Buckling Of Structures Research Articles

Shape optimization of reticulated shells with constraints on member instabilities

ABSTRACTReticulated shell structures are commonly adopted because of their large spanning capacity and novel appearance. These structures generally comprise slender beam members, which may cause the buckling of entire structures. A shape-finding method is proposed in this study to eliminate the bending moment while optimizing the member buckling capacity of reticulated shell structures. The bending stiffness of connections can be accurately considered in the topology optimization analysis. This study can systematically be adopted for designing, analysing and optimizing such structures. All of the work presented was conducted using a general finite element analysis program and can be easily mastered by engineering designers while avoiding tedious programming work.

Axial stress determination using highly nonlinear solitary waves.

This article presents a nondestructive evaluation (NDE) method to infer the neutral temperature and the axial stress in thick beams. The method relies on the propagation of highly nonlinear solitary waves generated at one end of a chain of spherical particles in a dry point contact with the beam to be evaluated. The waves are reflected back to the chain and the research hypothesis is that the axial stress, which influences the beam's stiffness, affects the amplitude and speed of the reflected waves. To verify this hypothesis a general finite element model of thermally stressed beams was developed and coupled to a discrete particle model able to predict the propagation of the waves along an L-shaped granular medium. The models were validated experimentally to quantify the repeatability of the setup, the sensitivity of the wave features on the thermal stress, and the independence of the wave features on the neutral temperature of the beam. The hypothesis was proven valid by both the numerical and the experimental results. In the future, these findings may be used to refine a NDE method to assess stress in columns, to infer the neutral temperature of continuous welded rails, and to prevent thermal buckling of critical structures.

MWCNT/epoxy strip-like sensors for buckling detection in beam-like structures

Buckling of slender structures constitutes a hazardous failure mechanism that can yield partial or total collapses. Nonetheless, given that buckling failure is characterized by a highly non-linear and sudden loss of stability, most off-the-shelf monitoring systems fail to detect buckling and very few research works in the literature can be found in this regard. Recent advances in the field of Nanotechnology have fostered the development of innovative composite materials with multifunctional properties, offering vast possibilities in the field of Structural Health Monitoring. Along these lines, the present work proposes a novel concept of smart beams for buckling detection applications. This consists of the deployment of carbon nanotube-reinforced epoxy strip-like sensors on the upper and bottom faces of a beam-like structure. Carbon nanotube-reinforced composites exhibit strain self-sensing capabilities, that is to say, these composites provide measurable variations in their electrical properties under the action of mechanical strains. In this way, the proposed sensing strips not only act as mechanical reinforcements, but also confer self-diagnostic properties to the system. The failure detection principle of the proposed smart beams consists of the assessment of the bending-induced variations of the normal strains during buckling. To do so, the electrical resistance of the sensing strips is continuously monitored through a two-probe resistivity measurement scheme. The present research furnishes detailed numerical parametric analyses to investigate the effectiveness of the proposed smart beams to detect buckling under uniaxially compression, as well as to evaluate the influence of design parameters such as filler volume fraction, boundary conditions and electrodes layouts. The macroscopic behaviour of the smart beams is simulated by a micromechanics-based piezoresistivity model and a multiphysics finite element code. The numerical results demonstrate that the buckling failure can be tracked through sudden disturbances in the electrical output of the smart strips.

Mechanically active materials in three-dimensional mesostructures.

Complex, three-dimensional (3D) mesostructures that incorporate advanced, mechanically active materials are of broad, growing interest for their potential use in many emerging systems. The technology implications range from precision-sensing microelectromechanical systems, to tissue scaffolds that exploit the principles of mechanobiology, to mechanical energy harvesters that support broad bandwidth operation. The work presented here introduces strategies in guided assembly and heterogeneous materials integration as routes to complex, 3D microscale mechanical frameworks that incorporate multiple, independently addressable piezoelectric thin-film actuators for vibratory excitation and precise control. The approach combines transfer printing as a scheme for materials integration with structural buckling as a means for 2D-to-3D geometric transformation, for designs that range from simple, symmetric layouts to complex, hierarchical configurations, on planar or curvilinear surfaces. Systematic experimental and computational studies reveal the underlying characteristics and capabilities, including selective excitation of targeted vibrational modes for simultaneous measurements of viscosity and density of surrounding fluids. The results serve as the foundations for unusual classes of mechanically active 3D mesostructures with unique functions relevant to biosensing, mechanobiology, energy harvesting, and others.

New experimental investigation of non-conventional dynamic biaxial plastic buckling of square aluminum tubular structures

Motivated by the biaxial plastic dynamic buckling behavior of tubular structures, the particular interest of this work was to develop a non-conventional energy-dissipating system. This system is based on the studying of the experimental behavior of square metallic tubular structures under dynamic complex loading paths. This non-conventional solution aims to enhance the strength properties of the used structures. Accordingly, the well-developed device (the ACTP: Absorption par Compression-Torsion Plastique) has been employed where biaxial loading paths (combined compression-torsion) with different complexities were generated from a uniaxial compressive load. Thus, five inclination angles (30°, 37°, 45°, 53° and 60°) were used to study their effect on the biaxial plastic buckling of square aluminum tubes. This new comprehensive experimental study aimed to investigate the effect of three loading boundary conditions (i.e., simple, intermediate and severe) on the tubes’ behavior.The dynamic deformation, mean collapse load (Pm) and energy absorbed are key parameters that require investigation. They are largely affected by the loading path complexity, applied strain rate and their interaction. The results showed that the higher the biaxial loading complexity provided by the ACTP, the greater the interaction between the loading complexity and the strain rate, and the greater the energy absorbed by the structures. Changes in local deformation mechanisms induced by the loading complexity thus lead to the energy absorbed enhancement. The latter was higher than 100%, in favor of the most complicated loading path (i.e., Biaxial-60°) compared to the classical uniaxial case for an axial deflection of 20 mm. Simple mathematical equations were proposed predicting the Pm of buckled tubes. These equations can describe the tubes’ behavior fairly well.

Investigation of the Dynamic Buckling of Spherical Shell Structures Due to Subsea Collisions

This paper is the first to present the dynamic buckling behavior of spherical shell structures colliding with an obstacle block under the sea. The effect of deep water has been considered as a uniform external pressure by simplifying the effect of fluid–structure interaction. The calibrated numerical simulations were carried out via the explicit finite element package LS-DYNA using different parameters, including thickness, elastic modulus, external pressure, added mass, and velocity. The closed-form analytical formula of the static buckling criteria, including point load and external pressure, has been firstly established and verified. In addition, unprecedented parametric analyses of collision show that the dynamic buckling force (peak force), mean force, and dynamic force redistribution (skewness) during collisions are proportional to the velocity, thickness, elastic modulus, and added mass of the spherical shell structure. These linear relationships are independent of other parameters. Furthermore, it can be found that the max force during the collision is about 2.1 times that of the static buckling force calculated from the analytical formula. These novel insights can help structural engineers and designers determine whether buckling will happen in the application of submarines, subsea exploration, underwater domes, etc.

Nonlinear vibrations and instability of a bistable shallow reticulated truss

In recent years, a growing interest has been observed on the static and dynamic behavior of multi-stable structures. In this paper, a shallow bistable regular pyramidal truss composed of n equally spaced bars is investigated, allowing a deeper understanding of the static and dynamic buckling of several shallow structures, from carbon nanostructures to large geodesic domes. First an exact formulation, considering large displacements, is presented and the nonlinear equations of motion for a nodal load acting along a generic oblique direction are derived. Based on this formulation, the parametric buckling analysis of the truss under vertical static load as a function of the shallowness parameter is conducted and the equations of motion of the preloaded structure are derived. Then, the fundamental vibration frequency and nonlinear frequency–amplitude relations are obtained as a function of the static preload. The structure exhibits a two-well potential function whose geometry controls the dynamics of the preloaded structure. A detailed parametric analysis of the preloaded truss under harmonic excitation is then conducted. Emphasis is placed on the influence of the load and geometric parameters on the coexisting stable and unstable dynamic responses associated with the two-well potential function. Detailed bifurcations analyses using continuation techniques show the existence of several in-well and cross-well stable and unstable coexisting solutions. Also they permit the identification of the stability boundaries under different loading conditions. These coexisting solutions lead to topologically involved portraits of the attractor-basin response. Thus, an analysis of the evolution and erosion of the basins of attraction is performed. In a dynamic environment excitation, noise has a palpable influence on the system response. So, in the last part of the paper the influence of random noise on the topology of the coexisting basins of attraction of the system is investigated. Notwithstanding the complex nonlinear dynamics of this class of bistable structures, the analysis of the truss global dynamics may provide hints for safe structural design and novel applications of multi-stable structures.

Buckling analysis of parallel eccentrically stiffened functionally graded annular spherical segments subjected to mechanic loads

ABSTRACTBased on the Donnell shell theory and smeared stiffeners technique, the governing equations of functionally graded annular spherical segments with functionally graded eccentrically stiffeners are derived to investigate the buckling of these structures under compression load and radial pressure. The segment is reinforced by parallel eccentrically functionally graded stiffener system. Approximate solutions are assumed to satisfy the simply supported boundary condition and Galerkin method is applied to obtain closed-form relations of linear buckling loads. Numerical results are given to evaluate effects of stiffeners, volume fraction index and dimensional parameters to the buckling of structure.

Axial buckling behavior of single-walled carbon nanotubes: Atomistic structural instability analysis

To reveal the atomistic scale mechanism of buckling of single-walled carbon nanotubes (SWCNTs) under axial compression, we carried out molecular dynamics (MD) simulation and atomistic structural instability (ASI) analysis. The ASI analysis enables us to reveal the development of ‘latent’ instability modes until buckling of structure, which cannot be clarified with MD alone. Euler-type buckling was found in relatively thin and long nanotube models, while buckling triggered by change of cross-sectional shape (i.e., radial buckling) was found in thick and short models. With growing radius, a crossover between the Euler-type and radial buckling modes was clearly found in the ASI analysis. Even when the deformation behavior was apparently the same, the buckling can be triggered by different instability modes. We also analyzed the structural instability using the Flügge theory based on linear continuum mechanics (LCM) to compare the results with the ASI analysis. Initial eigenvalues and development of instability modes were different between the ASI and LCM results, leading to a significant deviation of buckling strain and instability crossover points. We investigated the effect of chirality of CNT which is difficult to explain by LCM.

Controlled mechanical assembly of complex 3D mesostructures and strain sensors by tensile buckling

Recent research establishes methods of controlled mechanical assembly as versatile routes to three-dimensional (3D) mesostructures from patterned 2D films, with demonstrated applicability to a broad range of materials (e.g., semiconductors, polymers, metals, and their combinations) and length scales (e.g., from sub-microscale to centimeter scale). Previously reported schemes use pre-stretched elastomeric substrates as assembly platforms to induce compressive buckling of 2D precursor structures, thereby enabling their controlled transformation into 3D architectures. Here, we introduce tensile buckling as a different, complementary strategy that bypasses the need for a pre-stretched platform, thereby simplifying the assembly process and opening routes to additional classes of 3D geometries unobtainable with compressive buckling. A few basic principles in mechanics serve as guidelines for the design of 2D precursor structures that achieve large out-of-plane motions and associated 3D transformations due to tensile buckling. Experimental and computational studies of nearly 20 examples demonstrate the utility of this approach in the assembly of complex 3D mesostructures with characteristic dimensions from micron to millimeter scales. The results also establish the use of nonlinear mechanics modeling as a mechanism for designing systems that yield desired 3D geometries. A strain sensor that offers visible readout and large detectable strain range through a collection of mechanically triggered electrical switches and LEDs serves as an application example.

A multifunctional bistable laminate: Snap-through morphing enabled by broadband energy harvesting

The elastic instabilities associated with buckling in bistable structures have been harnessed toward energy-based and motion-based applications, with significant research toward energy harvesting and morphing. Often combined with smart materials, structural prototypes are designed with a single application in mind. Recently, a novel method of inducing bistability was proposed by bonding two piezoelectrically actuated macro fiber composites in a [Formula: see text] layup and releasing the voltage post cure to yield two cylindrically stable configurations. Since the macro fiber composites are simultaneously the actuator and host structure, the resulting efficiencies enable this bistable laminate to be multifunctional, with both broadband energy harvesting and snap-through morphing capabilities. This article experimentally characterizes the vibration-based energy harvesting performance of the laminate to enable morphing. Through frequency sweeps across the first two modes of both states, the laminate exhibits broadband cross-well dynamics that are exploited for improved power generation over linear resonant harvesters. Besides single-well oscillations, snap-throughs are observed in intermittencies and subharmonic, chaotic, and limit cycle oscillations. The maximum power output of each regime and their charge durations of an energy harvesting module are assessed. The laminate’s capabilities are then bridged by utilizing harvested energy in the charged module to initiate snap-through actuation.

Modal nudging in nonlinear elasticity: Tailoring the elastic post-buckling behaviour of engineering structures

The buckling and post-buckling behaviour of slender structures is increasingly being harnessed for smart functionalities. Equally, the post-buckling regime of many traditional engineering structures is not being used for design and may therefore harbour latent load-bearing capacity for further structural efficiency. Both applications can benefit from a robust means of modifying and controlling the post-buckling behaviour for a specific purpose. To this end, we introduce a structural design paradigm termed modal nudging, which can be used to tailor the post-buckling response of slender engineering structures without any significant increase in mass. Modal nudging uses deformation modes of stable post-buckled equilibria to perturb the undeformed baseline geometry of the structure imperceptibly, thereby favouring the seeded post-buckling response over potential alternatives. The benefits of this technique are enhanced control over the post-buckling behaviour, such as modal differentiation for smart structures that use snap-buckling for shape adaptation, or alternatively, increased load-carrying capacity, increased compliance or a shift from imperfection sensitivity to imperfection insensitivity. Although these concepts are, in theory, of general applicability, we concentrate here on planar frame structures analysed using the nonlinear finite element method and numerical continuation procedures. Using these computational techniques, we show that planar frame structures may exhibit isolated regions of stable equilibria in otherwise unstable post-buckling regimes, or indeed stable equilibria entirely disconnected from the natural structural response. In both cases, the load-carrying capacity of these isolated stable equilibria is greater than the natural structural response of the frames. Using the concept of modal nudging it is possible to “nudge” the frames onto these equilibrium paths of greater load-carrying capacity. Due to the scale invariance of modal nudging, these findings may impact the design of structures from the micro- to the macro-scale.

Critical Buckling of Prestress-Stable Tensegrity Structures Solved by Real-Coded Genetic Algorithm

Tensegrity structures are classified as kinematically determinate ones with two subcases and kinematically indeterminate ones with three subcases in view of their respective stability properties. How the stiffness of a tensegrity structure changes as the level of prestress changes is explored for different scenarios using six carefully chosen samples. For a tensegrity structure merely satisfying the prestress-stability condition, a new optimization model is presented to determine its critical buckling state corresponding to zero stiffness. A real-coded genetic algorithm (RCGA) is then developed to solve this problem, featured by the fact that a special sign-control technique is embedded in the fundamental genetic operations, making the individuals generated in each step fall into the admissible region automatically. The stability of the neutral equilibrium state for tensegrity structures violating the prestress-stability condition is also discussed. Several numerical examples are tested to validate the efficiency of the present approach.

Mechanical performance of wall structures in 3D printing processes: Theory, design tools and experiments

In the current contribution for the first time a mechanistic model is presented that can be used for analysing and optimising the mechanical performance of straight wall structures in 3D printing processes. The two failure mechanisms considered are elastic buckling and plastic collapse. The model incorporates the most relevant process parameters, which are the printing velocity, the curing characteristics of the printing material, the geometrical features of the printed object, the heterogeneous strength and stiffness properties, the presence of imperfections, and the non-uniform dead weight loading. The sensitivity to elastic buckling and plastic collapse is first explored for three basic configurations, namely i) a free wall, ii) a simply-supported wall and iii) a fully-clamped wall, which are printed under linear or exponentially-decaying curing processes. As demonstrated for the specific case of a rectangular wall lay-out, the design graphs and failure mechanism maps constructed for these basic configurations provide a convenient practical tool for analysing arbitrary wall structures under a broad range of possible printing process parameters. Here, the simply-supported wall results in a lower bound for the wall buckling length, corresponding to global buckling of the complete wall structure, while the fully-clamped wall gives an upper bound, reflecting local buckling of an individual wall. The range of critical buckling lengths defined by these bounds may be further narrowed by the critical wall length for plastic collapse. For an arbitrary wall configuration the critical buckling length and corresponding buckling mode can be accurately predicted by deriving an expression for the non-uniform rotational stiffness provided by the support structure of a buckling wall. This has been elaborated for the specific case of a wall structure characterised by a rectangular lay-out. It is further shown that under the presence of imperfections the buckling response at growing deflection correctly asymptotes towards the bifurcation buckling length of an ideally straight wall. The buckling responses computed for a free wall and a wall structure with a rectangular lay-out turn out to be in good agreement with experimental results of 3D printed concrete wall structures. Hence, the model can be applied to systematically explore the influence of individual printing process parameters on the mechanical performance of particular wall structures, which should lead to clear directions for the optimisation on printing time and material usage. The model may be further utilised as a validation tool for finite element models of wall structures printed under specific process conditions.

Investigation of natural frequencies of axially loaded thin-walled columns

The paper deals with correlation between natural frequencies of two steel thin-walled columns and the corresponding applied load. The structures are made of cold-formed lipped channel sections. The columns lengths were assumed to follow two buckling patterns – global flexural and flexural-torsional buckling. In the thicker structure two material models were considered – linearly-elastic and elastic-perfectly plastic. Numerical computations cover dynamic eigenvalue problem, linear buckling and geometrically (and materially) non-linear analysis. The correlation between squares of natural frequencies and the applied load is linear in both columns. The first natural frequencies drop to zero due to structural buckling. This method, called the Vibration Correlation Technique, allows to predict buckling loads on the basis of measured vibration frequencies of the structures. Plasticity does not affect the corresponding curves – the use of the presented technique is limited to the structures exhibiting elastic buckling behaviour.

How tall can gelatin towers be? An introduction to elasticity and buckling

The stability of elastic towers is studied through simple hands-on experiments. Using gelatin-based stackable bricks, one can investigate the maximum height a simple structure can reach before collapsing. We show through experiments and by using the classical linear elastic theory that the main limitation to the height of such towers is the buckling of the elastic structures under their own weight. Moreover, the design and architecture of the towers can be optimized to greatly improve their resistance to self-buckling. To this aim, the maximum height of hollow and tapered towers is investigated. The experimental and theoretical developments presented in this paper can help students grasp the fundamental concepts in elasticity and mechanical stability.

Equivalent unique global and local initial imperfection - Imperfection in EN 1993-1-1 and EN 1999-1-1 Clause 5.3.2 (11) - calculation procedure and discovered obstacles

Equivalent unique global and local initial imperfection is introduced in standards EN 1993-1-1, EN 1999-1-1 in clause 5.3.2 (11) and in Slovak national annex to EN 1993-1-1 NB. 5. However approach described in these standards needs further explanation to fully understand its background to reduce possibility of causing errors. Equivalent unique global and local initial imperfection and derived equivalent unique global and local initial imperfection method is based on obtaining amplitude for structural buckling mode, which can be than used as full-sized imperfection in assessment of structures loaded by combination of axial compression forces and bending moments. Equivalent unique global and local initial imperfection was firstly used and introduced by prof. Eugen Chladný from Slovak University of Technology in Bratislava. The origin of this method was based on the need of assessment of upper chords of open-deck truss bridges. The main idea is described in detail by Prof. Eugen Chladný and Magdaléna Štujberová in paper in magazine Stahlbau vol. 82. Equivalent unique global and local initial imperfection method in mentioned standards and paper is designated for plane structures like simple structural members or frame structures. This paper examines in plane behavior of structures with presented imperfection and calculation procedure, which allows fast examination of many different types of plane structures.

Penghitungan Beban Kritis pada Struktur Balok-Kolom Akibat Pembebanan Secara Aksial dan Lateral Berbasiskan Metode Beda Hingga

This paper is aimed to determine the critical buckling load of the beam-column structures for practical applications. Firstly, the critical buckling loads is calculated numerically based on finite difference method. For geometrical modeling, the structures are supported in three types of supports; pinned-roller, fixed-free, and fixed-pinned. Then, a geometry of the beam is varied in variation of slenderness ratio. The load is given in types of uniform distribution load, q, laterally and a concentrated load, P, axially. Using a computational program, the relationships of P/q to slendernes ratio, l, for all models are determined. The results then are approached by regression technique of a least square method in order to find a the best fit of equation. This equation is recommended as a practical equation for determining the critical buckling of the beam-column structures under variation of types of supports, slenderness ratio and rasio P/q.

The effect of eccentricity of load on the behavior of compressed composite columns in critical state

The study concerns a short thin‐walled channel section column made of carbon‐epoxy laminate. The tested column was subjected to compression, including an eccentric compression force relative to the gravity center of the cross‐section of the column toward its higher rigidity. The boundary conditions applied in the study reflected articulated, simple support of the column's ends. The scope of the study included determination of the effect of eccentric load on the structure's buckling mode and critical load. The critical load of the real structure was determined using approximation methods based on the post‐critical equilibrium paths of the structure obtained in experimental tests. At the same time, a numerical analysis by the finite element method was performed using the commercial simulation software Abaqus®. The numerical analysis involved solving an eigenproblem to determine the buckling mode of the structure and to determine the critical load of this structure under axial and eccentric load. The experimental results were used to verify the developed numerical models. The analysis enabled determination of the effect of eccentric load on the structure's buckling mode and critical load, which is of vital importance for the design of thin‐walled real structures that are coated with reinforcing thin‐walled stiffeners. Examples of such constructions include thin‐walled stressed‐skin structures such as an aircraft's main body section and wings, where the thin skin or outer covering is reinforced by a series of longitudinal profiles with open and closed sections. POLYM. COMPOS., 40:70–77, 2019. © 2017 Society of Plastics Engineers

Nonlinear dynamic stability analysis of Euler–Bernoulli beam–columns with damping effects under thermal environment

In this study, a unified nonlinear dynamic buckling analysis for Euler–Bernoulli beam–columns subjected to constant loading rates is proposed with the incorporation of mercurial damping effects under thermal environment. Two generalized methods are developed which are competent to incorporate various beam geometries, material properties, boundary conditions, compression rates, and especially, the damping and thermal effects. The Galerkin–Force method is developed by implementing Galerkin method into force equilibrium equations. Then for solving differential equations, different buckled shape functions were introduced into force equilibrium equations in nonlinear dynamic buckling analysis. On the other hand, regarding the developed energy method, the governing partial differential equation for dynamic buckling of beams is also derived by meticulously implementing Hamilton’s principles into Lagrange’s equations. Consequently, the dynamic buckling analysis with damping effects under thermal environment can be adequately formulated as ordinary differential equations. The validity and accuracy of the results obtained by the two proposed methods are rigorously verified by the finite element method. Furthermore, comprehensive investigations on the structural dynamic buckling behavior in the presence of damping effects under thermal environment are conducted.