Postbuckling Response Research Articles

Flexural buckling and post-buckling analysis of tapered columns in transient fire

This paper presents time-varying flexural buckling and post-buckling responses of tapered columns under a transient fire of which the temperature is non-uniform along the length. First, the transient temperature field within the column is determined by solving a dimensionally reduced transient heat transfer problem considering the convection-radiation boundary conditions and temperature-dependent (TD) material properties. Subsequently, the von Kármán geometric nonlinear theory is applied to obtain the nonlinear equilibrium equation. The nonlinear heat transfer and equilibrium equations are solved by the differential quadrature (DQ) method. According to a yield criterion, the ultimate load at which the material failure occurs is predicted. The influences of a transient localised fire on the temperatures, buckling and post-buckling responses of tapered steel tubular columns are discussed.

Nonlinear thermal post-buckling analysis of rectangular FG plates using CUF

In this study, a numerical analysis examines the thermal post-buckling response of rectangular functionally graded (FG) plates. Since the post-buckling is analyzed in a thermal environment, temperature distribution across the thickness is computed by an accurate numerical method. Here, for the geometrically nonlinear analysis because of assuming large displacement, FEM is utilized according to Carrera Unified Formulation (CUF), a precise formulation and higher-order deformation theory. The principle of virtual work is employed to achieve the nonlinear equilibrium equations. The governing equations are solved by operating the arc-length method. The influence of parameters such as geometric aspect ratio, length-to-thickness ratio, boundary conditions, type of temperature distribution, and volume fraction index on the FG plates’ temperature-deflection path has been investigated. In addition, a comparison among the present study method with another reference about thermal post-buckling and also linear buckling analysis (LBA) is performed. Nonlinear post-buckling analysis shows that the bifurcation point occurs only at fully clamped boundary conditions and low aspect ratios. It proved the results of LBA in the mentioned state are verifiable. The outcomes confirm the usefulness of utilizing CUF on thermal post-buckling since the temperature curves in the figures are at a lower level compared to the similar reference.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

Nonlinear thermo-mechanical axisymmetric stability of FG-GPLRC spherical shells and circular plates resting on nonlinear elastic medium

ABSTRACT An analytical approach for nonlinear thermo-mechanical buckling of functionally graded graphene platelet reinforced composite (FG-GPLRC) circular plates and shallow spherical shells resting on nonlinear elastic medium using the higher-order shear deformation theory (HSDT) and the nonlinearities of von Kármán is established in the present work. Assuming that the structures are axisymmetrically displaced for the central axis and external pressure and thermal loads are applied. The nonlinear elastic medium is used to model the behaviour of hardening or softening medium depending on the positive and negative values of the nonlinear parameter, respectively. By applying the Ritz energy method, the relation expressions of load-deflection are archived to investigate the postbuckling response and critical buckling load of the structures. Special effects on the nonlinear thermo-mechanical behaviour of plates and shells with different medium stiffnesses, different material parameters, and different geometrical dimensions are explored and discussed in numerical results.

3D heat conduction-induced postbuckling behaviour of thin-walled imperfect laminated cylindrical panels reinforced with graphene platelets

This work reports, in virtue of IGA (isogeometric analysis) method, 3D state heat conduction-induced postbuckling behaviour of thin-walled laminated cylindrical panels with geometric imperfection where GPLs (graphene platelets) are scattered in a functional gradation manner. Shear deformation theory of first-order suited for the numerical nonlinear and buckling analysis of thin-walled shells that takes geometric imperfection into account is established. To accurately describe the initial imperfection, imperfection function capable of modelling the amplitude and localization of geometric imperfection is employed. An equation based on the effective thermal conductivity of the GPL-strengthened cylindrical panel to determine the temperature distribution by 3D steady-state heat conduction is presented as well, and a new formula to dictate the volume fraction of GPLs which is valid regardless of the number of layers is proposed. The present isogeometric approach, through testing the analysis capacity via several benchmark problems, is substantiated to be accurate in predicting the critical thermal buckling temperature and in following the whole postbuckling trace. Effect of the initial imperfection on the nonlinear postbuckling response behaviour of the thin-walled GPL-reinforced cylindrical panels in 3D state heat conduction is discussed in detail through the further parameter analysis.

Assessment of failure mechanism of double-strap 3D-FML adhesively bonded joints under tensile and compressive loadings using cohesive zone modelling approach

Despite the widespread applications of Fiber Metal Laminates (FMLs), there is a clear lack of information regarding the performance of bonded joints mating 2D-FMLs and the recently-developed 3D-FMLs, particularly when considering their performances under compressive loading. Therefore, as a follow-up to the authors’ recent investigations, an extensive series of numerical analyses are conducted in the present study to simulate the tensile and compressive (i.e., buckling, and post-buckling) responses of double-strap adhesively bonded joints mating 3D-FMLs. The 3D-FMLs are joined using carbon fiber-reinforced plastic (CFRP) straps and structural epoxy resin. The damage initiation and evolution in the bonded regions are modelled by seven different FE models, which incorporated the mixed-mode trapezoidal Cohesive Zone Modelling (CZM) approach in conjunction with continuum elements. Firstly, the effects of different numbers of CZM layers and thicknesses on simulating the damage mechanism are investigated. Subsequently, the influence of CFRP straps’ thickness and length on the performance of the adhesive joints is parametrically studied. The distributions of peel and shear stresses along the length of the bonded regions are also systematically explored. Based on the results, the optimal joint configuration is established, and the most effective modelling framework is highlighted by comparing numerically predicted results against experimental results.

Re-examination of thermo-mechanical buckling and postbuckling responses of sandwich plates with porous FG-GPLRC core

The current study re-examines compressive postbuckling and thermal postbuckling responses of porous sandwich plates resting on an elastic foundation. The core of the sandwich plates is made of porous graphene platelets reinforced composite (GPLRC). Compressive postbuckling of the porous sandwich plates under uniaxial compression at elevated temperature and thermal postbuckling of the same plates under uniform thermal loading are studied. The GPLRC core is made of multilayers, and each layer may have different values of porosity coefficient to obtain a piecewise functionally graded (FG) pattern. The Young’s moduli along with the shear modulus of the porous GPLRC core are estimated through a generic Halpin–Tsai model in which the porous nature of the material is taken into consideration. The material properties of the metal face sheets and the porous GPLRC core are assumed to be temperature dependent. The governing equations for the postbuckling of the porous sandwich plates reinforced by GPLs are established based on the Reddy’s third order shear deformation theory (TSDT). In the modeling, we also consider the von Kármán nonlinear strain–displacement relationships, the plate-foundation interaction and the thermal effect. A two-step perturbation approach is utilized in solving the postbuckling problem. Numerical studies are performed to compare the results obtained from the present model and the equivalent isotropic model (EIM). The results confirm that the EIM is not suitable for analyzing buckling and postbuckling of porous plates.

Reply to “A comment on buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis” [Thin-Walled Struct. 184 (2023) 110505

The comment published by Dehrouyeh-Semnani (2023) is based on an incorrect assumption which results in completely erroneous conclusions. In the present paper, the invalidity of claims given by that comment are proved clearly to avoid any possible mistake for readers.

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

Advanced Finite Elements for Geometrically Nonlinear Analysis of Rectangular Plates under Various In-Plane Loadings Accounting for the Boundary Conditions of the Stiffeners

Highly flexible plate structures are subjected to various loading and edge conditions in many engineering applications, such as in the aerospace and civil industries. In this paper, the post-buckling response of rectangular plates subjected to axial and shear loadings is investigated by employing the Carrera Unified Formulation (CUF). As in previous works, only isolated plates have been considered, regardless of the effects of the stiffeners and boundary conditions imposed by the surroundings. However, in this research, different plate models are implemented, and the effects of boundary conditions imposed by the stiffeners are evaluated precisely. Comprehensive assessments are provided for the geometrically nonlinear equilibrium curves of the plate structures using the Newton–Raphson linearization method with the path-following constraint. Furthermore, the comparisons of different geometrically nonlinear assumptions according to the full Green–Lagrange nonlinear model and von Kármán nonlinear plate theory are presented. The importance of strain–displacement relationships and stiffeners’ effects in the nonlinear response of plate structures is highlighted. As a result, the accuracy and effectiveness of the presented CUF-2D model are demonstrated.

The Effects of Favorable and Unfavorable Geometric Perturbations on the Response of Pressure-Loaded Hemispherical Shells

Abstract Hemispherical shells subjected to external pressure loading are known to be sensitive to geometric imperfections. Dimple imperfections and their effects on the buckling and postbuckling response of spherical and hemispherical shells under externally applied pressure loads have been widely studied. The studies have shown that dimple imperfections are unfavorable, and their presence leads to a drastic lowering of buckling pressure, the severity being dependent on the radius-to-wall thickness ratio of the shell. Motivated by a plenary lecture presented by Hutchinson, we have conducted equilibrium analysis through finite element computations of externally pressurized hemispherical shells to understand if we can intentionally design shells with initial geometric perturbations that are favorable to resisting external pressure. We have studied dimple imperfections that can either increase or decrease the local curvature. We show that outward pointing dimples outperform inward-pointing dimples in such a structure and hence can be viewed as being favorable with regards to shell buckling under external pressure. With advances in precision manufacturing, the results presented here serve as a guide to designing shells with intentional perturbations to the initial shell geometry that can lead to favorable outcomes.

Buckling response of externally pressurised sealed and retrofitted steel cylindrical shells

Buckling response of steel cylindrical shells with no reinforcement or seal, cylindrical shells reinforced with CFRP, and cylindrical shells sealed against leakage was experimentally compared. Twelve cylindrical shells were tested; four had no reinforcement/seal, four were reinforced with CFRP, and four were sealed. The cylindrical shells had the same radius-to-thickness ratio of 445 but had various height-to-radius ratios of 2, 4, and 6. Results indicated that the buckling capacity of the shells decreased with increasing the shell height. Whilst a post-buckling response was established for the sealed shells, CFRP-reinforced shells had no post-buckling response. CFRP reinforcement significantly increased the shells' initial and overall buckling load, but the sealing was only effective in the initial buckling load.

Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory

In this article, the nonlinear postbuckling behavior of the magneto-electro-thermo-elastic (METE) laminated microbeams is presented. According to the modified couple stress theory (MCST) and Reddy's three-order shear deformation theory (RTSDT), in conjunctions with the von Karman geometric nonlinearity, the nonlinear static model of METE laminated microbeam is established. The nonlinear governing equations and the corresponding boundary conditions are derived by using the principle of virtual work principle. Afterwards, using an analytical method, the nonlinear postbuckling behavior of METE laminated microbeam with simple supported and clamped boundary conditions at the both ends is described. Moreover, numerical examples are exhibited to reveal the effects of the material length scale parameter, slenderness ratio, temperature rise, magneto-electric potential on the critical buckling load and the nonlinear postbuckling response. Also the distribution law of magneto-electric potential through the thickness direction of the microbeam with various lay-up modes is discussed.

Numerical study of the compression of tightly constrained slender rods

Compression of a slender rod, tightly confined in a bore, is an archetypal problem that has relevance in a wide variety of situations ranging from worm like macromolecular chains and microtubules to drill strings in wellbores. If persistence length quantifies the stiffness of the rod, these problems span over cases where persistence length is of the order of the contour length to where it is many orders larger. In spite of this wide spread, all of these problems adopt the Kirchhoff rod theory, whereby, with appropriate normalisations, the force displacement response becomes independent of stiffness. This is true for unconstrained rods but not for rods constrained in bores. In fact, when the bore dimensions are much smaller compared to the length of the rod, i.e. the rod is ‘tightly’ constrained, post Euler buckling responses are characterised by abrupt large configurational adjustments or so-called ‘jump-instabilities’. The large and repeated configurational adjustments render the numerical simulation of the post-buckling force displacement responses difficult. We have used a geometrically exact 3-dimensional beam theory with rough constraining walls in order to successfully simulate the force displacement response of constrained rods up to large strains. In particular, we have simulated rods with stiffness values ranging over 6 orders of magnitude — starting from values comparable to microtubules in the cytoskeleton to those applicable to large scale engineering structures. Our results show that, unlike what is expected from the Kirchhoff rod theory, the geometry of the bore, and its size, play a significant role in rendering the compression response stiffness dependent. Moreover, nature and degree of confinement also dictate the stable 3-dimensional configuration to which the rod eventually evolves. In circular bores, the rod evolves to an almost perfect helical shape while in square ones, it does not.

Three-dimensional postbuckling analysis of thick hyperelastic tubes

While the buckling of tubes under axial compression has been extensively studied, the postbuckling behavior of thick tubes remains elusive. In this paper, we conduct three-dimensional buckling and postbuckling analysis for thick hyperelastic tubes subjected to axial compression under finite deformation by the asymptotic expansion method. Our theoretical results successfully predict the deformation and stress-strain curves of buckled tubes near the critical loading, which are well validated by finite element analysis. Depending on the geometry, three kinds of postbuckling paths, including continuous buckling, snap-through and snap-back, are discovered. We summarize our results in two phase diagrams of the critical stretch for the onset of buckling and postbuckling paths with respect to the geometric parameters. In particular, we have observed that the postbuckling response can undergo a complex transition among different types of postbuckling paths, including continuous buckling, snap-through and snap-back, which is attributed to the competition between two modes of deformation, i.e., global deformation and local distortion. When a tube is long and thick, it prefers global deformation, and its cross section remains almost a plane after buckling, whereas when a tube is relatively short and relatively thin, it prefers local distortion and its cross section does not remain a plane any more after buckling. Our work provides understanding and insights into the buckling and postbuckling of thick tubes, and bridges the knowledge gap between postbuckling of thick columns and tubes.

A comment on “Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis” [Thin-Walled Struct. 159 (2021) 107231

The reported results by Fan et al. (2021) show that the size-dependent and -independent porous functionally graded (PFG) small-scale plate with unsymmetrical material distribution, under fully simply supported boundary conditions, and subject to uniaxial uniform compressive load remains flat and then loses its stability at a critical buckling load. In the present comment, it is proved that the above-mentioned small-scale plate does not remain flat, no matter how small are the in-plane load, at the beginning of in-plane loading, the small-scale plate undergoes lateral deformation. Consequently, the buckling and post-buckling responses predicted by Fan et al. (2021) do not occur for the small-scale plate. It should be pointed out that Aydogdu (2008) utilized the method proposed by Leissa (1986) for the unsymmetrically laminated plate under in-plane load to prove that the functionally graded plate with unsymmetrical material distribution and fully simply supported boundary conditions does not remain flat when it is exposed to an in-plane load. The work presented by Aydogdu (2008) was not reviewed and discussed by Fan et al. (2021).

Thermal Effect on The Post-buckling And Mechanical Response of Single-Walled Carbon Nanotubes: A Numerical Investigation

Thermal Effect on The Post-buckling And Mechanical Response of Single-Walled Carbon Nanotubes: A Numerical Investigation

Post-buckling analysis of shell-like structures using an implicit dynamic time integration scheme

PurposeThis paper aims to investigate post-buckling responses of shell-like structures using an implicit conservative-decaying time integration dynamic scheme.Design/methodology/approachIn this work, the authors have proposed the use of a four-node quadrilateral flat shell finite element with drilling rotational degree of freedom within the framework of an updated Lagrangian formulation mutually with an implicit conservative-dissipative time integration dynamic scheme.FindingsSeveral numerical simulations were considered to evaluate the accuracy, robustness, stability and the capacity of the considered time integration scheme to dissipate numerical noise in the presence of high frequencies. The obtained results illustrate a very satisfying performance of the implicit conservative-dissipative direct time integration scheme conjointly with the quadrilateral flat shell finite element with drilling rotation.Originality/valueThe authors have investigated the potential of the implicit dynamic scheme to deal with unstable branches after limit points in the non-linear post-buckling response of shell structures with no need for structural damping. The capability of the studied algorithm to study buckling and post-buckling behaviour of thin shell structures is illustrated through several numerical examples.

Bimetallic Thin-Walled Box Beam Thermal Buckling Response.

A beam model for thermal buckling analysis of a bimetallic box beam is presented. The Euler-Bernoulli-Vlasov beam theory is employed considering large rotations but small strains. The nonlinear stability analysis is performed using an updated Lagrangian formulation. In order to account for the thermal effects of temperature-dependent (TD) and temperature-independent (TID) materials, a uniform temperature rise through beam wall thickness is considered. The numerical results for thin-walled box beams are presented to investigate the effects of different boundary conditions, beam lengths and material thickness ratios on the critical buckling temperature and post-buckling responses. The effectiveness and accuracy of the proposed model are verified by means of comparison with a shell model. It is revealed that all of the abovementioned effects are invaluable for buckling analysis of thin-walled beams under thermal load. Moreover, it is shown that the TD solutions give lower values than the TID one, emphasizing the importance of TD materials in beams.

A non-linear Ritz method for progressive failure analysis of variable angle tow composite laminates

A Ritz formulation for non-linear analysis of damage initiation and evolution in variable angle tow composite plates under progressive loading is presented. The model is built on a few key items. It assumes first order shear deformation theory kinematics and non-liner strains in the von Kármán sense. The constitutive relationships are formulated in the framework of continuum damage mechanics at the ply level, so that each laminate layer can experience in-plane damage initiation and evolution, then reflected in material softening and loss of local stiffness. A Ritz polynomial expansion of the primary variables and the minimization of the total potential energy provide the discrete solution equations, which are then solved with an incremental-iterative approach for capturing damage evolution. A few tests have been successfully performed to validate the approach. Eventually, some original results about variable angle tow plates under progressive in-plane compression loads highlight the effect of damage on the post-buckling response.