Buckling Of Beams Research Articles

Re-examination of nonlinear vibration, nonlinear bending and thermal postbuckling of porous sandwich beams reinforced by graphene platelets

This paper re-examines the nonlinear free vibration, nonlinear bending and thermal postbuckling behaviors of porous sandwich beams with graphene platelets (GPL) reinforcements rested on elastic foundations. We remove the equivalent isotropic model (EIM) and introduce an inhomogeneous model. The shear modulus together with the Young’s moduli for porous GPLRC core is determined by a generic Halpin–Tsai model with porosity. Mechanical properties of both porous GPLRC core and metal face sheets are temperature dependent. Based on the framework of higher order shear deformation beam theory, the motion equations of porous sandwich beams with GPL reinforcements are established. In the modeling, the von Kármán kinematic nonlinearity, beam-foundation interaction and thermal effect are considered. By employing the two-step perturbation method, the analytical solutions are obtained for the nonlinear vibration and nonlinear bending as well as thermal postbuckling problems. Numerical investigations are performed for comparing the results obtained from the EIM and the present model. The outcomes reveal that the EIM is invalid for linear free vibration, thermal buckling and postbuckling analyses of porous sandwich beams with GPL reinforcements, but the EIM is valid in some cases for nonlinear bending and nonlinear vibration analyses of the same beam.

Steel-concrete composite beams strengthened with NSM CFRP systems at the hogging-moment regions

Compared with conventional strengthening methods, the near-surface mounted (NSM) technique is more suitable for strengthening steel–concrete composite beams, especially at the hogging-moment regions. This paper investigates the flexural performance of steel–concrete composite beams strengthened with NSM CFRP systems. Four-point bending tests are carried out on four composite beam specimens, including one control beam without strengthening and three strengthened beams with NSM CFRP systems. The failure mode, ultimate load-carrying capacity, deflection distribution, and material strain distribution of the composite beams are presented, based on which the effects of the anchorage method and the number of CFRP bars are studied. The test results demonstrate that the main failure mode of the strengthened beams is the buckling of the steel beam. Compared with the control beam, the load-carrying capacity of the strengthened composite beams with two and four CFRP bars is increased by 10% and 20%, respectively. The use of end anchorage has a slight effect on the flexural capacity of the strengthened beams, while significantly improves the ductility during the later stage of loading. In addition, a nonlinear finite element (FE) model is proposed and calibrated, based on which a parametric study is performed to investigate the effects of the length and section area of CFRP bars, the spacing of grooves, and the thickness of bottom steel flange on the ultimate load-carrying capacity of the strengthened beams.

A meshfree model of hard-magnetic soft materials

Hard-magnetic soft materials comprising a soft matrix embedded with hard-magnetic particles can exhibit large deformations under external magnetic fields, which have attracted much interest due to their non-contact activation, flexible programmability, and rapid response in various applications such as biomedical devices, soft robotics and flexible electronics. Precise predictions of large deformations of hard-magnetic soft materials would be a key for relevant applications. Given that the rational designs in the programmable shape morphing of ferromagnetic structures requires high precision, here we develop a meshfree model based on the radial point interpolation method to quantitatively predict large deformations of hard-magnetic soft materials. We apply the stabilized conforming nodal integration instead of direct nodal integration to eliminate the influence of zero-energy mode, and the central difference scheme is implemented for the resolution procedure. We explore the bending of a hard-magnetic beam and snap buckling of a bistable hard-magnetic beam under external magnetic stimuli. Uniform and non-uniform discretizations are compared to show the insensitivity of the radial point interpolation method to nodal mesh. To demonstrate the versatile programmability of our model in the design of magnetic robotics, we further simulate complex locomotion (crawling, walking and rolling) of hard-magnetic soft robots under external magnetic excitation. The results could be used to guide rational designs of ferromagnetic structures and soft continuum robots.

Explicit Finite Element Formulations for the Free Vibrations and Buckling of Partial-Interaction Composite Beams with Two Sub-components

Explicit Finite Element Formulations for the Free Vibrations and Buckling of Partial-Interaction Composite Beams with Two Sub-components

XFEM based method for buckling analysis of thin-walled beams

Buckling of thin-walled beams involves complex deformation modes, making their analyses challenging. Most of the methods available for the computation of buckling loads of these elements rely on beam theories, which provide high computational efficiency, yet are limited in accuracy and versatility. On the other hand, the finite element method provides reliable solutions for a variety of structures, albeit with higher computational costs.This paper presents a new approach for the buckling analysis of thin-walled beams. The proposed model is developed by utilizing the eXtended Finite Element Method (XFEM). Global enrichment functions, based on the solutions of beam theory, are added to a standard 3D Finite Element (FE) coarse mesh in the longitudinal direction to enhance its performance and reduce the number of degrees of freedom. We consider the enrichment functions to be a set of modes derived from dynamic free vibrations of a beam or those derived from buckling analysis. The enriched nodes can be viewed as connected lines along the sections of the beam, enabling the computation of lateral–torsional buckling.While our XFEM method can be viewed as a semi-analytical approach, it does overcome well-known limitations of similar methods, e.g., the Finite Prism Method (FPM). For example, our method can in principle be extended to beams with non-prismatic cross-sections (e.g., tapered beams), open cross-sections, or those with perforations. However, these extensions are not discussed in the current paper. Furthermore, the XFEM method can also be applied to beams with different boundary conditions along the beam, such as lateral restraints. Additionally, one can improve the solution by sectional and/or longitudinal mesh refinements.The performance of the proposed method is investigated, and the results are compared with solutions obtained from FE models with a very fine mesh. It is shown that the proposed approach enables modeling thin-walled beams for buckling analysis using coarse FE mesh. In some cases, the number of degrees of freedom in XFEM is less than 5% of those required in the 3D FE model to achieve the same error in computing the first buckling load.

Nonlinear model reduction to fractional and mixed-mode spectral submanifolds.

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace E of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type. A further limitation has been that in some problems, the nonlinear behavior of interest may be far away from the smoothest nonlinear continuation of the invariant subspace E. Here, we remove both of these limitations by constructing a significantly extended class of SSMs that also contains invariant manifolds with mixed internal stability types and of lower smoothness class arising from fractional powers in their parametrization. We show on examples how fractional and mixed-mode SSMs extend the power of data-driven SSM reduction to transitions in shear flows, dynamic buckling of beams, and periodically forced nonlinear oscillatory systems. More generally, our results reveal the general function library that should be used beyond integer-powered polynomials in fitting nonlinear reduced-order models to data.

Static and dynamic instability of functionally graded graphene origami-enabled auxetic metamaterial beams with variable thickness in fluid

Functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) structures have shown great potential for various engineering applications due to their exceptional mechanical and physical properties such as high strength-to-weight ratio, tuneable stiffness and strength, and negative Poisson's ratio (NPR). This paper aims to investigate the buckling and dynamic instability behaviours of FG-GOEAM beams with variable thickness immersed in a fluid, with a particular focus on the influence of NPR. The material properties such as Young's modulus and Poisson's ratio of the GOEAM are determined by using a genetic programming (GP)-based micromechanics model. Within the framework of the first-order shear deformation theory and by employing Hamilton's principle and modelling the fluid effect as added mass, the governing equations of motion are established and are then discretised by means of differential quadrature (DQ) method to obtain a linear system of Mathieu-Hill equations from which the principal instability regions of the FG-GOEAM beam are determined by Bolotin's method. A comprehensive parametric study is conducted to reveal the effects of GOri's folding degree, distribution, weight fraction, as well as fluid density and beam dimensions on the static and dynamic instability behaviours of the FG-GOEAM beam immersed in fluid. Numerical results show that the FG-GOEAM beam with NPR considerably outperforms its pristine metallic counterpart in terms of resistance against static buckling and dynamic stability.

Elastic local shear buckling of beams with sinusoidal corrugated webs

Steel beams with corrugated webs are used in the construction industry to increase the shear capacity of the web without the need to weld stiffeners to the web. Most research in the past has focused on beams with trapezoidal corrugated webs with little research available for beams with sinusoidal corrugated webs. The Eurocode is the only standard with shear design rules for beams with trapezoidal and sinusoidal corrugated webs. However, the equation for the elastic local shear buckling stress for beams with sinusoidal corrugated webs shown in the Eurocode was derived for a specific girder geometry and it is uncertain whether this equation can be used for any general beam with sinusoidal corrugated webs. In this paper, a finite element analysis is described to compute the elastic local shear buckling stress for beams with sinusoidal corrugated webs. The finite element analysis was done by modelling the beam as a cantilever acted upon by an edge shear stress at the free end. The aim is to compare the elastic local shear buckling stress with that predicted by the Eurocode. Over 600 beams with different geometric parameters of the sinusoidal web were analysed to investigate the elastic local shear buckling behaviour. The results show that the Eurocode is unconservative for some beams with sinusoidal corrugated webs.

Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads

Cantilever steel beams are an essential structural element in civil engineering fields such as bridges and buildings. However, there is very little research on the critical moment (Mcr) of cantilever beams subjected to a concentrated load (CL) or a combination of concentrated load and uniformly distributed load (CUDL) when the concentrated load is not limited to the free end. Therefore, the focus of the current paper is the calculation of Mcr for cantilever steel beams under CL and CUDL. This paper proposes a program and a simple closed-form solution for Mcr that are applicable to the elastic buckling analysis of cantilever I-beams under CL and CUDL. Based on the Rayleigh–Ritz method, a matrix equation and the corresponding procedure about Mcr under CL and CUDL are derived by using infinite trigonometric series for the buckling deformation functions. The value of Mcr and the corresponding mode of buckling can be obtained efficiently by considering the symmetry of the section, the ratio of two load values and the load action position. Experimental results and finite element calculations validate the numerical solutions of the procedure. A closed-form solution for Mcr is derived according to the assumption of a small torsion angle and the specific values of each coefficient in the closed-form solution of Mcr are calculated by the proposed procedure. The results show that the procedure and closed-form solution for Mcr presented in this paper have a high degree of accuracy in calculating the Mcr of the cantilever beam under CL and CUDL. The deviations between the results calculated by the proposed procedure and data from existing literature are less than 8%. These conclusions are capable of solving the calculation problem of Mcr for cantilever beams under CL or CUDL, which are both significant load cases in engineering. The study provides a reference for the design of cantilever steel beams.

Variable RPIM and HOCM coupling for non-linear buckling and post-buckling analysis of transverse FG sandwich beams

This paper presents an analytical-based approach to analyze non-linear buckling and post-buckling of transverse functionally graded sandwich beams (TFGSB). The proposed approach combines a high order continuation method (HOCM) and a variable radial point interpolation method (RPIM). First of all, the strong form of the beam buckling governing equations are established in the framework of first order shear deformation theory (FSDT) by taking into account the non-linear tangential term. Then, the spatial approximation of the unknown variables and their partial derivatives using the RPIM with variable shape parameters is the first step of this approach. The RPIM shape parameter is considered to change according to Franke formula, modified Franke formula or reciprocal Franke formula. The non-linear resulting system of discrete equations is linearized by using the Taylor series expansion, whilst the continuation method is employed to obtain the whole solution. To demonstrate the effectiveness of the proposed approach, numerical examples on non-linear buckling and post-buckling analysis of TFGSB are presented. These numerical tests are carried out to define the optimal RPIM with variable shape parameters and to investigate the sensitivity of the accuracy regarding boundary conditions, effects of power-law index, Young’s modulus, skin-core-skin thickness and the span-depth ratios. The results obtained through the proposed approach are compared with those computed by finite element method (FEM) and the available data in the literature.

Effect of thermal pre/post-buckling regimes on vibration and instability of graphene-reinforced composite beams

The aim of this research is to analyze the natural frequencies and instability of laminated nanocomposite beams in thermally pre/post-buckled configurations. The thick beam consists of ten composite layers, each reinforced with graphene platelets (GPLs). It is also assumed that the beam is in contact with a generalized linear/nonlinear elastic medium. Different patterns are used to simulate selected distribution profiles of material properties through the beam thickness. Reddy’s third-order shear deformation theory within the framework of the von-Kármán assumption is implemented to derive the kinematic relation. Hamilton’s principle is used to derive the equations of motion for the uniformly heated graphene-reinforced composite beam. In the absence of an external pressure load, the Duffing-type equation is obtained and solved for the nonlinear vibration problem of the beam. The analytical solutions for the vibration and stability problems of the thermally buckled beam are obtained using the perturbation-based method. This study presents new insights into the vibration and stability problems of laminated nanocomposite beams. The influence of important parameters of the nanocomposite beam such as the weight fraction of nanofillers, the distribution pattern of GPLs, slenderness ratio, temperature variation and foundation components on the nonlinear behavior of the beam is numerically presented and discussed.

Modeling of cyclically loaded square thin-walled CSFT columns stiffened by diagonal ribs using OpenSees

Among various proposed stiffening forms, the diagonal binding ribs, made of thin-walled steel plates with circular holes and welded to the steel tube, were found to particularly effective in improve the cyclic-bending and cyclic-shear behavior of square thin-walled concrete-filled steel tubular (CFST) columns. Although the mechanical behavior can be obtained from the experiments, numerical models are still needed to expand the range of investigation beyond what was tested physically and to facilitate the practical design. A 3D model using shell and solid elements is able to explicitly show the influence of different parameters, and simulate the details including the holes in diagonal ribs and the contact interaction between different parts. However, such analyses are relatively complicated, time-consuming, and are susceptible to numerical issues that cause lack of convergence. By contrast, a 2D model using fiber-based beam–column elements is far more efficient thus allowing for a wider range of investigation. In this work, the fiber models of flexure-dominated diagonal ribs stiffened square thin-walled concrete-filled steel tubular (DSCFST) columns were firstly developed in OpenSees and validated based on the effective stress–strain relationships of steel tubes and moment–rotation curves of RC bases. The fiber models of shear-dominated DSCFST columns were further developed by defining the nonlinear shear force–deformation relationships at the section level. These models were verified by the test results, and then the effect of the deformable RC base, tube local buckling, shear effect, and types of beam–column elements was analyzed in detail. Based on extensive parametric analyses, the effective stress–strain relationships applicable to various square thin-walled steel tubes stiffened by diagonal ribs and infilled concrete were derived, and the simplified shear force–deformation relationships were proposed. These results facilitate understanding of the behavior of DSCFST columns and lay a foundation for further system-level analysis.

Thermo-Mechanical Buckling and Non-Linear Free Oscillation of Functionally Graded Fiber-Reinforced Composite Laminated (FG-FRCL) Beams

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He’s variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature.

A Machine-Learning-Based Model for Buckling Analysis of Thermally Affected Covalently Functionalized Graphene/Epoxy Nanocomposite Beams

In this paper, a machine-learning model is utilized to estimate the temperature-dependent moduli of neat, thermally reduced graphene and covalently functionalized graphene/epoxy nanocomposites. In addition, the governed mathematical expressions have been used to solve the buckling problem of beams fabricated from such nanocomposites in the presence of a thermal gradient. In order to do so, an energy-based method including the shear deformable beam hypothesis is used. The beam structure is rested on the Winkler–Pasternak substrate. The reported verifications demonstrate the impressive precision of the presented ML model, as well as the buckling response of the under-study structures. Finally, in the framework of some numerical case studies, the impact of several parameters on the buckling of nanocomposite beams is depicted. The results of this study delineate that temperature has a vital role in the determination of the critical buckling load that the nanocomposite structures can endure.

Comparative study of the negative bending behaviour of corrugated web steel–concrete composite beams using NC, ECC and UHPC

Engineered cementitious composites (ECC) and ultra-high performance concrete (UHPC) could significantly enhance the cracking performance of the negative bending regions of composite beams. In this study, the negative bending behaviour of corrugated web composite beams using normal concrete (NC), ECC and UHPC was comparatively discussed. Firstly, the negative bending test was performed for three composite beams using different concrete. The results showed that the failure modes of the corrugated web composite beams were characterized by local buckling of the steel beam compressed flanges and severe cracking of the concrete slabs. The negative bending capacity of the composite beams using UHPC increased by 17.9% and 14.4% compared to those using NC and ECC, respectively. Besides, finite element simulation indicated that the negative bending performance of steel-ECC composite beam was particularly affected by steel strength and reinforcement strength, and that of steel-UHPC composite beam showed the most significant response to steel strength. Furthermore, the negative bending capacity formula in the code was found conservative in predicting corrugated web composite beams, especially for those using UHPC. With the consideration of the tensile strength of concrete and the omission of the contribution of corrugated webs, a reasonable formula for predicting the corrugated web composite beams using different concrete was proposed and proved to have satisfactory accuracy.

Interrogating the Configuration Space of Postbuckled Beams

This paper considers the global configuration space of postbuckled structures. Initially, a slender buckled beam is considered. The boundary conditions are pinned at one end and fully fixed at the other. The pin-end of the initially flat beam is moved toward the other end by a small amount. This causes the beam to buckle laterally (the beam is axially inextensional) with a characteristic shape. Since this is the outcome of a stable-symmetric pitchfork bifurcation, the buckled shape possesses a near mirror-image shape on the other side. Which of the two available equilibrium configurations is taken up depends on the usual, often small, symmetry-breaking effects commonly encountered in axially loaded systems. Given fixed conditions to maintain this initial buckled shape we then apply a moment at the pinned end. The relation between the applied moment and the change in shape is the primary focus of this study. The extent of the buckling (end shortening) is varied, with the magnitude of the moment, as a function of the angle, providing considerable information about the potential energy landscape in which the axially loaded system operates. The applied moment can be thought of as a probing mechanism whereby various equilibrium configurations are revealed, together with information regarding their stability and robustness. A similar approach is then used where the structure consists of two nominally identical beams attached at a right angle. In both cases, the opposite end from the actuation is clamped. An experimental study is conducted on 3D-printed specimens, and this is compared with a finite element analysis using ANSYS, and a large-deflection elastica analysis.

Prestressed Steel-Concrete Composite I-Beams with Single and Double Corrugated Web

Composite steel girders with concrete have been used for many years and advances in structural and fabrication technology have established their optimization. One of the changes in structural steel I-beams during the past few years has been the availability of web corrugation. The economic design of steel girders normally requires thin webs. Moreover, using externally prestressed tendons as a strengthening technique controls deflections and stresses. However, this strengthening technique causes shear buckling of the steel beams. In this study, the flexural behavior of externally prestressed composite steel-concrete I-beams with a single and double corrugated web was experimentally and numerically investigated. Three simply supported prestressed steel-concrete composite I-beams with single corrugated web (SCW) and double corrugated web (DCW) were tested under four-point loading. The tested beams were externally prestressed by using straight tendons along the full length. The experimental results showed that using SCW was more efficient in shear buckling resistance than DCW with the same equivalent web thickness. The ABAQUAS package was used to simulate the nonlinear behavior of the tested beams. The developed model was validated against the experimental results to carry out a parametric study in order to investigate the effect of various parameters on the behavior of the composite beams with SCW and DCW. Using stiffeners at the loading points as deviators to maintain the prestressed tendon positions increased the beam capacity and improved the beam performance.

Buckling and vibration of beams using Ritz method: Effects of axial grading of GPL and axially varying load

The present work discusses buckling and vibration characteristics of axially functionally graded (AFG) graphene platelet (GPL) composite beams exposed to axially varying loads (AVLs). Timoshenko beam composition with five different types of axial grading GPLs subjected to six different types of AVLs are studied. The effective elastic properties are obtained using Halpin-Tsai model and the equations of motion are obtained following the Hamilton’s principle. Then the equations are solved for buckling and vibration analysis using the Ritz method. Influences of nature of axial grading of GPLs and load, content of GPL, and structural boundary conditions are investigated through detailed parametric studies. It is found that the grading pattern of GPLs not only influences the buckling load, but also changes buckling mode shapes of the beam at specific type of AVL. Furthermore, results reveal that buckling and vibration characteristics of beam enhanced in case of AFGM-A type for most of the load cases studied. The proposed study will be helpful for the structural engineers to select the nature of graded distribution of GPLs for the given type of AVL and design the structural member.

An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.

Thermoelastic Dynamic Responses Near Buckling of the Non-conservative Gradient Porous Beam in Hygrothermal Environments

Thermoelastic Dynamic Responses Near Buckling of the Non-conservative Gradient Porous Beam in Hygrothermal Environments