Buckling Of Beams Research Articles

Finding transition state and minimum energy path of bistable elastic continua through energy landscape explorations

Mechanical bistable structures have two stable equilibria and can transit between them under external stimuli. Due to their unique behaviors such as snap-through and substantial shape changes, bistable structures exhibit unprecedented properties compared to conventional structures and thus have found applications in various fields such as soft robots, morphing wings and logic units. To quantitatively predict the performance of bistable structures in these applications, it is desirable to acquire information about the minimum energy barrier and an energy-efficient transition path between the two stable states. However, there is still a general lack of efficient methodologies to obtain this information, particularly for elastic continua with complicated, unintuitive transition paths. To overcome this challenge, here we integrate energy landscape exploration algorithms into finite element method (FEM). We first utilize the binary image transition state search (BITSS) method to identify the saddle point and then perform nudged elastic band (NEB) calculations with an initial guess based on the BITSS results to find the minimum energy path (MEP). This integrated strategy greatly helps the convergence of MEP calculations, which are highly nonlinear. Two representative cases are studied, including bistable buckled beams and a bistable unit of mechanical metamaterials, and the numerical results agree well with the previous works. Importantly, we numerically predict the complicated MEP of an asymmetric bistable unit of mechanical metamaterials and use experiments to demonstrate that following this MEP leads to successful transition between stable states while intuitive uniaxial compression fails to do so. Our work provides an effective numerical platform for identifying the minimum energy barrier and energy-efficient transition path of a bistable continuum, which can offer valuable guidance to the design of actuators, damping structures, energy harvesters, and mechanical metamaterials.

Elastic Lateral Torsional Buckling of Beams Under External Moments: A New Type of Nonconservative Problem

The important character of the applied moment with a fixed line of direction as implied in deriving the governing differential equations of LTB for elastic beams, however, is not represented in stating the boundary conditions based on conventional energy approaches. This paper sets out to consider the boundary conditions derived by conventional potentials in order to identify the ignored second-order moments that need to be added for maintaining the initial direction of the applied moment during LTB. By taking into account the non-zero work done by these ignored moments for modifying original potentials, the nonconservative nature of the elastic LTB of beam under external moments is clarified in two aspects — by discovering the path-dependent property of the non-zero work and by demonstrating that only trivial solution of LTB mode is expected from the analytical solution of the governing differential equations. Using the concept of dynamic instability, an analysis procedure combining the numerical solution of an eigen-value problem and the bisection method is proposed to identify the first appearance of imaginary or conjugate complex frequencies and to determine the value of the critical moment. Demonstrative examples are considered for illuminating the elastic LTB behavior using conventional energy methods and revised virtual work equation. It has been found that the critical moments given by conventional potentials are generally not consistent with each other unless the work done by the ignored second-order moments vanishes. However, consistent critical moment, under which the free vibrating amplitude of elastic beam is increased unboundedly with time, can be obtained by the kinetic approach based on any revised virtual work equation, indicating the necessity of considering the nonconservative nature of non-follower applied moment.

Experimental and numerical investigations towards the lateral torsional buckling of cellular steel beams

This paper deals with the response of so-called “cellular members” against lateral torsional buckling. These beams comprising regularly-spaced circular web openings are especially used for their high resistance-to-weight ratio, the possibility to integrate service pipes within their height, and for aesthetics. Such profiles usually exhibit a complex behaviour and many potential failure modes, including interactional local/global instability modes. Regarding global instability, the members are usually designed by means of approximate design rules, which often lead to an unduly conservative solution, with beams sometimes showing up to 150 % resistance reserves.The present research works aim at improving this situation, through the development of adequate design formulae. In this respect, the present paper reports on investigations led towards improved solutions for cellular members at the experimental and numerical levels. This first paper focuses on experimental activities and on the development and validation of dedicated shell F.E. models. The results of three bending tests on members spanning from 7.5 m to 11 m are reported. Cross-sectional dimensions, material properties and accurate initial geometrical imperfections were measured for each specimen and further introduced in the corresponding F.E. models, which are shown to provide predictions in close agreement with the experimental results. The companion paper [1] further makes use of the F.E. models within extensive parametric studies and proposes a new set of design rules that could be proposed for integration in Eurocode 3.

A dimensionless analytical analysis for buckling and lateral buckling interaction of thin-walled beams with open cross sections

This paper investigates the interaction between buckling and lateral buckling of thin-walled beams with arbitrary open cross-section using nonlinear modeling in large rotation and with warping. First, the equilibrium equations have been transformed into dimensionless ones, and within a nonlinear stability model, various dimensionless parameters are introduced to control the interaction between buckling and lateral buckling. In comparison to a reference of literature, this work focuses on the case of arbitrary sections and includes a greater number of higher-order terms in our analysis. These additions have an impact on the coefficients of the stability formula by expanding its applicability to non- symmetric sections. Then, the effect of the different dimensionless parameters, thus introduced on the stability curve, has been studied. The proposed solutions are validated and compared to numerical solutions obtained by a general finite element package including the warping developed by the authors using Asymptotic Numerical Method (ANM) to solve the initial nonlinear equations.

Lateral torsional buckling of I‐section simply supported beams with stepped height

AbstractVariable cross‐section beams are widely used, especially I‐section stepped beams with variable height, which have benefits such as saving materials to match the moment diagrams, reducing the structure height, and access for technical services. However, lowering the beam height also leads to a significant decrease in lateral torsional buckling resistance. Currently, the standards do not provide calculation rules for this type of beam yet, and some studies on lateral torsional buckling of beams have only focused on those with varying flange width and thickness while keeping the beam height constant. This article introduces an approximate formula to determine the critical moment of simply supported I‐section beams when the height varies. The proposed formula is validated by comparing with finite element simulation results obtained by COMSOL Multiphysics 5.6. The evaluation shows that the formula has a low coefficient of variation and high coefficient of determination, ensuring its reliability.

Lateral torsional buckling of unsymmetric beams by FEM with linear elements

As the exact solution of lateral torsional buckling of elastic prismatic beams is practically limited to the simple case of simply supported beams under equal end moments, other loading conditions and boundary conditions require more practical solutions to the problem. As the general solution of lateral torsional buckling of unsymmetric beams is a function of the sign of the bending moment along the axis of the beam, i.e. location of the shear center concerning the compression zone of the cross section, application of the current traditional empirical expressions such as coefficient of the moment, Cb, becomes highly inconsistent and unwarranted to capture the correct critical moment as acknowledged by the American Institute of Steel Construction, AISC. The finite element method, FEM, offers a feasible alternative to overcome these shortcomings. FEM is formulated in its simplest form of linear elements for the general case of lateral torsional buckling of unsymmetric cross sections. Finite element development shows that the characteristic equation is of the nonlinear quadratic eigenvalue problem type. Using the classical polynomial shape functions for beams, FEM proves to be extremely accurate and can overcome the high inconsistencies and discrepancies embedded in the application of the classical methods.

Non-linear analysis of the flexural-torsional stability of slender tropical glulam beams

The main objective of this manuscript is to model the flexural-torsional buckling of slender tropical glulam beams using the finite element method. The non-linear approach to quantify both the buckling strength and the lateral geometrical features of wooden beams is proposed. A very rigorous methodology is used which is based on the kinematic Vlasov’s theory to determine the displacement field of the beam section by considering an updated Lagrangian description. For the finite element discretization, the Author develop a single wood element characterized by 14 degrees of freedom, i.e. seven degrees for each nodal point, and derive non-linear stiffness matrices by using the virtual works principle. The model is implemented in MatLab and the numerical results are compared with results based on the classical linear stability theory of slender wood beams. The selected case-studies are a cantilever beam and a beam on two supports.

An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory

An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory

Efficient buckling and post-buckling analysis of porous FG sandwich beams by new RPIM-HOCM mesh-free approach

The main purpose of this paper is to analyze buckling and post-buckling behavior of porous functionally graded (FG) beams and sandwich beams using a new efficient optimal mesh-free high order algorithm. The proposed solver combines mesh-free approach based on radial point interpolation method with high order continuation method (RPIM-HOCM). It is assumed that the FG beam and FG layers of sandwich beam are made from a mixture of ceramic and metal such that bottom surface of the plate is metal-rich, whereas top surface is ceramic-rich. So, constitutive equation as well as material properties are considered to change smoothly and gradually in the thickness direction via a mathematical relationship distribution. Non-linear strong form of the beam buckling problem which is based on the first-order shear deformation theory (FSDT) is derived using the virtual displacements theorem. The proposed coupling (RPIM-HOCM) is presented and validated firstly by comparing obtained numerical results with those computed using a reference algorithm. Then, RPIM-HOCM solver is used to investigate the effect of power-law index, porosity distribution, span-to-height ratio, boundary conditions and skin-core–skin thickness ratio on critical buckling load and post-buckling behavior.

Test and design of cold-formed steel closed built-up beams with double sigma sections and triple lipped channel sections

Built-up section beams fabricated from several single sections are finding increasing use in the cold-formed steel (CFS) industry due to various advantages such as higher bending capacities, innovative new shapes, and elimination of eccentricity. However, a built-up beam is connected at discrete points along the longitudinal direction, which weakens the shear and flexural rigidities of the built-up section. The structural behaviour of built-up CFS beams is therefore more complicated than that of single section beams. This paper presents an experimental investigation on the moment capacity of screw fastened built-up CFS beams. To investigate the local, distortional, and flexural-torsional buckling of built-up CFS beams, four-point bending tests of 8 braced and 24 unbraced built-up C2C (fabricated from 2 sigma sections) section beams and C3C (fabricated from 3 lipped channels) section beams are conducted. Unbraced bending test specimens are tested at three lengths covering inelastic failure at short lengths to essentially elastic failure at long lengths. The strength predictions for the tested built-up beams are calculated using the current Direct Strength Method (DSM) in the AISI-S100 Specification. The comparison between the experimental results and DSM predictions indicates that the current DSM in AISI specification is overly conservative and unreliable for the types of built-up beam sections examined in this research, due to the incorrect failure mode prediction and underestimation of the critical elastic buckling stresses which is a key input to the DSM.

The effect of prebuckling deflections on the lateral‐torsional buckling of beams: numerical studies

AbstractIn this research the effect of prebuckling in‐plane deflections on the critical moment to lateral‐torsional buckling of beams is investigated. Though this topic is already discussed in the literature, the relevant papers show that there are some discrepancies in the proposals and results from the various authors. In this paper a numerical parametric study is reported, conducted on doubly symmetric simple supported I‐shaped beams. The critical moment values were calculated with and without considering the effect of prebuckling deflections. Various methods were considered, including analytical formulae, thin‐walled beam finite element, and shell finite element calculations. The results justify that the in‐plane deflection has a positive effect on the critical moments, and that the effect is essentially determined by the ratio of in‐plane and lateral rigidities. However, the results highlight that the moment‐increasing effect of the prebuckling deformations can be much smaller compared to the prediction of the analytical formulae, if a shell finite element model is employed.

Linear elastic buckling of composite beams – An analytical engineering approach

AbstractThis article outlines a concise and complete versatile novel analytical framework to quickly and effectively assess lateral shear and lateral torsional buckling of composite beams build of segments with different lengths. The adjective “composite” includes dissimilar beam profiles with symmetrical homogeneous and or inhomogeneous cross‐sections and distinct beam materials. The major aim is to provide engineers with an accurate, simple but elegant analytical beam buckling tool based on advanced solid theoretical propositions. The justification of the analytical theory is supplied by demonstration of executed numerical tests, which affirm the validity of the proposed analytical buckling models adapted to the considered beam buckling cases.

Digital‐Image‐Correlation for Simulating Cyclic Local Buckling in Steel Beams

AbstractNonlinear continuum finite element (CFE) analyses rely on accurate multiaxial constitutive law formulations along with reliable imperfection patterns for simulating nonlinear geometric instabilities in steel members under mechanical loading. Validations of these models usually rely on conventional measurements of macroscopic parameters (e.g., displacement and strain field). It is common to use global deformation characteristics (e.g., deduced moment versus chord rotation) for model validation purposes of steel members exhibiting inelastic cyclic local buckling. The reason is that measurements from conventional strain gauge readings are not deemed to be reliable well before the onset of local buckling. This is potentially problematic when the accurate prediction of local strain demands is essential. This paper discusses the experimental results from a steel beam instrumented with a digital image correlation (DIC) system and how optical measurements acquired from DIC can be used to benchmark a CFE model representation of this beam. It is shown that while different CFE model types simulate accurately the moment rotation relationship of the steel beam, local strain demand predictions could vary considerably between model types. The results suggest that DIC measurements can inform the selection of proper imperfection patterns to be used in CFE models for the reliable estimation of inelastic strain demands.

Lateral‐torsional buckling of beams made of mono‐symmetrical thin‐walled sections

AbstractThin‐walled cold‐formed sections are extensively used as primary structural elements in storage system structures or steel houses, or as secondary elements like roof and wall systems, floor girders, etc. Despite their many advantages, such as transportability and lightweight, these members are usually susceptible to lateral‐torsional buckling as they are not always laterally restrained.This paper deals with the elastic stability of such beams and, in particular, is focusing on cross‐sections that are symmetrical about their major axis only, like channel or sigma profiles, and loaded perpendicular to their axis of symmetry. Their stability has been studied analytically, in accordance to available provisions in the concerned codes, and numerically with two different finite element tools.

Thermal buckling and postbuckling of functionally graded multilayer GPL-reinforced composite beams on nonlinear elastic foundations

Under the influence of axial forces and uniform temperature variations, the thermal buckling and postbuckling of composite beams reinforced of functionally graded multilayer graphene platelets (GPLs) resting on nonlinear elastic foundations are examined. The Halpin-Tsai model is used to calculate the elastic modulus of each layer of GPL-reinforced composite (GPLRC). According to the virtual work principle, the nonlinear governing equations for the beam are obtained from the first-order shear deformation beam theory. The impact of axial force and nonlinear elastic foundation on thermal buckling and postbuckling is discussed using the differential quadrature method (DQM), and the analytical expression is given by the two-step perturbation method (TSPM). The effects of axial force, boundary conditions, slenderness ratio, GPL geometry, GPL weight fraction, GPL distribution pattern, and elastic foundation coefficient on thermal buckling and postbuckling are examined through parameter analysis.

Buckling and Free Vibration Analyses of Various Nanoparticle Reinforced Concrete Beams Resting on Multi-Parameter Elastic Foundations.

Given their considerable specific surface area and amorphous characteristics, nanoparticles exhibit excellent pozzolanic activity, and when undergoing a reaction with calcium hydroxide, this leads to the generation of a denser matrix by promoting the formation of a greater amount of C-S-H gel, thereby enhancing the strength and durability of the concrete and fortifying the overall structure. Indeed, the present study investigates a comparative study of the buckling and free vibration analyses of concrete beams reinforced with various types of nanoparticles. For its simplicity and accuracy, a higher-order shear deformation theory will be used to analytically model the reinforced concrete beam. Furthermore, the powerful Eshelby's model is used to derive the equivalent nanocomposite properties. The soil medium is simulated with Pasternak elastic foundation, including a shear layer, and Winkler's spring, interlinked with a Kerr foundation. The motion equations are derived using Hamilton's principle. Moreover, based on Navier's analytical methods, the closed-form solutions of simply supported beams have been obtained. Different parameters, such as type and volume percent of nanoparticles, geometrical parameters, choice of theory and soil medium, on the buckling and dynamic behavior of the beam, are exercised and shown. The major findings of this work indicate that the use of nanoparticles in concretes increases better mechanical resistance and amplifies the natural frequencies. In addition, the elastic foundation has a significant impact on the buckling and vibration performances of concrete beams.

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.

Lateral distortional buckling of continuous composite beams in fire situation: Experimental and numerical analyses

In continuous steel-concrete composite beams, hogging moments may occur near the internal supports. This type of bending causes compression in the bottom flange, which may buckle laterally in an ultimate limit state known as lateral distortional buckling. Despite the consideration that continuity may result in an economic design, no research had yet been conducted on the possibility of lateral distortional buckling in continuous composite beams in fire situation. This paper presents an experimental and numerical study on the lateral distortional buckling of steel concrete composite beams. Four real scale fire tests were conducted and proved that it may indeed occur. A parametric study was also developed. It has been shown that this buckling mode may lead to a reduction in the strength of composite beams of up to 50% when compared to the plastic moment. It was also noted that this buckling mode is usually accompanied by local buckling, which may occur even in compact sections, leading to a decrease in bearing capacity. These results indicate a need of reviewing the design codes to consider the possibility of lateral distortional buckling at high temperatures, as it may be more than at ambient temperature.

Free vibration and buckling of functional graded box beams and the effect of the cross-sectional warping shape

Box beams have been used widely in engineering fields. This paper proposes a higher-order shear deformation beam theory to investigate the transverse vibration and stability of box beams in pure bending, especially free vibration and buckling of functionally graded box beams. A warping shape function is constructed for the first time to take shear deformation and rotary inertia of the hollow-core rectangular cross section into account simultaneously during bending. The governing equation and boundary conditions are derived. The frequency equations for typical end supports are given. The characteristic equations for buckling and critical loads are determined. Numerical results are evaluated. Besides its simplicity and easy-to-implement, the proposed theory has sufficient accuracy on the mechanical behavior of the box beams. By comparing the obtained results with the simulation results based on three-dimensional finite element method, the accuracy of the numerical results obtained is verified. The effects of parameters such as aspect ratio, gradient index, and wall thickness on the natural frequencies and critical loads are explored in detail. When box beams become rectangular beams, our model reduces to the well-known Levinson beam model.

Exploring buckling and post-buckling behavior of incompressible hyperelastic beams through innovative experimental and computational approaches

The objective of this paper is to conduct a comprehensive investigation into the buckling and post-buckling behavior of hyperelastic beams through both computational and experimental means. Natural rubber is used in the construction of a beam with a square cross-section. To determine the mechanical properties of natural rubber, a uniaxial tensile test is performed in accordance with ASTM D412. In finite element modeling (FEM), the nonlinear behavior of rubber is modeled using hyperelastic theory and the Yeoh strain energy function. The Static-Riks method is also implemented using Abaqus for the analysis of nonlinear buckling. To validate the present investigation results with FEM, an experimental test of digital image correlation (DIC) is conducted. The critical buckling force obtained via numerical methods exhibits an error of nearly 5% when compared to the corresponding results obtained from experimental testing. In order to ascertain the impact of various design parameters on the buckling behavior of the system, a comprehensive parametric analysis has been conducted. The parameters studied include the cross-sectional thickness, length of the structure, eccentric loads, as well as the mechanical properties of the materials used in the system. Consistent with the FEM outcomes, the critical buckling force exhibited by the hyperelastic beam demonstrates a positive correlation with increasing levels of hardness, cross-sectional thickness, and eccentric loads. The buckling behavior of the system is adversely affected by increasing its length. To ultimately validate the precision and reliability of the model, a supervised neural network (NN) learning method is employed.