Effects Of Prebuckling Deformations Research Articles

Buckling of Bulk Structures With Finite Prebuckling Deformation

Abstract The prebuckling deformation of structures is neglected in most of the conventional buckling theory (CBT) and numerical method (CNM), because it is usually very small in conventional concepts. In the preceding paper (Su et al., 2019), we found a class of structures from the emerging field of stretchable electronics, of which the prebuckling deformation became large and essential for determining the critical buckling load, and developed a systematic buckling theory for 3D beams considering the effects of finite prebuckling deformation (FPD). For bulk structures that appear vastly in the advanced structures, a few buckling theories consider the effects of the prebuckling deformation in constitutive equations by energy method, which are significantly important but not straightforward and universal enough. In this paper, a systematic and straightforward theory for the FPD buckling of bulk structures is developed with the use of two constitutive models. The variables for the prebuckling deformation serve as the coefficients of the incremental displacements, deformation components, and stress in the buckling analysis. Four methods, including the CBT, CNM, DLU (disturbing-loading-unloading method) method and FPD buckling theory, are applied to the classic problems, including buckling of an elastic semi-plane solid and buckling of an elastic rectangular solid, respectively. Compared with the accurate buckling load from the DLU method, the FPD buckling theory is able to give a good prediction, while the CBT and CNM may yield unacceptable results (with 70% error for the buckling of an elastic semi-plane solid).

In-plane nonlinear buckling analysis of circular arches considering shear deformation

A new theory for the buckling and nonlinear analysis of circular arches with shear deformation is developed, the theory is based on the variational principle in which the nonlinear strain energy of all stress components are included. For arches subjected to a uniform constant-directed radial load, internal forces and deformations including shear deformation are determined, then the buckling of arches only considering uniform axial forces and incorporating the effect of pre-buckling deformations and all internal forces are respectively carried out, with the results of the latter are fully validated by FE predictions, thus the proposed theory is verified. The buckling of arches with shear deformation is then viewed as an interactive buckling between flexural and shear buckling, Simplified formulas for the critical loads and the critical axial forces of arches for 4 types of loads are presented, the comparison shows excellent agreement between the proposed formulas and FE analysis.

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.

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.

Out-of-Plane Buckling of Microstructured Beams: Gradient Elasticity Approach

AbstractThis paper deals with the lateral-torsional buckling of elastic microstructured beams. A linear gradient elasticity approach is presented. The elastic constitutive law is linear for the generalized beam model, expressed with first- and second-order moment variables in a sixth-order dimensional stress-strain space. The boundary conditions (including the higher-order boundary conditions) are derived from application of the variational principle applied to second-grade elastic models. The effect of prebuckling deformation is taken into consideration based on the Kirchhoff-Clebsch theory. Some analytical solutions are presented for a hinged-hinged strip beam. The lateral-torsional buckling moment is sensitive to small length terms inherent in the microstructured constitutive law. It is shown that the gradient parameter tends to increase the critical lateral-torsional buckling moment, a conclusion different from the one obtained with an Eringen-based nonlocal model. This tendency is consistent with tha...

Lateral–torsional buckling analysis of thin-walled beams including shear and pre-buckling deformation effects

In this paper, lateral–torsional buckling behavior of open-section thin-walled beams is investigated based on a geometrically nonlinear formulation, which considers the effects of shear deformations. A finite element numerical solution along with an incremental-iterative solution procedure is adopted to trace the pre-buckling as well as the post-buckling equilibrium paths. Formulation is applicable to a general type of open-section and load position effects are also included. Numerical results are validated through comparisons with experimental results and those based on other formulations presented in the literature. Comparisons have also been made between the results based on fully nonlinear analysis and linearized buckling analysis in order to illustrate the effects of pre-buckling deformations as well as the shear deformations on the buckling load predictions. Examples illustrate the influence of beam slenderness and moment gradient on the effects of pre-buckling deformations in predicting bucking loads.

On Lateral-Torsional Buckling of Non-Local Beams

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microor nanostructures. This paper deals with the lateral-torsional buckling of elastic nonlocal small-scale beams. Eringen's model is chosen for the nonlocal constitutive bendingcurvature relationship. The effect of prebuckling deformation is taken into consideration on the basis of the Kirchhoff-Clebsch theory. It is shown that the application of Eringen's model produces small-length scale terms in the nonlocal elastic lateraltorsional buckling moment of a hinged-hinged strip beam. Clearly, the non-local parameter has the effect of reducing the critical lateral-torsional buckling moment. This tendency is consistent with the one observed for the in-plane stability analysis, for the lateral buckling of a hinged-hinged axially loaded column. The lateral buckling solution can be derived from a physically motivated variational principle.

Structural optimization for specified nonlinear buckling load factor

The author’s method of design sensitivity analysis of nonlinear coincident buckling load factors and corresponding optimization method of finite dimensional elastic structures are shown to be applicable to a structure with moderately large number of degrees of freedom. The sensitivity formulations are written in general form for coincident buckling load factors including limit points and bifurcation points. Optimum designs are found for a spherical latticed dome subjected to a concentrated load and distributed loads, respectively. The results are compared with those by linear eigenvalue formulation. An approximate optimal solution against nonlinear buckling can be obtained by only scaling the solution under linear eigenvalue formulation even for the case where the effect of prebuckling deformation is very large. It is shown that the proposed method is practically applicable to optimum design with coincident critical points especially for the case of hill-top branching where bifurcation points are located at the limit point. In this case, imperfection sensitivity is not enhanced as a result of optimization. The reduction of maximum load factor due to a minor imperfection is shown to be in the same order as that to a major imperfection. Therefore symmetric imperfection as well as asymmetric imperfection should be considered in estimating the maximum loads.

Anisotropic effects in the compression buckling of laminated composite cylindrical shells

This paper addresses two aspects of buckling relevant to design. The first describes the preliminary experimental results that validate the presence of a newly reported spiral buckling mode; whilst the second concerned the effects of extension/twist anisotropy on the buckling loads of cylindrical shells under compression. The influence of designing laminates with antisymmetric lay-ups in comparison with a symmetrical lay-up is also investigated. Antisymmetric quasi-isotropic laminates, based on 0, 90 and ±45° angles produces greater buckling loads than the equivalent symmetric lay-up. Whilst all anisotropic coupling effects appear to lower the buckling loads, a laminate that is 48 plies thick is necessary to eliminate them in thin-walled shells. Such a laminate is at least 6 mm thick. Many designs require thinner laminates, and in so doing, mitigate the need for guidelines on efficient anisotropic lay-ups. Neglecting the effect of prebuckling deformation, it is found that extension/twist coupling is less deleterious than flexural/twist coupling in this respect. Therefore, antisymmetric laminates appear preferable for initial buckling of quasi-isotropic laminates.

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.

Elastic Buckling of Circular Plates Allowing for Prebuckling Deformation

The effect of prebuckling deformation on the elastic buckling of circular plates is studied. Based on Trefftz's initial stress theory, the energy functional is obtained from which the governing differential equation for equilibrium can be derived via calculus of variations. Exact and simple approximate closed-form critical- load expressions are obtained after solving the two-point boundary value problem. The effect of the nonlinear terms involving in-plane displacements in the Green-Lagrange strain expressions on the solutions is also investigated. For circular plates with relatively large thickness-to-radius ratios, the allowance for prebuckling deformation affects the critical load of clamped plates more than simply supported plates. This occurs because the prebuckling deformation is greater in clamped plates, since they undergo higher loads before buckling.

Inelastic Asymmetric Buckling of Ring-Stiffened Cylindrical Shells under External Pressure

A variational principle and associated criteria on bifurcation and stability of elastic-plastic solids, which are expressed in terms of Lagrangian variables, are described. The principle is employed to study the effects of prebuckling deformation and yield surface corners on the inelastic asymmetric buckling of a ring-stiffened cylindrical shell under external pressure. The shell is made of a work-hardening material having incremental type constitutive relationships associated with either Mises' yield surface or a yield surface with corners. An incremental Rayleigh-Ritz numerical method is employed to determine the nonlinear deformation process and asymmetric bifurcation pressure. Numerical results are obtained and compared with available experimental results

Inelastic buckling process of axially compressed cylindrical shells subject to edge constraints

In this paper the effects of prebuckling deformation caused by edge constraints, and non linear strain displacement relationships, on the inelastic buckling process of axially compressed cylindrical shells are studied. By Drucker's postulate of positive work in plastic deformation, a modified incremental theory of plasticity on the stress-strain relationships expressed in terms of Kirchhoff stress and Green strain rates is developed. A variational principle in the Lagrangian description and for quasi-static problems of finite plasticity is developed, and the existence of an extremum principle for a material having a sufficiently great hardening rate is shown. In addition a criterion for the stability of a body under dead load is established.The variational principle in conjunction with the incremental Rayleigh-Ritz technique is used to determine the deformation process of an idealized cylindrical shell composed of four thin load carrying sheets made of a general work hardening material. The theoretical predictions obtained numerically by digital computer compare favorably with available experimental results.

円筒殻の圧縮座屈荷重に対する座屈前変形の影響

An analytical investigation is carried out to determine the effect of prebuckling deformation on the compressive buckling load of cylindrical shells. The boundary condition of one end is assumed to be clamped, and a semi-infinite approach is used to clarify the essential property of the effect. A finite difference approach and electronic digital computers are used to obtain the solution. The present paper is a continuation of the author's previous paper to discuss the accuracy in the numerical calculation. The effects of the interval of the finite difference, the number of the points and the significant figures of the electronic computer are discussed. As a conclusion, the author's previous result that the buckling stress is 0.9268×(the classical buckling stress) is confirmed.