical shell has been subjected to compressive loads leading to buckling phenomena in geometrically nonlinear regime. The non-uniformity of strain field has been inserted in the formulation for considering the microscale effects. The material properties of the shell are considered to be inhomogeneous with graded distribution. After solving the governing equations using DQM, it is realized that if the nanoscale shell is subjected to electrical and magnetic fields, the post-buckling path may be changed with the value of electrical voltage and magnetic potential. Also, strain gradient effects have remarkable influence on post-buckling curves and critical voltages.

Post-buckling path and free vibration of a symmetric laminated plate vertically coupled with fluid under in-plane load

In this paper, the effect of compressive in-plane load on the equilibrium paths and vibration characteristics of a vertical symmetric laminated plate coupled with fluid is studied theoretically both in the pre- and post-buckling state. The Third-order shear deformation theory and von Karman nonlinear strain displacement relationship are utilized to describe the nonlinear deformation of the plate consists of a number of fiber-reinforced layers. The Hamilton's principle is introduced to formulate the plate-fluid coupled governing equations. A two-step theoretical approach is developed. Firstly, based on the Cylindrical arc-length method, we solve the nonlinear static equations to trace the snap-back behavior and pass the limit points along the complex equilibrium paths obtaining the stable buckling deflection. Then the coupled vibration characteristics are calculated using the tangent stiffness at the stable positions. The present theoretical results are validated by the numerical results. The effects of added mass and hydrostatic load are compared and analyzed. It is noted that, in the post-buckling state, the number of limit points along the static unstable equilibrium path increases with the compressive in-plane load getting larger. The variation trends of the natural frequencies and modes under different in-plane loads in pre- and post-buckling state are investigated.

Postbuckling responses of porous FGM spherical caps and circular plates including edge constraints and nonlinear three-parameter elastic foundations

An analytical investigation on nonlinear stability of porous functionally graded spherical caps and circular plates under uniform external pressure and elevated temperature, respectively, is presented in this article. Porosities are evenly and unevenly distributed within the structures and the effective properties of functionally graded material (FGM) are determined according to a modified rule of mixture. Governing equations are established within the framework of classical theory taking into account geometrically nonlinear terms, initial geometric imperfection, and interactive pressure from nonlinear three-parameter elastic foundation. Both temperature dependence of material properties and tangential elasticity of edge constraint are included. Approximate analytical solutions are assumed to satisfy clamped condition of boundary edge and Galerkin method is applied to derive load-deflection relations from which buckling loads and postbuckling paths are determined. Numerical results indicate that porosities have deteriorative and beneficial influences on load-carrying capacity of structures under external pressure and elevated temperature, respectively. Furthermore, the snap-through buckling response of pressurized FGM spherical cap can be alleviated and even eliminated by virtue of elastic foundations with suitable stiffness parameters.

Isogeometric analysis for buckling and postbuckling of graphene platelet reinforced composite plates in thermal environments

This work aims at examining the buckling and postbuckling behaviour of laminated composite plate in which GPLs (graphene platelets) are functionally dispersed under various kinds of thermal loadings. For this purpose, IGA (isogeometric analysis) method of which merits encompass the precise geometric representation, smoothness of higher-order, cost-efficiency and high accuracy is taken advantage of. Shear deformable plate theory into which a hyperbolic shear shape function and the von Kármán type nonlinear kinematics are incorporated is used to derive the nonlinear equations of equilibrium for the GPL reinforced composite plate subjected to thermal gradients. On the basis of the total Lagrangian formulation, the governing equation for the thermal buckling is established and the nonlinear thermal postbuckling path is traced together with the modified Newton-Raphson iteration. Three schemes of the symmetric GPL distributions are considered. The IGA approach proposed here, by checking the performance capacity via the multiple numerical tests, is substantiated to successfully follow the nonlinear buckling and postbuckling path of the nanocomposite plate in thermal environments. The further parametric investigations highlight the impacts of the thermal load type, GPL weight fraction, GPL arrangement pattern, plate geometric parameter, plate constraint condition and surface temperature contour on the thermal buckling and postbuckling responses of the GPL strengthened composite plate.

Robust improvement of the asymmetric post-buckling behavior of a composite panel by perturbing fiber paths

The buckling behavior of structures is highly sensitive to imperfections, i.e., deviations from the geometry and material properties of the ideal structure. In this paper, an approach is presented in which the effects of spatially varying fiber misalignments in composite structures are assessed through random field analysis and are subsequently used to improve the structure while simultaneously making it more robust to fiber misalignments. Effects of misalignments are quantified by applying random fields on the structure, which represent fiber misalignments. Using analyses of the effect of the random local stiffness changes due to fiber misalignments, a pattern of the relative influence these local changes have on the buckling load is created. By applying a small change to local fiber orientation corresponding to this pattern to the original structure, the performance of the design is improved. Additional stochastic analyses are performed using the improved design, reanalyzing the effects local fiber misalignments have on the structural performance and the subsequent changes in robustness. Stochastic results show an overall increase in the mean buckling load and a reduction in the coefficient of variation in the analysis of the perturbed structure. The approach is applied to a composite panel exhibiting asymmetric post-buckling behavior, i.e., having an unstable post-buckling branch and an (initially) stable branch. Results show that perturbations in the fiber path can nudge a structure into a more stable post-buckling path by promoting a post-buckling path using local changes in structural stiffness. The robustness of improved designs can also increase, making structures less susceptible to local fiber misalignments.

The study on stability of composite panel structure under axial compression load

The axial compressive load test and finite element simulation are made to study the mechanical properties of aircraft stiffened composite panel. Firstly, the numerical simulation model of composite stiffened panel structure is constructed, the accuracy and feasibility of the simulation model are verified by calculating the value of MAC (Modal Assurance Criterion) between the computational free modal and experimental free modal. Then, the nonlinear post-buckling numerical simulation is carried out to extract the buckling load, the post-buckling load path and the failure load. Then, the axial compression test of the panel structure is carried out by the testing machine, and the load-displacement curve and the load-strain curve of the loading position are obtained. Finally, the correctness of the analysis method is proved by comparing the numerical simulation value with the experimental analysis value. The experimental research method of the mechanical properties of the composite stiffened panel under axial load is explored, which can be used as a reference for the study of the bearing capacity of the stiffened panel.

Post-buckling of bamboo reinforced composite plates

Composite materials are preferred as today’s material for aerospace structural applications largely because composites provide high specific strength and stiffness. To their advantages, natural fibers such as bamboo, kenaf, jute and flax have been researched significantly for having high effective strength and stiffness and environmental friendly advantages in being renewable, biodegradable and sustainable. While many structures used in aerospace applications are thin and as such susceptible to buckling problem, this paper presents the study on the buckling and post-buckling of bamboo reinforced composite plates (BRCPs), applying the finite element method software of ANSYS APDL. A compressive load was given to the BFRC and linear buckling analysis was conducted first. The determined critical load from this analysis was then used in the non-linear post-buckling analysis to give the post-buckling path of the BRFC. In both linear and non-linear analysis, the effect of BRFC thickness was considered while both angle-ply and cross-ply orientations were used. The results of deflection of a composite plate showed an excellent agreement with past results. The study shows that as the length to thickness ratio (l/t) was increased, critical loads increased as well. Further, the critical loads for the angle-ply BFRCs were higher than the cross-ply counterparts of the same thickness. Similar pattern of behaviour can be seen for the post-buckling paths of the BFRCs.

Thermal Postbuckling of Temperature-Dependent Functionally Graded Nanocomposite Annular Sector Plates Reinforced by Carbon Nanotubes

In this study, the thermal buckling and postbuckling of functionally graded (FG) nanocomposite annular sector plates reinforced by carbon nanotubes (CNTs) are numerically analyzed. The effective material properties of FG nanocomposite are temperature-dependent (TD) and evaluated via the modified micromechanical method and rule of mixture. Based on the higher-order shear deformation theory (HSDT) and using the principle of virtual work and variational differential quadrature (VDQ) approach, the unified weak form of discretized nonlinear governing equilibrium equations is derived. Then, by using the linear part of equations and solving the derived eigenvalue problem, the critical temperature rise and associated mode shapes are obtained, which are used as the initial guess in solving the nonlinear thermal postbuckling problem. The pseudo-arc-length method and an iterative solver are employed to obtain the nonlinear thermal postbuckling equilibrium path of the FG nanocomposite annular sector plates. The influences of geometrical parameters, boundary conditions (BCs), CNT volume fraction, and CNT distribution pattern on the critical temperature rise and thermal postbuckling behavior of the FG nanocomposite annular sector plates are evaluated and discussed. Also, comparisons are made between the results considering the TD and temperature-independent (TID) properties. It is demonstrated that for higher values of sector angle, the effect of sector angle on the critical temperature rise and thermal postbuckling path is negligible. Moreover, by increasing the sector angle, the effect of BCs of straight edges vanishes, and the critical temperature rise and thermal postbuckling curves of for BCs of CSCS and SCSC approach those for CCCC and SSSS ones.