Buckling Mode Shapes Research Articles

Experimental and numerical buckling analysis of CNT reinforced polymer composite plates

A combined experimental and numerical buckling analysis is performed for Carbon nanotube-reinforced composite (CNTRC) laminates under uniaxial in-plane compressive load. A finite element simulation program is developed in MATLAB environment and the recorded buckling responses are verified with the experimental results. Closely aligned numerical and experimental results exhibits that inclusion of 0.3% CNT increases buckling strengths of CNTRC composite laminates, regardless of their geometrical parameters. Clamped edge laminates exhibit higher buckling mode shapes, influenced by aspect ratio and edge restraint. Microscopic analysis of CNTRC's fractured surface under compression shows enhanced buckling strength attributed to CNTs’ bridging and pullout effects in the matrix, suppressing crack initiation and propagation. Field Emission Scanning Electronic Microscope (FESEM) is utilized for this investigation.

Influence of geometric imperfections on the seismic performance of unanchored liquid storage tanks

This paper investigates the inelastic buckling of imperfect unanchored cylindrical steel tanks subjected to seismic loads. A pushover-based seismic analysis was conducted considering the impulsive hydrodynamics pressures on the wall and base plate of the tank. Material and geometric nonlinearities were considered in the seismic analysis of the tanks. Three types of imperfections were analyzed: imperfections due to impacts, imperfections located at the bottom of the tank wall, and imperfections with the first buckling mode shape. The buckling analysis was performed for two tank geometries (one slender and one broad), and the effect of the imperfections was correspondingly evaluated. The considered imperfection amplitudes were established according to the New Zealand seismic design code for storage tanks. Additionally, amplitudes that exceed the normative limit were evaluated to further analyze the sensitivity to imperfection. The analysis revealed that geometric imperfections reduce the peak ground acceleration needed to induce buckling failure. Specifically, the critical peak ground acceleration for the slender tank decreased from 0.195 g to 0.180 g for the slender tank and from 0.600 g to 0.535 g for the broad tank. This buckling capacity reduction due to geometric imperfections were about 8 % and 11 % for the slender and the broad tanks, respectively.

Some Unfamiliar Structural Stability Aspects of Unsymmetric Laminated Composite Plates.

It is widely recognized that certain structures, when subjected to static compression, may exhibit a bifurcation point, leading to the potential occurrence of a secondary equilibrium path. Also, there is a tendency of deflection increment without a bifurcation point to occur for imperfect structures. In this paper, some relatively unknown phenomena are investigated. First, it is demonstrated that in some conditions, the linear buckling mode shape may differ from the result of geometrically nonlinear analysis. Second, a mode jumping phenomenon is described as a transition from a secondary equilibrium path to an obscure one as a tertiary equilibrium path or a second bifurcation point. In this regard, some non-square plates with unsymmetric layer arrangements (in the presence of extension-bending coupling) are subjected to a uniaxial in-plane compression. By considering the geometrically linear and nonlinear problems, the bucking modes and post-buckling behaviors, e.g., the out-of-plane displacement of the plate versus the load, are obtained by ANSYS 2023 R1 software. Through a parametric analysis, the possibility of these phenomena is investigated in detail.

An experimental parametric study assessing the effect of a certain initial geometric imperfection on cylinder buckling

Given the sensitivity of axially-loaded cylinder buckling to geometric imperfections, the advent of additive manufacturing provides a compelling platform to assess subtle changes in initial configuration from an experimental perspective. The cylindrical form is ubiquitous in many load-bearing contexts, from soda cans to submarines to rocket fuel tanks. Despite the relative simplicity of the shape, it is striking how the assessment of buckling loads continues to pose strong challenges to the designers of these load-bearing components, often in a situation where weight-saving is a key constraint. Furthermore, this type of buckling typically corresponds to a sudden, severe, catastrophic event, and in addition to posing serious consequences in practice, this also presents challenges for those conducting high-fidelity experiments, especially in terms of repeatability. Any conditions that deviate from a pristine cylindrical shape, and purely axial loading, are the root-cause of the buckling variability. Typically, the slight deviations from the pristine (perfect) conditions are somewhat random and unpredictable in nature. 3D-printing now provides the experimentalist with the ability to produce cylindrical specimens to a very high degree of geometric fidelity, thus allowing a degree of control of the form of subtle geometric imperfections. This paper focuses attention on a specific form of deviation from a pristine cylindrical shape, in which the longitudinal sides follow a mild cosine wave form. The amplitude of this shape is varied (in both directions, i.e., inwards and outwards), and its effect on both the buckling load and buckling mode shape assessed experimentally, using the outcome of finite element analysis as a guide.

Buckling mode constraints for topology optimization using eigenvector aggregates

Buckling-constrained structural design problems have conventionally prioritized optimizing the buckling load factor with less consideration given to the buckling mode shape. In this work, mode shape constraints are imposed within a topology optimization problem using an eigenvector aggregate constraint that is a weighted sum of homogeneous quadratic functions of the linearized buckling eigenvectors. A generalized formulation of the eigenvector aggregate is introduced, extending previous work. A new adjoint-based derivative evaluation technique is derived that is valid even in the presence of repeated eigenvalues. Numerical examples, including a clamped beam, a compressed column, and a square plate, demonstrate the effectiveness of the proposed approach. The results show the ability of the eigenvector aggregate to handle repeated eigenvalues, enable design space exploration, and capture mode shape switching.

A size-dependent electro-mechanical buckling analysis of flexoelectric cylindrical nanoshells

In this paper, a buckling analysis for flexoelectric cylindrical nanoshells under axial compression is performed by considering the higher-order shear deformation theory and non-uniform pre-buckling deformation. Size-dependent critical buckling stresses and buckling mode shapes are obtained by the Galerkin's method with newly proposed displacement functions. Numerical results are compared with existing solutions and excellent agreements are observed. Furthermore, a comprehensive parametric study of boundary conditions, geometrical parameters and applied electric voltage is carried out to reveal the influence of flexoelectric effect on the size-dependent buckling characteristics of flexoelectric cylindrical nanoshells.

Optimization-free design of stiffened thin-walled structures guided by data-rich buckling modes

Regarding the buckling resistance design of thin-walled structures, the conventional iterative optimization approaches pose significant challenges for engineering professionals and create a high barrier for structural designers. This study introduces a novel approach for the anti-buckling design of stiffening ribs, which is based on the solved linear buckling modes rather than on the buckling load factors (BLFs) as in traditional iterative optimization methods. Specifically, the design process begins with a linear buckling analysis of the pre-stiffened part to extract the first-order eigenmode shape. Subsequently, normal vectors are calculated for both the pre-buckling geometric model and the buckled mode shape. By analyzing the normal deflection angle, a group of characteristic points is identified, and these points would be used to serve as guidelines for determining the layout of the stiffening ribs. We validate the effectiveness and efficiency of the proposed protocol through numerical examples spanning from 2D to 3D space considering different loading conditions. Our approach eliminates the need for tedious parametric modeling and high computational cost associated with conventional optimization-based methods. It is particularly well-suited for engineering applications that solely require linear buckling analysis.

DFT-based finite element analysis of compressive response in armchair phosphorene nanotubes

In this paper, the finite element method is utilized to evaluate the behavior of the armchair phosphorene nanotubes under the compressive loading. The energy equations of the molecular and structural mechanics are used to obtain the elemental properties. The critical compressive forces of various armchair phosphorene nanotubes are computed with clamped-clamped and clamped-free boundary conditions. Results show that the stability of armchair phosphorene nanotubes increases with increasing nanotube aspect ratio, particularly under clamped-clamped boundary conditions. Finally, the buckling mode shapes of armchair phosphorene nanotubes under different boundary conditions are compared. Our work offers valuable insights into how these nanotubes respond to mechanical stress, helps determine elemental properties, and investigates the effects of nanotube geometry and different boundary conditions on their stability. This knowledge has broad applications in fields like nanotechnology, materials science, and nanomechanics, advancing the understanding of nanoscale materials and their potential for various practical uses.

Parametric study on the assessment of the local buckling behavior of perforated square hollow sections with non-uniform wall thickness under axial compression

The aim of this rigorous parametric study is to explore the influence of perforations on the local buckling behavior of square hollow sections (SHSs) possessing non-uniform wall thickness. A finite element procedure followed in the current study has been first validated against existing test results documented for the local buckling behavior of the perforated SHS with uniform web and flange segment thickness under axial compression. The linear elastic eigenvalue buckling and elastoplastic buckling analyses have been implemented using the Abaqus engineering finite element code. The verification of the numerical procedure has been achieved by favorably comparing the finite element results with the existing test results in terms of the first local buckling mode shape and load-end shortening curves of the perforated SHS with uniform wall thickness. . The verified numerical procedure has been applied to the problem of finding the perforation effect on the local buckling response of the SHS with non-uniform thickness. Finite element analyses have been performed for four various web width-to-perforation diameter ratios ranging from 0.3 to 0.9. Finite element analysis results have revealed that the presence of perforations does not influence the local buckling mode shape of the SHS but considerably affects the critical local buckling loads. The results have put forth that increasing perforation diameter leads to a more pronounced and drastic decrease in the critical local buckling load. The outcomes of the study have also shown that the critical post-buckling load of the SHS with non-uniform wall thickness is less susceptible to perforations compared to the SHS with uniform wall thickness. The results obtained in the context of this parametric study have been made available to practical engineering for use in actual design of the perforated SHSs.

Evaluation of the Applicability of Digital Photogrammetry-Based Initial Imperfections on NLFEM Ultimate Strength Analysis of Ship- Type Stiffened Plates

In this study, the effects of initial imperfections on the ultimate strength values of ship-type stiffened plate structures are discussed and analysed from the point of view of how the initial imperfection forms are obtained. Ship stiffened panels are subjected to complex loading conditions during their operational lifespan. Accurate prediction of their strength and failure modes necessitates a thorough understanding of the effects of various imperfections on their behavior. Two primary sources of initial imperfections are considered: the buckling mode shapes resulting from linear eigenvalue buckling analysis and measurements based on digital photogrammetry. Buckling mode shapes, arising from the manufacturing process are extracted using linear static structural analysis. Digital photogrammetry is employed to capture and quantify imperfections by analysing high-resolution images of the physical structure. Numerical investigation is conducted by incorporating these two types of initial imperfections into non-linear finite element method (NLFEM) calculations. The buckling mode imperfections are applied as geometric perturbations, while the photogrammetry-based imperfections are incorporated as statistically representative deviations from the ideal geometry. Stiffened panel's structural response is analysed under longitudinal uniaxial compression. A new NLFEM project schematic has been utilized instead of the procedure outlined in the technical circular S.P 01/19 by Türk Loydu, which has been discontinued after ANSYS® Workbench™V.2019R2. This is proven by two validation case studies in ANSYS® Workbench™2022R2 version for considering buckling mode initial imperfections. A case study is then conducted using a 3D model of a stiffened plate panel, which is fabricated in a shipyard located in Trabzon, created by Photomodeler V. 2023.3.0.238 employing the digital photogrammetry method.NLFEM analysis is carried out for both initially deflected model after eigenvalue buckling analysis and naturally deflected model after welding operations. The comparative ultimate strength results are quite consistent, and this shows that the digital photogrammetric modelling method can be used in the analysis of ship structural elements.

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.

Conical shells made by 4D printing of composites

This paper presents a new, novel method for the manufacturing of composite cones. The method utilizes the principle of 4D printing of composites. The method does not require the use of a mold of conical shape, which is essential using conventional manufacturing methods such as filament winding or hand lay up. Instead, only a flat mold is required. In this technique, layers with different fiber orientations are laid on a flat mandrel. Upon curing and cooling to room temperature, the flat stack of composite layers will curl into the shape of a conical shell. When the edges of the shell are joined together by adhesive bonding, one has a structure that has the shape of a cone. Cones made using this new technique show axial buckling strengths that are comparable to those of cones made from similar materials with the same weight, using filament winding method. The buckling load and buckling mode shape are determined using finite element method. The results agree well with experimental results.

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

Elastic local buckling coefficients of I-shaped beams considering flange–web interaction

Since using high–strength and high–performance steels has become a more common structural design practice in building and bridge construction, there is an increased potential for designing extremely thin-walled I-shaped beam members to enhance the efficiency of the steel beam design. For a more accurate design approach for these thin-walled I-beam members, particularly when they are near noncompact limits, it is necessary to explore more rational methods for determining the elastic local buckling strength of I-beams. This study aimed to investigate the local buckling behavior and strength of I-shape structural sections by considering flange-web interactions through three-dimensional finite element analysis. The goal was to provide a more reasonable estimation of local buckling strength under uniform bending. To evaluate the local buckling behavior of flange and web panels and explain it reasonably, this study adopted the ratio of flange-web slenderness (λf/λw) and height-to-width ratio (H/bf) of I-shaped beams which can affect buckling mode shapes and local buckling strength of I-shaped beams induced by flange local and web bend bucklings. Finally, this study presented design equations for both flange local and web-bend buckling coefficients considering λf/λw and H/bf. It was expected that the presented local buckling coefficients (kf) for flanges and webs could lead to a more reasonable design, as compared to the existing AISC design provisions.

An Analytical Thermal Buckling Model for Semiconductor Chips on a Substrate.

Semiconductor chips on a substrate have a wide range of applications in electronic devices. However, environmental temperature changes may cause mechanical buckling of the chips, resulting in an urgent demand to develop analytical models to study this issue with high efficiency and accuracy such that safety designs can be sought. In this paper, the thermal buckling of chips on a substrate is considered as that of plates on a Winkler elastic foundation and is studied by the symplectic superposition method (SSM) within the symplectic space-based Hamiltonian system. The solution procedure starts by converting the original problem into two subproblems, which are solved by using the separation of variables and the symplectic eigenvector expansion. Through the equivalence between the original problem and the superposition of subproblems, the final analytical thermal buckling solutions are obtained. The SSM does not require any assumptions of solution forms, which is a distinctive advantage compared with traditional analytical methods. Comprehensive numerical results by the SSM for both buckling temperatures and mode shapes are presented and are well validated through comparison with those using the finite element method. With the solutions obtained, the effects of the moduli of elastic foundations and geometric parameters on critical buckling temperatures and buckling mode shapes are investigated.

In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation

Abstract In this study, four absolute nodal coordinate formulation (ANCF)-based approaches are utilized in order to predict the buckling load of Lee’s frame under concentrated load. The first approach employs the standard two-dimensional shear deformable ANCF beam element based on the general continuum mechanics (GCM). The second approach adopts the standard ANCF beam element modified by the locking alleviation technique known as the strain-split method. The third approach has the standard ANCF beam element with strain energy modified by the enhanced continuum mechanics formulation. The fourth approach utilizes the higher-order ANCF beam element based on the GCM. Two buckling load estimation methods are used, i.e., by tracing the nonlinear equilibrium path of the load–displacement space using the arc-length method and applying the energy criterion, which requires tracking eigenvalues through the dichotomy scheme. Lee’s frame with different boundary conditions including pinned–pinned, fixed-pinned, pinned-fixed, and fixed–fixed are studied. The complex nonlinear responses in the form of snap-through, snap-back, and looping phenomena during nonlinear postbuckling analysis are simulated. The critical buckling loads and buckling mode shapes obtained through the energy criterion-based buckling method are obtained. After the comparison, higher-order beam element is found to be more accurate, stable, and consistent among the studied approaches.

Effect of porosity and viscoelastic boundary conditions on FG sandwich beams in thermal environment: Buckling and vibration studies

The present study is carried out to investigate the combined effect of porosity and temperature on the buckling and vibration attributes of FG sandwich beams in the thermal environment using FE formulation. The modeled sandwich beam consisting of the viscoelastic core material is restrained by viscoelastic boundary conditions (VBCs). The FG face layers and core are subjected to temperature-dependent material properties. Complex stiffness model is adopted for VBCs. Porosity patterns such as H, V, X, and O are incorporated into FG face layers. The Lagrange equation is used to derive the sandwich beam’s equilibrium equations of motion in static and dynamic conditions. The derived equilibrium equations are solved for buckling and vibration of the beam using the FE solution. Lagrange and Hermite shape functions are assumed for axial and transverse displacements. Critical buckling temperature (CBT), natural frequency (NF), and loss factors (LF) are obtained for various temperatures and boundary stiffness values (BSVs). Transverse buckling and vibration mode shapes are extracted for changing BSVs. The behavior of NF and LF at buckling temperature is also discussed. The existence of porosities ameliorates the buckling characteristics of the sandwich beam. VBCs expedite the vibration damping of sandwich beams alongside the viscoelastic core. The natural frequency and loss factor reach zero and infinity, respectively, when the temperature reaches CBT.

Buckling responses and instability mode landscapes of composite sandwich panels with extreme auxeticity based on higher-order shear and normal deformation theory

Sandwich structures have been a subject of intense scrutiny and utilization in engineering applications. However, there has been a limited understanding of the mechanical behaviors of novel sandwich structures composed of auxetic metamaterials. This study aims to contribute to the body of knowledge by examining the buckling responses of sandwich beams or wide plates with an orthotropic anti-tetra chiral lattice (ATCL) aluminum cores and symmetric angle-ply carbon-fiber/epoxy face sheets, both of which can exhibit highly negative Poisson’s ratios or extreme auxeticity. The linear stability analysis of the sandwich structures under axial compression is conducted by applying the minimum potential energy principle and solving the eigenvalue problem using a numerical method. To achieve an accurate representation of arbitrary two-dimensional buckling morphologies, the first- and fifth-order shear and normal deformation models are employed to simulate the transverse kinematics of the face sheets and cores, while the finite-element model is utilized to discretize the longitudinal displacement of all constituents. The investigation reveals new insights into the interrelationship between global buckling, facial wrinkling, and shear crimping in conventional sandwich structures. The effects of material auxeticity on buckling responses are analyzed across an extensive range of geometrical configurations for the first time, unveiling five unprecedented buckling mode shapes, sophisticated buckling mode landscapes, and Poisson’s ratio-sensitive buckling criteria. These findings lay a crucial foundation for designing and optimizing auxetic sandwich structures, bringing diverse possibilities in lightweight constructions, acoustic insulation, energy absorption, and dynamic and shock mitigation.

Thermal Postbuckling Characteristics Of Angle-ply Laminated Truncated Circular Conical Shells

Here, the nonlinear thermoelastic pre- and post-buckling characteristics of angle-ply laminated composite conical shells subjected to uniform temperature rise are studied using semi-analytical finite element approach. The finite element formulation is based on first-order shear deformation theory and field consistency principle. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton-Raphson iteration procedure coupled with displace- ment control method to trace the prebuckling followed by postbuckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, ply-angles, number of layers and number of circumferential waves on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the deformation shape analysis.

Buckling optimization of curved grid-stiffened helicoidal composite panel including the factor of mode shape in multiple load cases

The buckling mode of helicoidal composite panel are non-axisymmetric, resulting its different capability to resist loads in different orientation and the irregular buckling mode after curvilinearly stiffened. A method was proposed to design asymmetric curved stiffeners considering the local displacement gradient of the buckling mode shape. The displacement gradient depends on the bending stiffness and yields a better optimization. The basic unit of the stiffener family includes two stiffeners that pass through the midpoint of the panel; the remainder can be formed from this unit via translation. A simply supported curved grid-stiffened panel using uniaxial compression or shear loads under multiple load cases was addressed, and an integrated design framework was developed to obtain the designs. The curved grid-stiffened helicoidal composite panel yields a weight reduction of up to 13.26% compared to angle-grid designs under uniaxial compression and 11.68% for shear loads in multiple load cases.