Buckling Mode Shapes Research Articles

Analyzing nonlinear mechanical-thermal buckling of imperfect micro-scale beam made of graded graphene reinforced composites

This research is devoted to analyzing mechanical-thermal post-buckling behavior of a micro-size beam reinforced with graphene platelets (GPLs) based on geometric imperfection effects. Graphene platelets have three types of dispersion within the structure including uniform-type, linear-type and nonlinear-type. The micro-size beam is considered to be perfect (ideal) or imperfect. Buckling mode shape of the micro-size beam has been assumed as geometric imperfection. Modified couple stress theory has been used for describing scale-dependent character of the beam having micro dimension. Via an analytical procedure, post-buckling path of the micro-size beam has been derived. It will be demonstrated that nonlinear buckling characteristics of the micro-size beam are dependent on geometric imperfection amplitude, thermal loading, graphene distribution and couple stress effects.

A rational analytical approach for buckling analysis of orthotropic double-nanoplate-systems

A novel analytical Hamiltonian-based approach is proposed for buckling analysis of orthotropic double-nanoplate-systems (DNPSs) under uniaxially compression embedded in an elastic medium. In the Hamiltonian system, the governing equations for in-phase and out-of-phase buckling are established in a unified form based on Eringen’s nonlocal plate theory. The buckling analysis of the orthotropic DNPS is reduced to an eigenproblem in the symplectic space. Analytical buckling equations and buckling mode shape functions can be obtained by the symplectic eigensolutions and boundary conditions simultaneously. Comparison studies demonstrate the accuracy and efficiency of the proposed method. Key influencing factors which may benefit the design of complex 3D mesostructures are studied in detail. Some new results are given also.

Recommendations on the geometrical imperfections definition for the resistance of I-sections

The resistance and design of steel sections such as I and H-shapes usually depends in great extents on their ability to resist buckling, as a result of high width-to-thickness b / t ratios in the plates constituents. Consequently, these sections remain quite sensitive to imperfections.Information on imperfections, a key data for F.E. simulations, is however quite lacking in design standards, either for material imperfections or for geometrical imperfections. Accordingly, an extensive numerical study on hot-rolled and welded I and H-sections was carried out with the aim of providing reliable and efficient recommendations for the F.E. modelling of initial geometrical imperfections. Several sets of imperfections were investigated including the use of (i) sine-shape functions and of (ii) the 1st buckling mode shape. Reliable recommendations based on modification of nodes coordinates by means of sine-shape functions could be proposed afterwards.

Buckling of a Timoshenko beam bonded to an elastic half-plane: Effects of sharp and smooth beam edges

The problem of a compressed Timoshenko beam of finite length in frictionless and bilateral contact with an elastic half-plane is investigated here. The problem formulation leads to an integro-differential equation which can be transformed into an algebraic system by expanding the rotation of the beam cross sections in series of Chebyshev polynomials. An eigenvalue problem is then obtained, whose solution provides the buckling loads of the beam and, in turn, the corresponding buckling mode shapes. Beams with sharp or smooth edges are considered in detail, founding relevant differences. In particular, it is shown that beams with smooth edges cannot exhibit a rigid-body buckling mode. A limit value of the soil compliance is found for beam with sharp edges, below which an analytic buckling load formula is provided without loss of reliability. Finally, in agreement with the Galin solution for the rigid flat punch on a half-plane, a simple relation between the half-plane elastic modulus and the Winkler soil constant is found. Thus, a straightforward formula predicting the buckling loads of stiff beams resting on compliant substrates is proposed.

The mechanical properties and structural instability of single- and double-walled boron-nitride nanotubes functionalized with 2-methoxy-N,N-dimethylethanamine (MDE) using molecular dynamics simulations

As an inorganic quasi-1D nanostructure with non-cytotoxic properties, boron-nitride nanotubes have been the center of interest to researchers due to their high potential biocompatible applications. Due to the importance of functionalization in designing novel devices, the mechanical properties and buckling behavior of functionalized single- and double-walled boron-nitride nanotubes with 2-methoxy-N,N-dimethylethanamine (MDE) are investigated using molecular dynamics (MD) simulations. The calculated results demonstrate that Young’s moduli of BNNTs reduce, while critical buckling force and critical strain of BNNTs increase as MDE are attached to BNNTs. Moreover, it is observed that by increasing the MDE weight percentage, critical buckling force and critical strain of functionalized BNNTs reduce, unlike Young’s modulus. It is also observed that variations of the aforementioned parameters corresponding to DWBNNTs are less sensitive to MDE weight percentage compared to SWBNNTs and they lie between its inner and outer constituent functionalized SWBNNTs. Moreover, snapshots of buckling mode shapes of functionalized BNNTs are presented.

Accurate nonlinear stability analysis of functionally graded multilayer hybrid composite cylindrical shells subjected to combined loads

An accurate nonlinear buckling model for functional graded multilayer hybrid composites cylindrical shells under combined loads is proposed. A two-steps micromechanical approach including Halpin-Tsai and Mori-Tanaka method is employed to obtain the effective material properties of hybrid composites. A unified shell theory including Donnell's shell theory, first-order shear deformation theory and high-order shear deformation theory are adopted to derive the governing buckling equations. The axisymmetric pre-buckling deformation is taken into consideration in the solution procedure. Highly accurate critical buckling loads and analytical buckling mode shapes are obtained by Galerkin's method. The accuracy of the present solutions is validated by comparing with results of existing literature and numerical simulations. Effects of influencing parameters of reinforcements and their interactions on the structural stability of the hybrid composites cylindrical shells are revealed.

A modal finite element approach to predict the lateral buckling failure of the tensile armors in flexible pipes

This paper proposes a finite element (FE) modal approach to determine the critical loads and buckling mode shapes associated with the lateral buckling of the tensile armors in flexible pipes. This approach is based on pre-curved and twisted beam finite elements that are restrained to slide on the surface of a torus. The influence of the pipe's bending is addressed by assuming that the pre-buckling configuration of the tensile armors are geodesics or loxodromes of a torus and, moreover, friction effects are approximated using elastic springs. The proposed FE approach is employed in the classical buckling analyses of different tensile armors of several flexible pipes that have been previously studied in the public literature. The conducted analyses indicated that the buckling modes are sinusoidal and the pre-buckling configuration of the tensile armors in bent pipes affects their lateral buckling load. Moreover, friction between layers increases the lateral buckling resistance of the armor. Finally, comparisons with experimental and numerical results available in the literature indicates that the proposed approach is promising.

Accurate Buckling Analysis of Magnetically Affected Cantilever Nanoplates Subjected to In-plane Magnetic Fields

Cantilever-based nanostructures have attracted widespread attention in nanoengineering. As key components of cantilever-based nanosensors, some cantilever nanoplates are exposed to magnetic fields in working conditions. This paper performs the buckling analysis of magnetically affected cantilever nanoplates resting on elastic foundations. The aim of the presented research is to discuss the effects of magnetic fields on the buckling characteristics of cantilever nanoplates. Based on Eringen’s nonlocal elasticity theory, the exact buckling solutions of the cantilever nanoplates are obtained by using symplectic approach incorporating a superposition technique. To resolve the difficulties in dealing with the complex boundary value problems of cantilever nanoplates, the buckling problem is divided into two sub-problems which can be analytically solved by the symplectic approach. Exact solutions of the buckled cantilever nanoplate are derived by a superposition of the obtained solutions for sub-problems. Comparisons are presented to show the accuracy and stability of the proposed method. Parametric studies of the in-plane magnetic field and elastic foundation on the buckling characteristics of cantilever nanoplate are also given. Highly accurate critical buckling loads and buckling mode are derived analytically. This study has shown that applying magnetic field is an effective way to control the critical buckling loads and buckling mode shapes of cantilever nanoplates.

New analytic buckling solutions of rectangular thin plates with two free adjacent edges by the symplectic superposition method

This paper deals with a classic but very difficult type of problems, i.e., pursuing analytic buckling solutions of biaxially loaded rectangular thin plates with two free adjacent edges that are characterized by having both the free edges and a free corner. The primary challenge is to find the solutions satisfying both the governing high-order partial differential equations (PDEs) and non-Lévy-type boundary constraints. Here, an up-to-date symplectic superposition method is developed for the issues, which yields the analytic solutions by converting the problems to be solved into the superposition of two elaborated subproblems that are solved by the symplectic elasticity approach. The distinctive merit of the method is that a direct rigorous derivation helps to access the analytic solutions without any assumptions/prior knowledge of the solution forms, which is attributed to the implementation in the symplectic space-based Hamiltonian system rather than in the classic Euclidean space-based Lagrangian system. As the important outputs, comprehensive new analytic results are obtained, with 1200 critical buckling loads and 100 buckling mode shapes presented, which are all well validated by the refined finite element analysis. The rapid convergence and favorable accuracy of the present method make it competent as a benchmark one for similar problems.

Buckling Analysis of Tapered Continuous Columns by Using Modified Buckling Mode Shapes

Elastic critical buckling load of a column depends on various parameters, such as boundary conditions, material, and cross-section geometry. The main purpose of this work is to present a new method for investigating the buckling load of tapered columns subjected to axial force. The proposed method is based on modified buckling mode shape of tapered structure and perturbation theory. The mode shape of the damaged structure can be expressed as a linear combination of mode shapes of the intact structure. Variations in length in piecewise form can be positive or negative. The method can be used for single-span and continuous columns. Comparison of results with those of finite element and Timoshenko methods shows the high accuracy and efficiency of the proposed method for detecting buckling load.

Buckling strength of thin-walled steel cylindrical shells with stiffened plates

Thin-walled steel shell members are structurally superior as structural members. However, the thin-walled steel shell with stiffeners has been only studied in a part by reason of excelling dynamically on ordinary thin-walled shell without stiffeners. In this paper an investigation in regard of the increase of the buckling capacity on steel cylindrical shells with out- and inside stiffened plates is generated under pure axial compression. Then, so as to clarify the effect of the buckling capacity on thin-walled steel cylindrical shells with out- and inside stiffened plates, exact buckling strength and buckling mode shapes are generated by using the extended transfer matrix method. The buckling strength of cylindrical shell with out- and inside stiffened plates are obtained during the combination of half wave number (n) for perpendicular direction to axis and half wave number (m) for axis direction. Depending on circumstances, the local buckling strength of thin-walled shells with stiffened plates is sometimes failing by local buckling of stiffened plate to differ from local buckling of shell-panel in comparison with that of ordinary thin-walled shells without stiffener. Therefore, this study aims to plan increase of the local buckling capacity on the thin-walled steel cylindrical shells with out- and inside stiffened plates due to pure axial compression.

On the mechanical stability and buckling analysis of carbon nanotubes filled with ice nanotubes in the aqueous environment: A molecular dynamics simulation approach

In this article, molecular dynamics (MD) simulations are utilized to investigate the buckling behavior of carbon nanotubes (CNTs) containing ice nanotubes in the vacuum and aqueous environment. The obtained results show that unlike the critical strain, the critical buckling load of CNT containing ice nanotube is higher than that of pure CNT in the vacuum. It is also indicated that the sensitivity of critical buckling load and the critical strain of CNT containing ice nanotube to the variation of length decreases when the nanostructure is subjected to the aqueous environment. Additionally, it is observed that the calculated critical buckling load and the critical strain of CNTs filled with ice nanotubes in the aqueous environment are respectively larger and smaller than those obtained in the vacuum. It is further observed that CNTs lose their symmetric buckling mode shape as they are filled with ice nanotubes in the vacuum. The results of these simulations can be used as a benchmark for further studies in designing novel potential applications such as proton electronic-based nanoelectromechanical systems (NEMS).

Thermal Buckling Analysis and Optimization of Curvilinearly Stiffened Plates with Variable Angle Tow Laminates

This paper presents results from thermal buckling analysis and optimization of stiffened composite panels with variable angle tow (VAT) laminates and curvilinear stiffeners. Considering the meshing difficulties for curvilinearly stiffened VAT laminates due to both spatially dependent fiber path orientation and arbitrarily shaped stiffeners in the traditional finite element analysis, the present work models the plate and stiffeners separately without the need to place finite element nodes along the stiffener/plate and stiffener/stiffener interfaces. The displacement compatibility conditions are enforced at these interfaces in the finite element analysis. Convergence and verification study results show that the present method can accurately predict the thermal buckling behaviors of VAT laminates and curvilinearly stiffened VAT laminates. Parametric studies show that either VAT laminates or curvilinear stiffeners alone can improve the buckling performance as compared with using straight fiber laminates and straight stiffeners. VAT laminates and curvilinear stiffeners are found to redistribute the in-plane stress resultants and tailor the buckling mode shape, respectively, to improve the total buckling response. Optimization studies using curvilinear stiffeners and VAT laminates together for maximizing buckling temperature for a stiffened plate are conducted.

An Analysis of elastic and inelastic lateral torsional buckling of web-tapered I beams using the finite element method

Elastic and inelastic lateral torsional buckling of simply web tapered I beam is studied using the finite element method. The length of the beams is 8000 mm for elastic lateral torsional buckling and 4000 mm for inelastic lateral torsional buckling. The depth of cross section at one end is 500 mm and at the other end it is varied. The thickness of the flange is 16 mm and the thickness of the web is 10 mm. The section of the beams is compact. The beams are loaded by end moments. The ratio of end moments at one end and the moments at the other end are varied. The beams are assumed to have geometric imperfection and the distribution of imperfection follows the shape of the first buckling mode of elastic lateral torsional buckling of the beam. The amplitude of the imperfection is taken one milimeter at the top flange in lateral direction. The load is increased until the beams collapse. The ultimate load is considered as the critical moments of the beams. The results of the analysis are compared to nominal lateral torsional buckling moments using the method presented in the AISC Design Guide 25 (Design Guide for web tapered members). It can be concluded that usually the critical moments of the collapse analysis are close to the critical moments of the design guide. But sometimes it is less than the design guide and it means design guide is sometimes is not on the safe side.

Free vibration and buckling analyses of functionally graded annular thin sector plate in-plane loads using GDQM

In the present study, buckling and free vibration analyses of annular thin sector plate made of functionally graded materials (FGMs) resting on visco-elastic Pasternak foundation, subjected to external radial, circumferential and shear in-plane loads is investigated. Material properties are assumed to vary along the thickness according to an power law with Poisson\'s ratio held constant. First, based on the classical plate theory (CPT), the governing equation of motion is derived using Hamilton\'s principle and then is solved using the generalized differential quadrature method (GDQM). Numerical results are compared to those available in the literature to validate the convergence and accuracy of the present approach. Finally, the effects of power-law exponent, ratio of radii, thickness of the plate, sector angle, and coefficients of foundation on the fundamental and higher natural frequencies of transverse vibration and critical buckling loads are considered for various boundary conditions. Also, vibration and buckling mode shapes of functionally graded (FG) sector plate have been shown in this research. One of the important obtained results from this work show that ratio of the frequency of FG annular sector plate to the corresponding values of homogeneous plate are independent from boundary conditions and frequency number.

Accurate nonlinear buckling analysis of functionally graded porous graphene platelet reinforced composite cylindrical shells

By considering the pre-buckling effect and in-plane constraint, an accurate nonlinear buckling analysis of a functionally graded porous graphene platelet reinforced composite cylindrical shells under axial compressive load is performed. The stability equation is established according to a unified shell theory including the classical thin shell theory and the high-order shear deformation theory. Three types of porosity distributions and graphene platelet reinforced patterns are considered, and the modified Halpin–Tsai model and rule of mixtures are employed to determine their effective material properties. Explicit expressions of buckling equations for clamped or simply supported boundary conditions are obtained by the Galerkin's method. Highly accurate critical buckling loads and analytical buckling mode shapes are obtained simultaneously. A comparison between theoretical prediction and experiment is presented to verify the present method and very good agreement is reported. The influences of material properties on the buckling behaviors are also extensively investigated. It is recommended that the symmetric dispersion pattern is the optimal material distributions for both graphene platelets and porous, and the largest possible weight fraction, specific surface area and average thickness of graphene platelets could induce a better anti-buckling performance for the nanocomposite shell.

Ultimate strength formulations for FPSO stiffened panels under combined compression and shear with initial imperfections and damage

Abstract Accidental collision of striking objects, such as a supply vessel, into the side panel of a FPSO highly influences its ultimate strength assessment. An empirical formula for predicting the ultimate strength of damaged stiffened panels under combined loading of shear and longitudinal compression is empirically derived in this work based on curve fitting of quasi-static nonlinear finite element (FE) analyses. Initial imperfections are introduced by scaling the first buckling mode shape and the damage is caused by residual deformation from a rigid sphere indentation. A pure shear loading is applied at several levels followed by compression loading using the modified Riks method for a number of sphere indentation damages. The suggested formula for a typical FPSO stiffened side shell panel presented an excellent correlation with the nonlinear FE results and can be particularly useful in the preliminary design phase (no damage) and for a quick estimation of the panel residual strength with indentation damage.

Stability and failure analyses of delaminated composite plates subjected to localized heating

A layerwise B-spline finite element formulation is used to study the stability and first-ply failure of composite plates with embedded delaminations and subjected to localized thermal heating over the plate surface with uniform temperature rise through its thickness. The step functions are used to represent jump discontinuities in displacement fields at the zone of delamination between layers of composite plates. As the applied thermal loading is nonuniform, in the first step the prebuckling stress distribution within the plate is evaluated by solving the thermoelasticity problem. Subsequently, thermal buckling temperature is computed using the plate prebuckling stress distribution. Postbuckling response of the plate is obtained considering the von Kármán type geometric nonlinearity. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The localized thermal load at which the first-ply failure of the lamina occurs has been detected by Tsai-Wu quadratic interaction criterion. The effects of the area of heating region, delamination size and its position in the thickness direction and plate boundary condition on the composite plate critical buckling temperature and on first-ply failure load are reported. The present model can capture global, local and combined global–local buckling mode shapes.

The role of n-3 polyunsaturated fatty acids in the formation of the psychophysiological status of ski racers

A layerwise B-spline finite element formulation is used to study the stability and first-ply failure of composite plates with embedded delaminations and subjected to localized thermal heating over the plate surface with uniform temperature rise through its thickness. The step functions are used to represent jump discontinuities in displacement fields at the zone of delamination between layers of composite plates. As the applied thermal loading is nonuniform, in the first step the prebuckling stress distribution within the plate is evaluated by solving the thermoelasticity problem. Subsequently, thermal buckling temperature is computed using the plate prebuckling stress distribution. Postbuckling response of the plate is obtained considering the von Kármán type geometric nonlinearity. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The localized thermal load at which the first-ply failure of the lamina occurs has been detected by Tsai-Wu quadratic interaction criterion. The effects of the area of heating region, delamination size and its position in the thickness direction and plate boundary condition on the composite plate critical buckling temperature and on first-ply failure load are reported. The present model can capture global, local and combined global–local buckling mode shapes.

An optimization of composite lattice cylinder using the approximate method

Recently, the approximate methods based on continuous models have been developed to perform structural analysis of composite lattice structures due to their relative simplicity and computational efficiency. This paper defines the modified effective stiffness considering the directionally dependent mechanical properties to an intersection of ribs and mode shape function of a composite lattice cylinder. It subsequently presents an approximate method based on the continuous model of conducting a buckling analysis of the composite lattice cylinder with various boundary conditions under uniform compression. This method considers the coupled buckling mode as well as the global and local buckling modes. The validity of the present method is verified by comparing the results of the finite element analysis. In addition, a parametric analysis is performed to investigate the effects of the design parameters on the critical load and buckling mode shape of the composite lattice cylinder based on the present method. Finally, we apply the present method to perform the optimization of a composite lattice cylinder for a high-speed vehicle to minimize the mass. Consequently, it is concluded that the present method is very suitable to optimization of composite lattice cylinders due to their relative simplicity and computational efficiency.