Unstable Post-buckling Behavior Research Articles

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

Cyclic loading test and lateral resistant behavior of flat-corrugated steel plate shear walls

Corrugated plate shear walls (CPSWs) can achieve large initial lateral stiffness, high resistance, and good energy dissipation capacity. However, severe post-buckling resistance degradation of the infill corrugated panel compromises the lateral resistant behavior of CPSWs with a large width-to-height ratio. To solve this problem, this paper proposed an innovative flat-corrugated steel plate shear wall (FCPSW), in which the infill panel is composed of a middle trapezoidal corrugated steel plate and two flat steel plates covering both sides by bolts. Cyclic loading tests were carried out for a FCPSW specimen and a CPSW specimen, to reveal the hysteretic behavior, failure mode and the restraining mechanism between the corrugated plate and the flat plates. Then, finite element analysis (FEA) was adopted to simulate the tests and investigate the effect of key parameters on the lateral resistant behavior of the infill flat-corrugated panel in FCPSWs under cyclic loading. The study showed that, the infill flat-corrugated panel utilized the interaction between the corrugated plate and the flat plates, to effectively reduce the out-of-plane buckling deformation and improve the lateral performance. The normalized residual resistance of the FCPSW specimen can be increased by 50% compared to the ordinary CPSW specimen. The material yield stress, the bolt spacing, the corrugation length, the corrugation inclined angle and the thickness ratio of the infill plates were the key factors affecting the hysteretic performance of the flat-corrugated panel, while effects of the width-to-height ratio and the corrugation ratio was reduced compared to the conventional CPSWs. The design suggestions proposed can avoid unstable post-buckling behavior of the infill panel in FCPSWs and provide a simple and feasible method in the preliminary design.

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

Interactive buckling in long thin-walled rectangular hollow section struts

An analytical model describing the nonlinear interaction between global and local buckling modes in long thin-walled rectangular hollow section struts under pure compression founded on variational principles is presented. A system of nonlinear differential and integral equations subject to boundary conditions is formulated and solved using numerical continuation techniques. For the first time, the equilibrium behaviour of such struts with different cross-section joint rigidities is highlighted with characteristically unstable interactive buckling paths and a progressive change in the local buckling wavelength. With increasing joint rigidity within the cross-section, the severity of the unstable post-buckling behaviour is shown to be mollified. The results from the analytical model are validated using a nonlinear finite element model developed within the commercial package Abaqus and show excellent comparisons. A simplified method to calculate the local buckling load of the more compressed web undergoing global buckling and the corresponding global mode amplitude at the secondary bifurcation is also developed. Parametric studies on the effect of varying the length and cross-section aspect ratio are also presented that demonstrate the effectiveness of the currently developed models.

Buckling and Post-buckling Behavior of Beams With Internal Flexible Joints Resting on Elastic Foundation Modeling Buried Pipelines

The buckling and post-buckling behavior of axially loaded Winkler beams with flexible internal hinges is addressed, aiming to provide a background for the investigation of upheaval buckling of buried pipelines equipped with flexible joints for their protection against activation of reverse seismic faults. In order to acquire qualitative understanding of the interaction between the hinge stiffness and the soil stiffness for different cases, the beams under investigation are considered as either simply-supported or clamped. At first, elastic critical buckling loads and corresponding eigenmodes are numerically obtained using linearized buckling analysis, and eigenmode cross-over is investigated considering soil and hinge rotational stiffness. Geometrically nonlinear analyses with imperfections (GNIA) are then performed, indicating for most cases descending post-buckling paths, thus unstable post-buckling behavior, with the exception of cases of very soft soil. The sensitivity of the response to initial imperfection shape and magnitude is also addressed, to identify their impact on the post-buckling behavior. Beam buckling behavior is moreover examined by considering the beam being surrounded by soil exhibiting different stiffness in the upward and the downward direction. The results are compared to the case of a continuous beam, in order to highlight the impact of internal hinges on the beam overall buckling behavior.

Linear and Nonlinear Buckling Response and Imperfection Sensitivity of Cable-Stayed Masts and Pylons

An approximate analytical method for quantifying the lateral resistance offered by inclined cables to guyed towers, such as the pylons of cable-stayed bridges, is presented first. For this purpose, an analytical expression was derived for the evaluation of the pylon’s effective buckling length. Taking into account the cables’ inclination and prestressing and approximating the nonlinearity caused by sag using Dischinger’s formula, the cables were substituted by horizontal translational springs of equivalent stiffness. Then, an analytical solution of the linear buckling load of the equivalent spring model was obtained. The presented analytical method can be very useful at the preliminary design stage.Then, the approximate results were compared to the ones extracted from geometrically nonlinear finite element analysis with imperfections (GNIA) of both the more exact cable–pylon model and the spring–pylon model. The results show that the critical buckling load is predicted with very good accuracy for a wide range of values of cable inclination and pre-tensioning and indicate an unstable post-buckling behavior. In- and out-of-plane buckling of pylons with single or multiple levels of cables, in either a fan or a harp arrangement, with either a fixed or a hinged base, is addressed. A parametric analysis showed a strong imperfection sensitivity leading to significantly smaller ultimate loads of the imperfect structures compared to the linear buckling loads.

Local–global mode interaction in stringer-stiffened plates

A recently developed nonlinear analytical model for axially loaded thin-walled stringer-stiffened plates based on variational principles is extended to include local buckling of the main plate. Interaction between the weakly stable global buckling mode and the strongly stable local buckling mode is highlighted. Highly unstable post-buckling behaviour and a progressively changing wavelength in the local buckling mode profile are observed under increasing compressive deformation. The analytical model is compared against both physical experiments from the literature and finite element analysis conducted in the commercial code Abaqus; excellent agreement is found both in terms of the mechanical response and the predicted deflections.

Re-interpreting simultaneous buckling modes of axially compressed isotropic conical shells

Elastic stability of shell structures under certain loading conditions is characterized by a dramatically unstable postbuckling behavior. The presence of simultaneous ‘competing’ buckling modes (corresponding to the same critical buckling load) is understood to be largely responsible for such behavior. In this paper, within the framework of linear bifurcation eigenvalue analysis and Donnell shallow shell theory, the presence of simultaneous buckling modes in axially compressed isotropic cones is determined using the semi-analytical method of Galerkin. The results are presented in the plane of the dimensionless reciprocal meridional and circumferential buckling half wavelengths, and are compared with the locus of simultaneous buckling modes of axially compressed cylinders, described by the so-called Koiter circle. By using an optimizing procedure, it is shown that the cluster of simultaneous buckling modes in cones is well described by the Koiter circle of an equivalent cylinder with appropriate length and radius. Such optimizing values of length and radius allow us to gain some insight into the simplifying treatment of the buckling of cones through the concept of equivalent cylinder.

Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model

This paper presents an investigation on the buckling and postbuckling of microtubules (MTs) subjected to a uniform external radial pressure in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The small scale parameter e 0 a is estimated by matching the buckling pressure of MTs measured from the experiments with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling pressure and postbuckling behavior of MTs are very sensitive to the small scale parameter e 0 a. The results reveal that the 13_3 microtubule has a stable postbuckling path, whereas the 13_2 microtubule has an unstable postbuckling behavior due to the presence of skew angles.

A continuum mechanics nonlinear postbuckling analysis for single-walled carbon nanotubes under torque

Abstract Nonlinear buckling and postbuckling analysis of single-walled carbon nanotubes subjected to torsional load are examined by continuum mechanics elastic shell model. The boundary layer theory of shell buckling and the singular perturbation technique is applied to solve the governing equations based on the Karman–Donnell-type nonlinear differential equations. The buckling loads and postbuckling equilibrium paths are obtained theoretically. For more detailed descriptions on the postbuckling behaviors of single-walled carbon nanotubes, postbuckling relations between twist, end-shortening, transverse deflection and the torque are investigated for nanotubes with different geometric parameters. Numerical results obtained show that single-walled carbon nanotubes under the torsional load have unstable postbuckling behavior, and the postbuckling equilibrium path is higher for the thinner nanotube than that of the thicker nanotubes. Furthermore, a new conclusion drawn from the postbuckling analysis on single-walled carbon nanotubes with different Young's modulus and the wall thickness is significant to definite the mechanical parameters of single-walled carbon nanotubes.

Torsional buckling and postbuckling equilibrium path of double-walled carbon nanotubes

Based on a continuum elastic double-shell model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell, involving the van der Waals interaction between the inner and outer nanotubes, the governing equations for the buckling and postbuckling behavior of double-walled carbon nanotubes subjected to torsional load are presented based on the Karman–Donnell-type nonlinear differential equations. Applying the boundary-layer theory of shell and employing singular perturbation technique, accurate solutions are obtained to determine the buckling loads and postbuckling equilibrium paths. Through detailed calculations, main postbuckling characteristics, i.e. connections of the angle of twist, axial end-shortening and transverse deflection with applied torque, are clarified for a range of nanotube geometrics. Results reveal that under the torsional load double-walled carbon nanotube has an unstable postbuckling behavior. And it is confirmed that the postbuckling equilibrium path becomes higher when the nanotube becomes thinner.

Postbuckling prediction of double-walled carbon nanotubes under axial compression

Abstract An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube and the double-walled carbon nanotube both have an unstable postbuckling behavior.

積層ゴム支承の座屈荷重に与える幾何学的非線形の影響

This paper evaluates the effect of geometrical nonlinearity on the buckling load of laminated rubber bearings. In the past research, the author proposed a nonlinear buckling model for shear-bending columns, which revealed the fact that the reduction of lateral stiffness of rubber bearings in a large horizontal displacement depends on their configuration. It has been believed that the low-rise elastomeric columns have a more stable load-displacement characteristics than slender ones. The model, however, clarified the risk of the unstable post-buckling behavior latent in the flat configuration of bearings. On the other hand, the slenderness of elastomeric columns plays an important role to maintain their load supporting capacity in a large deformed area. These analytical discoveries are numerically investigated and compared with some experimental observations.

On the stability of Kelvin cell foams under compressive loads

It has been previously shown that the nonlinearity exhibited in the compressive response of open cell foams is governed by cell ligament buckling. Significant insight into this behavior can be gained by idealizing such foams as periodic, space-filling Kelvin cells assigned several of the geometric characteristics of actual foams. The cells are elongated in the rise direction; the ligaments are assumed to be straight, to have Plateau border cross sections, and nonuniform cross sectional area distribution. The mechanical response of such foams can be established using models of a characteristic cell assigned appropriate periodicity conditions. The ligaments are modeled as shear deformable beams. The periodicity of this microstructure allows the use of Bloch wave theory to conduct the search for the critical state efficiently. The method tailored to the present microstructure is outlined. It is subsequently used to establish the critical states for uniaxial and a set of triaxial loadings. A rich variety of buckling modes are identified which are affected by the anisotropy and the mutliaxiality of the applied loads. Under some loadings the critical modes have long wavelengths which are shown to lead to unstable postbuckling behavior involving localization. Under other loading conditions the modes are either local to the characteristic cell or involve an assemblage of a few such cells. For the cases analyzed local modes were found to have a stable postbuckling response.

Postbuckling prediction of double-walled carbon nanotubes under hydrostatic pressure

An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to external hydrostatic pressure. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under hydrostatic pressure. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube has a stable postbuckling path, whereas the double-walled carbon nanotube has an unstable postbuckling behavior due to the presence of van der Waals interaction forces.

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.

Effects of ply-arrangement on compressive failure of layered structures

Using detailed finite element models, compressive failure mechanisms of composite structures consisting of many laminae are analyzed. It is assumed that the structures contain interlaminar delamination and their failure mode to be characterized by either buckling or delamination growth. Our primary goal is to identify the effects of delamination and ply-arrangement on the multi-layered structures. Up to 32 laminae are distinctly modeled in this investigation. Our study considers two most basic geometry; one is flat panels under compressive load and the other is cylindrical shells subjected to external pressure. In both cases, the energy release rate and mixed-mode stress intensity factors are computed to quantify the crack driving force. The results are used to determine dominant failure initiation mode, structural buckling or delamination growth. Regardless of the structural type and total number of layers, a significant reduction in the load carrying capacity may occur when interlaminar delamination exists. In the flat panels, interlaminar delamination can generate unstable post-buckling behavior and lower the steady-state post-buckling load. However, delamination growth does not likely to occur during the pre-buckling stage. For the cylindrical shells, delamination growth may initiate prior to structural buckling. The location of delamination also plays an important role in defining the critical crack initiation load. When the delamination is located within 25% to 40% of shell thickness measured from the outer surface, the crack initiation load can be as low as half of the buckling load. In both types of structures, as the total number of plies increases, the layer effect diminishes. The overall deformation and failure behaviors of panels with large number of layers approach those of the infinite-layer model with homogenized material properties.

Interactively Induced Localized Buckling in Sandwich Structures With Core Orthotropy

Recent work on the interactive buckling behavior of compressed sandwich structures (Hunt and Wadee, 1998) is extended to orthotropic core materials. The variation of the material properties can tend to maximize the interactive effect such that overall and secondary localized modes are triggered almost simultaneously, giving rise to highly unstable post-buckling behavior in systems of practical dimensions.

Identifying failure mechanisms of composite structures under compressive load

Various failure mechanisms involving both local and global deformation mechanisms of layered structures, consisting of differently oriented orthotropic laminae, are investigated with a large deformation finite element analysis. Under large compressive loads, fiber-reinforced composite structures can fail by buckling, kink band formation and/or delamination growth. Our aim is to identify the dominant mode which leads to the structural failure under a given boundary condition and geometrical shape. It is assumed that these structures contain initial interlaminar flaws represented by embedded delaminations. Such flaws can play a significant role in defining overall structural integrity and deformation mechanisms. Our study considers two structures with distinguished shapes, both consisting of four layer laminae. The first structure is a flat panel under compressive load and the second structure is a cylindrical shell subjected to external pressure. In both cases, the energy release rate and mixed-mode stress intensity factors are computed to quantify the crack driving force and used to measure likelihood of delamination growth. In the flat panel model, two buckling modes, a global and a local, are observed under quasi-static loading conditions. Depending upon the delamination length and laminate stacking orientations, the interaction of these two modes can produce an unstable post-buckling behavior. It is found that the energy release rate exceeds experimentally estimated fracture toughness values only after buckling occurs. In the cylindrical shell study, lower critical buckling loads are observed for models with longer interlaminar delamination as in the flat panel model. However, unlike the flat panel case, the energy release rate surpasses the critical toughness well before the applied pressure reaches the buckling load of the flawed cylindrical shell. This behavior implies that a shell containing an embedded defect along an interface can fail by delamination growth and therefore has a failure load lower than its critical buckling load. Also, for thicker cylindrical shells, the compressive internal stress exceeds the predicted stress required for kink band formation prior to structural buckling. The competition among these failure modes, buckling, delamination growth and compressive kinking in fiber-reinforced composite structures, is studied and a design diagram is introduced here. This diagram predicts a likely dominant failure mode for given structural dimensions and material properties. In addition, unstable delamination growth along the lamina interface under dynamic loading conditions is simulated using an iterative computational procedure. The results predict that the crack tip velocity can reach a fraction of the material wave speed leading to total structural failure in a very short time. For the cylindrical shell, the inertial effect on the pressure-deformation relation is investigated by comparing it with the static results.

Interlaminar dynamic crack propagation

Effective computational procedures are established and used to characterize dynamic crack propagation along material interfaces between isotropic and orthotropic materials. We first simulate a dynamic fracture experiment in which crack propagation occurs along an interface between PMMA and steel in a dynamically loaded bend specimen. In the analysis, the dynamic energy release rate and mixed-mode stress intensity factors are extracted from finite element field solutions using suitably formulated conservation integrals. These variables form the basis for defining an interface fracture criterion under dynamic conditions. In order to propagate the crack according to such a criterion, an iterative procedure is utilized to determine the correct crack tip velocity history. An important computed result, which is consistent with experimental observation, is that the energy release rate decreases as the crack propagation accelerates. The physical interpretation of this result is that less energy is absorbed by the moving crack as its velocity increases. In the second part of our study, the computational procedures are modified for the dynamic fracture analysis of a thin composite panel consisting of differently oriented orthotropic laminae. Here we investigate the interplay between delamination and buckling of the more complex structure. It is assumed that the panel contains an initial finite interlaminar crack and is subjected to a uniaxial compressive load. Without any crack extension, two buckling modes are observed under quasistatic conditions. One is characterized by an overall panel buckling and the other is dominated by a local ligament buckling near the crack. Coupling of the two modes produces not only a lower critical buckling load but also an unstable post-buckling behavior. Therefore, an embedded delamination is seen to create imperfection sensitivity, leading to limit load type behavior. Such a condition is potentially troublesome for compression loaded thin composite panels. In addition, the large compressive load may also trigger a dynamic propagation of the embedded delamination. We have simulated this unstable dynamic crack growth using the iterative method and a simplified delamination criterion. In the simulation, the energy release rate, the mixed-mode stress intensity factors and the resulting crack tip speed are obtained. These results predict that an existing embedded delamination can weaken and significantly alter the post-buckling behavior of composite panel.