Buckling Mode Shapes Research Articles

GBT Analysis of Steel-Concrete Composite Beams: Recent Developments

This paper reports the most recent developments concerning Generalized Beam Theory (GBT) formulations, and corresponding finite element implementations, for steel-concrete composite beams. These formulations are able to perform the following types of analysis: (i) materially nonlinear analysis, to calculate the beam load-displacement response, up to collapse, including steel plasticity, concrete cracking/crushing and shear lag effects, (ii) bifurcation (linear stability) analysis, to obtain local/distortional bifurcation loads and buckling mode shapes of beams subjected to negative (hogging) bending, accounting for shear lag and concrete cracking effects and (iii) long-term service analysis including creep, cracking and arbitrary cross-section deformation (which includes shear lag) effects. The potential (computational efficiency and accuracy) of the proposed GBT-based finite elements is illustrated through several numerical examples. For comparison purposes, results obtained with standard finite strip and shell/brick finite element models are provided.

Mode transitions in buckling and post-buckling of stretched-twisted strips

The complex buckling behavior of one of the simplest structures, an elastic thin strip under combined stretch and twist loading represents a fascinating example for structural stability analysis. For stretched-twisted strips, in contrast to solely stretched strips, not just the computation of the post-buckling process but also the simulation of the pre-buckling behavior and the determination of the bifurcation loading state requires nonlinearities to be taken into account.Under twist, not only shear stresses but also membrane normal stresses, with buckling relevant compressive stress contributions, develop. These compressive membrane stresses have components in the longitudinal and – maybe surprisingly – also in the transversal direction, i.e., normal to the long axis of the strip. Depending on the length to width ratio of the strip and on the ratio of the intensity of torque to tension loading, it is either the longitudinal or the transversal compression stress distribution, which is predominantly responsible for the loss of stability of the trivial equilibrium state. This is the reason for the transition from one characteristic of buckling mode shapes to a completely different characteristic of buckling modes in dependence of the loading conditions. Of course, within each of these two characteristics, the buckling modes vary in terms of wave numbers in dependence of the mentioned parameters.While these kinds of mode transitions has been studied recently elsewhere, the focus of the present paper is, in addition, laid on further mode transitions appearing in the post-buckling domain due to instabilities of equilibrium states either in the form of further bifurcations or snap-through jumps.Capturing all these effects by numerical simulations requires a well thought out model and a careful control of the computational analysis.

Size optimization method for controlling the buckling mode shape and critical buckling temperature of composite structures

The present study develops a novel size optimization method to control the buckling mode shape and associated buckling temperatures of plates. From a structural stability point of view, predicting the buckling temperature and mode shape of structures is one of the most important research topics in engineering. However, coming up with optimized engineering structures through engineering intuition for controlling these aspects is challenging. To address this limitation, the present study proposes the combination of finite element simulation and a size optimization scheme. Based on the idea that the structural buckling temperature and mode shape of a plate are mainly influenced by the thickness of the plate, in the optimization process, the thickness values of the divided sections of the target plate are set as the design variables. The buckling mode shape and buckling temperature are set as the objective functions, subjected to the total volume of the target plate. By applying the size optimization scheme, it is possible to determine the optimal thickness distributions for inducing the desired buckling mode shapes and buckling temperature values. The validity of the proposed size optimization method has been verified using several numerical examples.

Efficient Design of Lightweight Reinforced Tensegrities Under Local and Global Failure Constraints

AbstractTensegrities are prestressable trusses that have been proven to support various load distributions with minimum mass. This article presents a novel efficient method for designing lightweight tensegrities under local and global failure constraints. Local failure includes buckling and material yielding of individual members in the tensegrity. Global failure refers to global buckling of the tensegrity, where it loses stability without undergoing local failure at its individual members. The formulation and numerical approach to determine the critical global buckling forces and mode shapes of tensegrities with arbitrary shape and topology are first provided. Next, the design method considering local and global failure is presented, which starts with the local sizing of the member areas of the given tensegrity for the prevention of local failure. The method then determines the dominant failure mode by comparing the external forces and the critical global buckling force of the locally sized structure. If the critical global buckling force is larger than the external force, the dominant mode is a local failure and the locally sized design is returned as the minimum mass design. Conversely, if global failure is the dominant mode, different global reinforcement approaches are applied to raise the critical buckling force of the structure until it matches the external force, preventing global buckling. These reinforcement approaches include increasing the areas of the members and increasing the prestress in the tensegrity. Representative examples are provided to demonstrate the effectiveness of the design method considering box and T-bar tensegrities.

Thermal buckling and dynamic characteristics of composite plates under pressure load

The effect of geometrical nonlinearities due to pressure load on the thermal buckling and dynamic characteristics of composite plates are investigated in this paper, which is the main contribution of this research work. The mechanical behavior of the plate is described with the first-order shear deformation theory. The geometrical nonlinearity due to both thermal effect and pressure load is introduced in the finite element model of the plate via additional stiffness matrices. Thermal buckling and modal analysis of a four-sided simply supported rectangular composite plate under different pressure fields are conducted. Numerical results show that both the mode frequencies and critical buckling temperature of the plate rise with the increase of the pressure. The vibrational mode shapes change with the gradient pressure load field. The maximum buckled deflection point moves from the center to the place where is easier to reach compressive stress state under uniform thermal load. The pressure distribution has a significant effect on the buckling mode shapes of the plate.

Semi-analytical optimal solution for maximum buckling load of simply supported orthotropic plates

The buckling characteristic of rectangular orthotropic plates under axial compression with simply supported boundary has been extensively studied. To enhance the structural efficiency, the stacking sequence of laminate should be optimized to maximize the critical buckling load. In this study, the discrete stiffness parameters are employed to yield the theoretical optimal ply orientation corresponding to the maximum critical buckling load at a layer level, and the buckling mode shapes are calculated analytically. Subsequently, the derivation is performed based on optimal ply orientation to derive the optimal stacking sequence of the laminate, in which a sign vector is adopted to minimize the bending-twisting coupling effects. As a result, two laminate optimal design problems are solved: maximizing critical buckling load with fixed thickness, as well as minimizing the thickness with buckling constraint. Two numerical examples are adopted to verify the yielded optimal solutions. Lastly, the theoretical optimal ply orientations exhibiting different load ratios and aspect ratios are presented.

Numerical Simulation of Buckling and Post-Buckling Behavior of a Central Notched Thin Aluminum Foil with Nonlinearity in Consideration

In thin notched sheets under tensile loading, wrinkling appears on the sheet surface, specifically around the cracked area. This is due to local buckling and compression stresses near the crack surfaces. This study aims to numerically study the buckling behavior of a thin sheet with a central crack under tension. A numerical model of a notched sheet under tensile loading is developed using the finite element method, which considers both material and geometrical nonlinearity. To overcome the convergence problem caused by the small thickness-to-length/width ratio and to stimulate the buckling, an imperfection is defined as a small perturbation in the numerical model. Both elastic and elasto-plastic behavior are applied, and the influence of them is studied on the critical buckling stress and the post-buckling behavior of the notched sheet. Numerical results for both elastic and elasto-plastic behavior reflect that very small perturbations need more energy for the activation of buckling mode, and a higher buckling mode is predominant. The influences of different parameters, including Poisson’s ratio, yield limit, crack length-to-sheet-width ratio, and the sheet aspect ratio are also evaluated with a focus on the critical buckling stress and the buckling mode shape. With increase in Poisson’s ratio. First, the critical buckling stress reduces and then remains constant. A higher yield limit results in increases in the critical buckling stress, and no change in the buckling mode shape while adopting various crack length-to-sheet-width ratios, and the sheet aspect ratio changes the buckling mode shape.

Finite element based post-buckling analysis of refined graphene oxide reinforced concrete beams with geometrical imperfection

The present paper researches post-buckling behaviors of geometrically imperfect concrete beam resting on elastic foundation reinforced with graphene oxide powders (GOPs) based on finite element method (FEM). Distribution of GOPs are considered as uniform and linearly graded through the thickness. Geometric imperfection is considered as first buckling mode shape of the beam, the GOP reinforced beam is rested in initial position. The material properties of GOP reinforced composite have been calculated via employment of Halpin-Tsai micromechanical scheme. The provided refined beam element verifies the shear deformation impacts needless of any shear correction coefficient. The post-buckling load-deflections relations have been calculated via solving the governing equations having cubic non-linearity implementing FEM. Obtained findings indicate the importance of GOP distributions, GOP weight fraction, matrix material, geometric imperfection, shear deformation and foundation parameters on nonlinear buckling behavior of GOP reinforced beam.

Numerical study of slender I-girders with one longitudinal stiffener under patch loading

Slender plate I-girders are steel members widely used in bridges, viaducts and heavy industrial buildings, which are subjected to combined bending and shear, making the web panel susceptible to stability problems. Therefore, longitudinal stiffeners are sometimes placed along the web. This paper aims at investigating the load carrying capacity of girders submitted to patch loading (common in bridge launching), using finite element models. A nonlinear stability analysis was carried out considering the initial geometric imperfection as the first buckling eigenmode, with amplitude defined according to EN 1993-1-5 recommendations. The performance of girders with different stiffener locations was evaluated, discussing the structural behavior and associated failure modes. If the stiffener is effective and properly placed, the failure mode is the local buckling of the two web subpanels (which increases the patch loading resistance), although it was verified that the optimal position for patch loading is mainly governed by the web crippling resistance of the upper subpanel. The numerical results are in accordance with the experimental results available in the literature. For the studied girder configurations, the optimal stiffener position for patch loading resistance is closer to the loaded flange when compared to girders under dominant bending. Depending on the cross-sectional shape of the buckling mode, the stiffener can be unfavorable to the patch loading resistance for certain range of positions. Furthermore, the definition of positive or negative cross-sectional imperfections, i.e., the inversion of the curvature of the buckling mode shape, has impact on the load capacity for larger patch loading lengths.

Analytical buckling solution of magneto-electro-thermo-elastic cylindrical shells under multi-physics fields

A new analytical buckling solution of a cylindrical shell made of two-phase magneto-electro-thermo-elastic (METE) composites under multi-physical fields is obtained by a Hamiltonian-based approach. Two types of technologically important distribution model are considered: continuous fibers and laminates. Based on the Reissner’s shell theory, the exact solution expanded into symplectic series is rigorously obtained from governing equations under the Hamiltonian description which has four possible forms of explicit expressions. Accurate critical buckling loads and analytical buckling mode shapes for various boundary conditions are obtained. A comprehensive comparison is presented to verify the proposed solution and very good agreement is reported. Effects of geometrical parameters, boundary conditions, cases of eigenfunctions, volume fractions and external magneto-electro-thermal loadings on buckling behaviors of the shell are investigated also.

Buckling Analysis of Circular Shells for RDE Reactor Pressure Vessel by Finite Element Study and Analytical Model

One of the basic design considerations for circular shells of Reactor Pressure Vessel (RPV) components of Reaktor Daya Eksperimental (RDE) is buckling mode shape of geometric structure imperfections. Geometric structure is prone to a large number of imperfections due to the manufacturing difficulties. Moreover, determination of critical loads of circular shell structure can be properly approximated only through randomness in the geometry. As at the basic design phase this is possible and expensive to measure the geometric structure imperfections in situ or in laboratory, a simulation of computer code should be provided. In this paper, a contribution is made to numerical simulation by finite element study and analytical model by Monte Carlo technique approach of such circular shells for RPV components. For this reason, computer code SolidWorks is used in the implementation finite element analysis and Fortran code is developed for analytical model analysis. The main objective of this investigation is to provide a comprehensive analysis of unstable structure due to the buckled geometric imperfection in the design calculations involved. A method of solution attempts to use the linearized equilibrium equations to interpret geometric imperfections as structure of circular shells associated with the RPV components of RDE. A number of calculation results carried out on a computer code are presented to illustrate the design of RPV components that are safe from the buckled shape failure. It is found that both of the computer code results are very similar with greatest difference value of vibrational amplitude of 0.00699 % and smallest value for buckling load factor of 12.51 on upper circular shell under 750 °C conditions.

Static test of a variable stiffness thermoplastic composite wingbox under shear, bending and torsion

AbstractAutomated manufacturing of thermoplastic composites has found increased interest in aerospace applications over the past three decades because of its great potential in low-cost, high rate, repeatable production of high performance composite structures. Experimental validation is a key element in the development of structures made using this emerging technology. In this work, a $750\times640\times240$ mm variable-stiffness unitised integrated-stiffener out-of-autoclave thermoplastic composite wingbox is tested for a combined shear-bending-torsion induced buckling load. The wingbox is manufactured by in-situ consolidation using a laser-assisted automated tape placement technique. It is made and tested as a demonstrator section located at 85% of the wing semi-span of a B-737/A320 sized aircraft. A bespoke in-house test rig and two aluminium dummy wingboxes are also designed and manufactured for testing the wingbox assembly which spans more than 3m. Prior to testing, the wingbox assembly and the test rig were analysed using a high fidelity finite element method to minimise the failure risk due to the applied load case. The experimental test results of the wingbox are also compared with the predictions made by a numerical study performed by nonlinear finite element analysis showing less than 5% difference in load-displacement behaviour and buckling load and full agreement in predicting the buckling mode shape.

Buckling load for telescopic cylinder using successive approximation method

In this manuscript, a new way of determining the buckling load of the two-stage both end-hinged, single acting hydraulic cylinder has been engrossed using successive approximation method. The method has been extended from single stage hydraulic cylinder to the two-stage hydraulic cylinder mounted with pin support at both ends. This method includes a few numbers of iterations required in order to achieve good accuracy or in other words getting the minimum error. Further, finite element analysis of the hydraulic cylinder has also been performed for the validation purpose. The buckling mode shape for three different modes and their corresponding buckling load has also been obtained using the finite element method. The approximation of the buckling load corresponding to the first buckling mode shape has been performed and observed load lies in the close vicinity of theoretical calculation. The critical load obtained using a successive approximation method after two iteration shows the minor deviation of 3.63%.

Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load

This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.

The effect of chitosan adsorption on the stability characteristics of single- and double-walled boron-nitride nanotubes under compressive force using molecular dynamics simulations

In this study, the buckling behavior of functionalized single- and double-walled boron-nitride nanotubes (SWBNNTs and DWBNNTs) with a monomer of chitosan using molecular dynamics (MD) simulations is explored. The effect of chemical adsorption of chitosan molecule on the critical buckling force and strain is investigated. The results show that the critical buckling force considerably increases as the chitosan is attached to the side wall of boron-nitride nanotube which is more considerable for larger radii of nanotube. Moreover, increasing the number of walls reduces the sensitivity of boron-nitride nanotube to the functionalization compared with similar SWBNNTs. Further, it is shown that critical buckling of functionalized BNNTs increases by rising the weight percentage of chitosan. Considering the critical strain, it is observed that functionalization reduces the critical strain of functionalized BNNTs which is more pronounced in the case of SWBNNTs with bigger radii. Moreover, the buckling mode shape of functionalized BNNTs is presented.

Size-dependent static and dynamic analysis of Reddy-type micro-beams by strain gradient differential quadrature finite element method

Abstract This paper proposes a strain gradient differential quadrature finite element method to analyze the size-dependent static and dynamic behaviour of Reddy-type micro-beams. This element has 6 of freedom per node and avoids the exploitation of shape functions. A sixth-order differential quadrature-based geometric mapping scheme is constructed to realize the higher-order continuity requirements of kinematic variables. And then, it is combined with the minimum total potential energy principle to derive the motion equation of a generic element. Afterwards, several numerical examples are provided to establish the validity of the developed element. Finally, we utilize this method to analyze the static bending, free vibration, and linear buckling characteristics of uniform and stepped micro-beams. Numerical results show that the current element has prominent convergence and adaptability advantages over the classical shape function-based element. Besides, the size-dependence of vibration and critical buckling mode shapes of micro-beams is demonstrated in the graphical form for the first time.

Strain gradient differential quadrature Kirchhoff plate finite element with the C2 partial compatibility

This paper constructs a four-node Kirchhoff plate element considering dilatation, deviatoric stretch and rotation gradient effects to address the general boundary value problems of size-dependent isotropic thin micro-plates. This element benefits from the merits of differential quadrature method (DQM) and finite element method (FEM) and possesses four nodal displacement parameters at each node, i.e., deflection, its two first partial derivatives and one second mixed partial derivative with respect to two in-plane coordinates. To guarantee the C2 partial compatibility among neighboring elements, we establish a novel DQ-based geometric mapping scheme relating the deflection values at Gauss-Lobatto quadrature points to the displacement parameters at four nodes. By applying the DQ rule, the Gauss-Lobatto quadrature rule and the developed mapping scheme, the total potential energy of a generic gradient-elastic Kirchhoff plate element is represented as a function of nodal displacement parameters. The element formulation is derived using the minimum total potential energy principle. Several numerical examples are provided to demonstrate the validity of the proposed method and explore the static bending, free vibration and critical buckling behavior of thin micro-plates. It is validated that the size-dependence of vibration and critical buckling mode shapes of thin micro-plates can be observed in some cases.

Torsional Buckling of Functionally Graded Multilayer Graphene Nanoplatelet-Reinforced Cylindrical Shells

Exact solutions for the torsional bifurcation buckling of functionally graded (FG) multilayer graphene platelet reinforced composite (GPLRC) cylindrical shells are obtained. Five types of graphene platelets (GPLs) distributions are considered, and a slope factor is introduced to adjust the distribution profile of the GPLs. Within the framework of Donnell’s shell theory and with the aid symplectic mathematics, a set of lower-order Hamiltonian canonical equations are established and solved analytically. Consequently, the critical buckling loads and corresponding buckling mode shapes of the GPLRC shells are obtained. The effects of various factors, including the geometric parameters, boundary conditions and material properties on the torsional buckling behaviors are investigated and discussed in detail.

Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method.

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

New stability design methodology through overall linear buckling analysis

ABSTRACTTwo new stability design methods are presented: the Overall Strength Reduction Method (OSRM) and the Overall Imperfection Method (OIM). Both methods are based on the linear buckling analysis (LBA) of global structural models and use the standard reduction curves. The OSRM is formulated in the classic way using generalized slenderness and reduction factors while the OIM uses equivalent amplitude for the buckling mode based geometrical imperfection. It is shown that both methods have the same mechanical background which is built on two essential components: (1) the generalized form of an Ayrton‐Perry (or Perry‐Robertson) type strength reduction method for basic reference members and (2) a generalized transformation technique connecting the real members of the global structural model with the proper basic reference member. The first component is a mechanically consistent extension of the model behind the traditional buckling curves used only for specific stability cases like flexural buckling of columns or lateral‐torsional buckling of beams so far. The second component can be regarded as the generalization of the well‐known effective length approach. The new design methods cover all types of buckling modes (flexural, torsional, flexural‐torsional, lateral‐torsional or any interaction), which can be calculated by LBA of structural models composed of uniform or non‐uniform members with arbitrary cross‐sections and support conditions and subjected to any complex loading (e.g. biaxial bending with moment gradient, direct torsion effects etc.). There is no need for any additional input for the calculations (like effective length factors, unrestrained lengths of beams, moment gradient factors, equivalent length for the imperfection amplitude etc.) since all the necessary information is received from the elastic critical load and buckling mode shape. This paper clarifies the mechanical interpretation and proper calculation of all the components of the two methods and shows some comprehensive validation study on the performance.