Prebuckling Deformations Research Articles

Nonlinear torsional buckling and postbuckling analysis of cylindrical silicon nanoshells incorporating surface free energy effects

In the present study, a size-dependent shell model is developed which can afford to describe the nonlinear torsional buckling and postbuckling characteristics of cylindrical nanoshells in the presence of surface stress effects. To accomplish this purpose, the Gurtin–Murdoch theory of elasticity together with the von Karman geometric nonlinearity is implemented into the first-order shear deformation shell theory. A linear variation through the thickness is considered for the normal stress component of the bulk to satisfy the balance conditions on the free surfaces of the nanoshell. By means of the virtual work principle, the non-classical governing differential equations are constructed in which the transverse displacement and Airy stress function are considered as independent variables. Thereafter, a boundary layer theory is employed including the effect of surface stress in conjunction with the nonlinear prebuckling deformations and the large postbuckling deflections. Subsequently, an efficient solution methodology based on an improved perturbation technique is put to use to obtain the size-dependent critical torsional buckling loads and the associated postbuckling equilibrium paths. It is observed that the torsional load exhibits a significant increase after reaching the minimum postbuckling load. Also, it is revealed that the effect of surface stress becomes negligible at high values of the deflection.

Buckling of spinning functionally graded graphene reinforced porous nanocomposite cylindrical shells: An analytical study

This paper investigates the buckling behavior of functionally graded graphene reinforced porous nanocomposite cylindrical shells with spinning motion and subjected to a combined action of external axial compressive force and radial pressure. The weight fraction of graphene platelet (GPL) nanofillers and porosity coefficient are constant in each concentric cylindrical shell but vary layer-wise through the thickness direction, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the shell thickness. The first-order shear deformation theory incorporated with the von Kármán's geometrical nonlinearity is employed to describe the pre-buckling deformation. The governing equations of the cylindrical shell are established by using the minimum potential energy principle where the centrifugal effect due to the spinning motion of the cylindrical shell is considered. The pre-buckling deformation is derived by adopting the Galerkin method, then the axial critical buckling force, radial critical buckling pressure and critical buckling hydrostatic pressure are obtained with the effect of pre-buckling deformation being taken into account. Special attention is given to the effects of the porosity coefficient, the weight fraction, the dispersion pattern, the geometrical size of the GPL and spinning speed of the cylindrical shell on the pre-buckling deformation and different types of critical buckling loads of the porous nanocomposite cylindrical shell.

Distortional lateral torsional buckling of beam-columns including pre-buckling deformation effects

The present study develops a unified distortional lateral torsional buckling finite element formulation for the elastic analysis of beam-columns with wide flange doubly symmetric cross-sections. The solution captures several non-conventional features including the softening effect due to web distortion, the stiffening effect induced by pre-buckling deformations, the pre-buckling nonlinear interaction between strong-axis moments and axial forces, the contribution of pre-buckling shear deformation effects, the destabilizing effects due to loads being offset from the shear centre, and the presence of transverse stiffeners on web distortion. In the present theory, it is possible to evoke/suppress any combination of features and thus isolate the individual contribution of each effect or quantify the combined contributions of multiple effects. The effects of beam span-to-depth, flange width-to-thickness, web height-to-thickness, and flange width-to-web height ratios on the critical moments are investigated. Comparisons with conventional lateral torsional buckling solutions that omit distortional and pre-buckling effects quantify the influence of distortional and/or pre-buckling deformation effects. The theory is also used to investigate the influence of P-delta effects of beam-columns subjected to transverse and axial forces on their lateral torsional buckling. The solution is adopted to quantify the beneficial effects of transverse stiffeners in controlling/suppressing web distortion in beams.

Imperfection Sensitivity: A Review of Buckling Behavior of Cones, Cylinders, and Domes

It is generally accepted that the presence of imperfections in pressure vessel components can significantly reduce their buckling strength. In fact, the discrepancies between theoretical predictions and experimental results have been attributed to various kinds of existing and unavoidable imperfections. This is not a new problem but despite of substantial research effort in this area over the recent decades, it is far from being satisfactorily resolved. This review provides insight into the past findings and current activities related to the role of different types of imperfections on the buckling strength. It aims to contribute to a better understanding of the influence of imperfections on the structural stability of cones, cylinders, and domes when these are subjected to external loading conditions. The review concentrates not only on the prominent role of initial geometric imperfections of the shell's generator but also on less known defects. This includes uneven axial length of cylinders, eccentricities, and nonuniformities of applied load, inaccurately modeled boundary conditions, corrosion of the wall, influence of material discontinuity or crack, and effect of prebuckling deformation. The study examines: (i) how the data were obtained (analytically, experimentally, and/or numerically), (ii) the type of material from which the shell structures were made, and (iii) the importance of findings of the previous works. Metallic and composite components are considered.

Simplified expressions for elastic lateral torsional buckling of wooden beams

A beam finite element formulation is developed for the elastic lateral torsional buckling analysis of wood beams with rectangular cross-sections. The formulation accounts for moment gradient, load height, and pre-buckling deformation effects. The validity of the finite element and underlying variational principle is established through comparisons with critical moments obtained by 3D FEA simulations and experimental results. The validated variational expression is then adopted to develop approximate analytical expressions that separately characterize the effects of moment gradient, partial twist restraints that may exist at beam-ends, load height, and pre-buckling deformation effects. Dimensionless coefficients were formulated to characterize each effect for simple beams and cantilevers under common loading scenarios. The derived coefficients are then consolidated into a unified framework that accounts for all effects combined to predict the elastic lateral torsional buckling capacity of wooden beams.

Nonlinear Theory of Sandwich Shells with a Transversely Soft Core Containing Delamination Zones and Edge Support Diaphragm

Sandwich shells with transversely soft cores and skin layers supported along the edges by a thin deformable diaphragm with a ruled middle surface are considered. Delamination zones are assumed on the contact surfaces between the core and skin layers. The refined geometrically nonlinear theory has been constructed for these type of structures at small deformations and moderate displacements, which can describe prebuckling deformation and reveal all possible buckling mode shapes of skin layers (antisymmetric, symmetric, mixed bending and mixed bending-shear, as well as arbitrary mode shapes comprising all listed modes) and of the reinforcing diaphragm. This theory is based on the introduction of interaction forces between the core and skin layers and between the core and reinforcing diaphragm at each point of the contact surfaces as unknowns. The previously proposed variant of the generalized Lagrangian variational principle has been utilized in the derivation of the governing equations, static boundary conditions for the shell and reinforcing diaphragm, and kinematic coupling conditions between the core and skin and between the core and support diaphragm.

Postbuckling behavior of shear deformable anisotropic laminated cylindrical shell under combined external pressure and axial compression

Structural design of composite shells are more challenging than conventional metals due to the complex mechanical behavior and damage mechanisms which composite materials exhibit. Postbuckling analysis for a moderately thick anisotropic laminated cylindrical shell subjected to combined loadings of external pressure and axial compression is presented which extends the boundary layer theory of shell buckling. The governing equations are based on Reddy’s higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity. Both nonlinear prebuckling deformations and initial geometric imperfections of the shell are taken into account. A two-step singular perturbation method is used to determine interactive buckling loads and postbuckling equilibrium paths. A verification study is conducted, and the validity of the formulation is established through comparison with results of nonlinear finite element software such as ABAQUS®. The internal physical mechanism of the shell geometric parameters on the buckling load and the postbuckling equilibrium path is obtained. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different load-proportional parameters. The analytical model can provide an effective tool to investigate postbuckling of composite shell structures.

Out-of-plane creep buckling analysis on slender concrete-filled steel tubular arches

Concrete-filled steel tubes (CFST) are becoming a popular structural solution for arch bridges because of their high compressive strength and efficiency in construction. For long span CFST arch bridges, the time-dependent behaviour of the core concrete may affect the stability of CFST arches. Despite this, only limited research has been carried out to date on their creep buckling behaviour. In this context, this study aims to investigate the influence of the prebuckling deformation induced by time effects on the out-of-plane stability of single parabolic CFST arches with fixed ends and subjected to uniformly distributed loads applied along the span by means of the finite element method using ABAQUS. The time-dependent behaviour of the concrete has been described using the Eurocode 2 model and implemented in the analysis using the integral type creep law. The nonlinear material property and the confinement effects under ultimate condition have also been taken into account and implemented in ABAQUS with UMAT subroutines. The accuracy of the proposed analysis method has been validated against the experimental results of an out-of-plane buckling test of a 1:10 scaled CFST arch reported in the literature. An extensive parametric study has been then carried out and it has been found that the ultimate capacity of the arches can be decreased by up to 18% due to the prebuckling deformation induced by time effects. Finally, designing equations are proposed based on the finite element analysis results to predict the ultimate loads of CFST arches accounting for time effects.

Buckling of bimorph functionally graded piezoelectric cylindrical nanoshell

Functionally graded piezoelectric materials are of interest for a wide variety of nanotechnological applications. Buckling of functionally graded piezoelectric nanotubes, as ubiquitous event, recently attracted increasing interest. Also, in this article, the buckling response of bimorph functionally graded piezoelectric cylindrical nanoshells attended to lateral pressure is investigated on the basis of nonlocal strain gradient theory. The formulation developed on the basis of the first-order shear deformation theory with von Kármán kinematic nonlinearity. The material properties are considered to vary across thickness direction according to power law distribution. The electric potential is considered to be quadratic through thickness direction. Due to bifurcation analysis, the prebuckling deformation is initially investigated; then, in the case of simply supported edge condition, the buckling response of a bimorph functionally graded piezoelectric nanotube is studied and the influences of various parameters on buckling pressure are illustrated.

Nonlinear vibrations of pre- and post-buckled lipid supramolecular micro/nano-tubules via nonlocal strain gradient elasticity theory

The unique geometry with high surface ratio makes lipid micro/nano-tubules as an excellent self-assembled supramolecular structure in various biological applications such as controllable release systems and drug delivery. In the present study, the size-dependent nonlinear vibrations of axially loaded lipid micro/nano tubules associated with the both prebuckling and postbuckling domains are explored comprehensively. To accomplish this purpose, the nonlocal strain gradient theory of elasticity including simultaneously two entirely different features of size dependency is utilized within the framework of the third-order shear deformable beam model. With the aid of Hamilton's principle, the non-classical governing differential equations of motion are established incorporating the nonlinear prebuckling deformations and the large postbuckling deflections. At the end, the Galerkin method in conjunction with an improved perturbation technique is employed to initiate explicit analytical expressions for nonlocal strain gradient nonlinear frequency of pre- and post-buckled lipid micro/nano-tubules. It is seen that by taking the nonlocal size effect into consideration, the influence of geometrical parameters of the lipid micro/nano-tubule on the nonlinear vibration characteristics within the both prebuckling and postbuckling domains decreases and the frequency-deflection curves are more close to each other. However, the strain gradient size dependency has an opposite effect and leads to increase the gap between the frequency-deflection curves of axially compressed lipid micro/nano-tubules with different geometrical parameters.

Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells

Abstract The present study deals with the size-dependent nonlinear buckling and postbuckling characteristics of magneto-electro-elastic cylindrical composite nanoshells incorporating simultaneously the both of hardening-stiffness and softening-stiffness size effects. To accomplish this purpose, the nonlocal strain gradient elasticity theory is applied to the classical shell theory. Via the virtual work's principle, the size-dependent governing differential equations are constructed including the coupling terms between the axial mechanical compressive load, external magnetic potential and external electrical potential. The nonlinear prebuckling deformations and the large postbuckling deflections are taken into consideration based upon the boundary layer theory of shell buckling. Finally, an improved perturbation technique is employed to achieve explicit analytical expressions for nonlocal strain gradient stability curves of magneto-electro-elastic nanoshells under various surface electric and magnetic voltages. It is seen that a positive electric potential and a negative magnetic potential cause to increase both of the nonlocality and strain gradient size dependencies in the nonlinear instability behavior of axially loaded magneto-electro-elastic composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role.

Thermal buckling response of laminated and sandwich plates using refined 2-D models

This paper discusses the thermal buckling analysis of composite plates and sandwich panels by means of a Ritz-based variable-kinematic formulation. Main feature of the proposed formulation consists in the representation of the structure by means of sublaminates, i.e. arbitrary groups of plies composing the panel. Each sublaminate is associated with an independent, arbitrary kinematic description, so that the use of refined, high-order theories can be restricted to specific regions, such as the core of sandwich panels. Monolithic plates can be studied as a special case where the structure is modeled using only one sublaminate. Presented are the critical temperatures, with and without accounting for the nonlinear pre-buckling effects, for a set of monolithic and sandwich configurations. When pre-buckling effects are neglected, the problem is solved as a standard eigenvalue problem. On the other hand, the introduction of pre-buckling effects leads to a nonlinear eigenvalue problem, which is solved with an iterative procedure. The results are validated against 3D solutions, and highlight the importance of accounting for pre-buckling deformations, especially in the case of sandwich panels. As demonstrated, high-fidelity predictions are obtained while keeping at minimum the amount of degrees of freedom.

Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory

Microtubules including tubulin heterodimers arranging in a parallel shape of cylindrical hollow plays an important role in the mechanical stiffness of a living cell. In the present study, the nonlocal strain gradient theory of elasticity including simultaneously the both nonlocality and strain gradient size dependency is put to use within the framework of a refined orthotropic shell theory with hyperbolic distribution of shear deformation to analyze the size-dependent buckling and postbuckling characteristics of microtubules embedded in cytoplasm under axial compressive load. The non-classical governing differential equations are deduced via boundary layer theory of shell buckling incorporating the nonlinear prebuckling deformation and microtubule-cytoplasm interaction in the living cell environment. Finally, with the aid of a two-stepped perturbation solution methodology, the explicit analytical expressions for nonlocal strain gradient stability paths of axially loaded microtubules are achieved. It is illustrated that by taking the nonlocal size effect into consideration, the critical buckling load of microtubule and its maximum deflection associated with the minimum postbuckling load decreases, while the strain gradient size dependency causes to increase them.

Thermo-electro-mechanical size-dependent postbuckling response of axially loaded piezoelectric shear deformable nanoshells via nonlocal elasticity theory

The current study addresses the size-dependent nonlinear thermo-electro-mechanical instability of piezoelectric nanoshells under a combined action of axial compressive load, lateral electric field and uniform changes in temperature. The non-classical formulations given herein are built upon the nonlocal elasticity theory within the framework of the shear deformation shell theory. On the basis of the minimum potential energy of the system and using a boundary layer theory of shell buckling, the nonlocal-based governing equations are deduced incorporating the nonlinear prebuckling deformations and initial geometric imperfection sensitivity. After that, a perturbation-based solution methodology is put to use in order to anticipate the size-dependent thermo-electro-mechanical postbuckling response of piezoelectric nanoshells corresponding to different nonlocal parameters, applied electric voltages and temperature changes. It is displayed that for the both local and nonlocal models with and without initial geometric imperfection, a lateral electric field coming from a positive applied voltage leads to increase the axial stiffness of piezoelectric nanoshell in such a way that the critical load and width of postbuckling domain increase, but no considerable change occurs in the minimum load of the postbuckling regime.

Nonlinear buckling and postbuckling behavior of cylindrical shear deformable nanoshells subjected to radial compression including surface free energy effects

The objective of the present investigation is to predict the nonlinear buckling and postbuckling characteristics of cylindrical shear deformable nanoshells with and without initial imperfection under hydrostatic pressure load in the presence of surface free energy effects. To this end, Gurtin-Murdoch elasticity theory is implemented into the first-order shear deformation shell theory to develop a size-dependent shell model which has an excellent capability to take surface free energy effects into account. A linear variation through the shell thickness is assumed for the normal stress component of the bulk to satisfy the equilibrium conditions on the surfaces of nanoshell. On the basis of variational approach and using von Karman-Donnell-type of kinematic nonlinearity, the non-classical governing differential equations are derived. Then a boundary layer theory of shell buckling is employed incorporating the effects of surface free energy in conjunction with nonlinear prebuckling deformations, large deflections in the postbuckling domain and initial geometric imperfection. Finally, an efficient solution methodology based on a two-stepped singular perturbation technique is put into use in order to obtain the critical buckling loads and postbuckling equilibrium paths corresponding to various geometric parameters. It is demonstrated that the surface free energy effects cause increases in both the critical buckling pressure and critical end-shortening of a nanoshell made of silicon.

Asymmetric thermo-inertial buckling of annular plates

The present research deals with the nonaxisymmetric buckling behaviour of isotropic homogeneous annular plates subjected to simultaneous effects of uniform temperature rise and constant angular speed. First-order shear deformation plate theory is used to obtain the complete set of governing equations and the associated boundary conditions. Pre-buckling deformations and stresses of the plate are obtained using the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The resulting equations are solved using a hybrid method, including the exact trigonometric functions through the circumferential direction and generalised differential quadrature method through the radial direction. The resulting eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Numerical results reveal that, only for annular plates with exterior edge clamped, rotation may enhance the critical buckling temperature of a plate under special circumstances. Furthermore, asymmetric stability analysis should be performed to extract the critical state and buckled shape of a rotating annular plate subjected to uniform heating. Otherwise, the critical buckling temperature is overestimated and the buckling pattern is wrongly predicted.

Geometrically Nonlinear Behavior of Axisymmetric Thin Spherical Shells

This article presents a geometrically nonlinear formulation of a two node axisymmetric shell element. The geometrically nonlinear formulation is based on the Total Lagrangian approach and the material behavior is assumed to be linearly elastic. The spherical arc-length procedure is used to obtain the pre-buckling, buckling and post-buckling deformation path. Some numerical examples are solved to demonstrate geometrically nonlinear behavior of elastic thin spherical shells subjected to various loads.

Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments

In this work, modeling and analysis for size-dependent buckling and postbuckling behavior of cylindrical nanopanels under axial compression in thermal environments are presented. To this end, a non-classical panel model based upon the Gurtin–Murdoch elasticity theory in conjunction with the first-order shear deformation shell theory is developed which considers efficiently the effects of surface free energy. In order to capture the large deflections, the von Karman geometrical nonlinearity is taken into account. The size-dependent governing differential equations are deduced to a boundary layer type that incorporates the nonlinear prebuckling deformations and the large postbuckling deflections. Afterwards, by means of an efficient solution methodology using a perturbation solving process, the size-dependent critical buckling loads and associated postbuckling equilibrium paths are obtained corresponding to various geometrical parameters and thermal environments. It is seen that for the both classical and non-classical panel models, the wide of snap-through phenomenon is larger for nanopanels with higher ratio of length to width. Also, it is found that thermal environment has no influence on the value of minimum load of the postbuckling domain and associated maximum deflection.

Localized Lateral Thermal Buckling of Pipelines With a Circular Layout

A numerical strategy is developed and used to investigate the localized lateral buckling of circular pipelines under thermal loading and friction. The constitutive relations for circular pipelines are derived for thermal stresses and finite strain based on a hyperelastic constitutive model. The prebuckling lateral expansion and localized postbuckling deformation are investigated. A critical included angle for circular profiles is studied. Beyond the critical included angle, increasing the included angle of the pipeline or changing the boundary conditions does not influence the localized buckling behavior. Parametric studies are performed and the results are validated with ansys.

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical first-order shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.