On the finite element formulation of bifurcation mode of creep buckling of axisymmetric shells

In this paper, a finite element formulation is given in detail for the creep buckling of an axisymmetric shell. A special emphasis is placed on the bifurcation mode of creep buckling. A bifurcation point is determined by examining the shape of the potential energy in the vicinity of an axisymmetric equilibrium state obtained from a creep deformation analysis in the prebuckling stage. To illustrate the capability of the finite element formulation, a numerical example is presented for the creep buckling of a shallow spherical shell subjected to a uniform external pressure. In this analysis, not only the axisymmetric snap-through type but also the asymmetric bifurcation one are considered as buckling modes.     

A novel snap-through buckling behaviour of axisymmetric shallow shells with possible application in transducer design

In this study, the snap-through buckling behaviour of axisymmetric shells, subjected to axisymmetric horizontal peripheral load or displacement for various shell parameters and various boundary conditions, is investigated. Results obtained seem not to have been reported previously. An application of peripheral displacement type of loading is seen in metal-ceramic composite transducers developed by sandwiching a piezoelectric (PZT) ceramic between two metal end caps which serve as mechanical transformers for converting and amplifying the lateral displacement of the ceramic into an axial motion normal to the metal cap. In our numerical search, we have observed that snap-through and snap-back buckling is possible for shallow spherical caps for a very narrow range of the shell parameter used. When a hole is opened around the apex of the cap, buckling is possible for a larger range of the shell parameter. Obtaining the displacement amplification and the blocking or generative force for various material and geometric properties is necessary for the possible application of the findings in transducer design. The numerical results are presented in graphical forms.     

On compliance and buckling objective functions in topology optimization of snap-through problems

This paper deals with topology optimization of static geometrically nonlinear structures experiencing snap-through behaviour. Different compliance and buckling criterion functions are studied and applied for topology optimization of a point loaded curved beam problem with the aim of maximizing the snap-through buckling load. The response of the optimized structures obtained using the considered objective functions are evaluated and compared. Due to the intrinsic nonlinear nature of the problem, the load level at which the objective function is evaluated has a tremendous effect on the resulting optimized design. A well-known issue in buckling topology optimization is artificial buckling modes in low density regions. The typical remedy applied for linear buckling does not have a natural extension to nonlinear problems, and we propose an alternative approach. Some possible negative implications of using symmetry to reduce the model size are highlighted and it is demonstrated how an initial symmetric buckling response may change to an asymmetric buckling response during the optimization process. This problem may partly be avoided by not exploiting symmetry, however special requirements are needed of the analysis method and optimization formulation. We apply a nonlinear path tracing algorithm capable of detecting different types of stability points and an optimization formulation that handles possible mode switching. This is an extension into the topology optimization realm of a method developed, and used for, fiber angle optimization in laminated composite structures. We finally discuss and pinpoint some of the issues related to buckling topology optimization that remains unsolved and demands further research.     

Effect of non-linear modal interaction on the dynamic instability of axially excited cylindrical shells

The non-linear vibrations and instabilities of cylindrical shells under pulsating axial loads are investigated using Donnell’s shallow shell equations. Based on physical and mathematical reasoning, a meaningful low dimensional model is derived and used together with Galerkin method to obtain a set of coupled non-linear equations of motion. Particular attention is paid to the investigation of modal interaction between non-linear vibration modes with equal or nearly equal natural frequencies and to its influence on parametric instability and snap-through buckling. A parametric analysis using continuation techniques identifies the relevant bifurcations connected with the modal interaction and highlights their relevance in design.     

FASOR — a program for stress,buckling and vibration of shells of revolution

FASOR (Field Analysis of Shells of Revolution) is a user-oriented code for the analysis of stiffened, laminated axisymmetric shells. Very general shell geometries are allowed in that the reference surface meridian may form a branched, multi-circuit figure. Modes of response treated are linear asymmetric and geometrically nonlinear axisymmetric prebuckling, and asymmetric buckling and vibration under static axisymmetric loads. Bifurcation buckling under asymmetric loads is also treated by using a symmetrized prebuckling state based on the linear response of a user-specified meridian. For each mode of response, the user may specify any combination of orthotropic or anisotropic material properties with classical or transverse shear deformation shell theories.FASOR employs a numerical integration method (called the field method) whereby a numerically unstable linear boundary-value problem (all modes of response reduce to a sequence of such problems) is converted into two successive numerically stable initial-value problems. In this context, numerical stability means that round-off errors introduced at each step of the integration process tend to decay out. As a consequence, solution accuracy is controlled essentially by a single number, the truncation error tolerance, which is satisfied by automatically adjusting the size of each integration step. The field method thus eliminates the need for mesh generation required by finite element and finite difference methods, and the associated problem of numerical convergence. It also provides for automatic determination of response storage points so as to obtain a uniformly valid discrete approximation of the continuous response.In this paper the field method is briefly described, basic aspects of the mathematical model are discussed, the organization of input data is presented, and input and plot output are given for specific examples.     

Computation of lower-bound elastic buckling loads using general-purpose finite element codes

This paper investigates ways to have a computational implementation of a lower bound approach for the buckling of imperfection-sensitive shells using general purpose finite element codes. This approach was developed by Croll and others, and has been mainly employed by developing special purpose programs or analytical solutions. However, it is felt that this limits the possibilities of the user, and this shortcoming is addressed in the paper. First, the formulation is presented in a way to highlight what computations can be done following a reduced energy approach. Then, a methodology is implemented in conjunction with a general purpose program to compute the lower bound buckling load for cylindrical shells with different geometric configurations under uniform pressure. The accuracy of the procedure and the difficulties in the implementation, depending on the finite element chosen for the discretization are shown. Results demonstrate that the proposed reduced energy model can predict the lower bound load for cylindrical shells under uniform pressure distributions.     

Dynamic buckling of axisymmetric spherical caps with initial imperfections

Dynamic buckling loads are obtained for axisymmetric spherical caps with initial imperfections. Two types of loading are considered, namely, step loading with infinite duration and right triangular pulse. Solutions of perfect spherical caps under step loading are in excellent agreement with previous findings. Results show that initial imperfections do indeed have the effect of reducing the buckling capacity for both dynamic and static responses, although they are affected in a different manner. From the solutions obtained for triangular pulse situations, it is revealed that pulse duration has a very significant impact on the magnitude of the dynamic buckling load. When comparing these solutions with those of step loading, it is concluded that the step loading with infinite duration is the limiting case of a triangular pulse, and that the step loading provides the most severe loading situation for dynamic analysis.     

Dynamic buckling of orthotropic shallow spherical shells

The dynamic axisymmetric behaviour of clamped orthotropic shallow spherical shell subjected to instantaneously applied uniform step-pressure load of infinite duration, is investigated here. The available modal equations, based on an assumed two-term mode shape for the lateral displacement, for the free flexural vibrations of an orthotropic shallow spherical shell is extended now for the forced oscillations. The resulting modal equations, two in number, are numerically integrated using Runge-Kutta method, and hence the load-deflection curves are plotted. The pressure corresponding to a sudden jump in the maximum deflection (at the apex) is considered as the dynamic buckling pressure, and these values are found for various values of geometric parameters and one value of orthotropic parameter. The numerical results are also determined for the isotropic case and they agree very well with the previous available results. It is observed here that the dynamic buckling load increases with the increase in the orthotropic parameter value. The effect of damping on the dynamic buckling load is also studied and this effect is found to increase the dynamic buckling load. It is further observed that this effect is more pronounced with increase in the rise of the shell.     

Stability analysis of ring stiffened shells of revolution

A finite element formulation is presented for the general instability of ring stiffened shells of revolution subjected to external pressure. Linear bifurcation buckling theory is used. A rigorous derivation for the potential due to the hydrostatic loading including follower force effect is presented. Comparison with results obtained by earlier research workers in this field is given. Substantial reduction in buckling pressures due to follower force effect is reported.     

Elastic buckling of double walled cylindrical storage tanks—numerical and analytical estimates

This paper is concerned with estimating the elastic buckling pressures of large liquid natural gas (LNG) storage tanks which are used by the British Gas Corporation for seasonal demand peak shaving. They consist of two concentric ring stiffened cylindrical shells separated by substantial thermal insulation which maintains the LNG within the inner shell at − 165°C with minimal boil off. There is natural gas vapour above the LNG and throughout the tank interior which is normally at just above atmospheric pressure. The shell walls increase in thickness from the top to the bottom and are fabricated from very thin steel or aluminium alloy plates (diameter to thickness ratio ~4000 at the top) since they are usually in hoop tension, but under certain conditions this can become compressive making elastic buckling a possible mode of failure. The individual buckling pressures for the two shells can be estimated using standard procedures but in these LNG tanks the annular insulation transfers loads between the shells enhancing their individual strengths. A numerical method using the finite difference code BOSOR4 and a simple analytical method have been used to estimate these pressures.     

Influence of edge conditions on the modal characteristics of cross-ply laminated shells

The dynamic and static behavior of cross-ply laminated shells are investigated using the third-order shear deformation shell theory of Reddy. The theory is a modification of the Sanders shell theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the shell and does not require shear correction coefficients. The Lévy-type exact solutions for bending, buckling and natural vibration are presented for doubly curved, cylindrical and spherical shells under various boundary conditions.     

Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

     

Asymmetric buckling of bimodulus thick annular plates

An annular element with Lagrangian polynomials and trigonometric functions as shape functions is developed for asymmetric finite element stability analysis. The annular element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. Using the asymmetric finite element model, the asymmetric static buckling of bimodulus thick annular plates subjected to a combination of a pure bending stress and compressive normal stress is investigated. The obtained results of non-dimensional critical buckling coefficients are shown to be very accurate when compared with the exact solutions. The effects of various parameters on the buckling coefficients are studied. The bimodulus properties are shown to have significant influences on the buckling coefficients.     

Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps

This investigation deals with the axisymmetric static and dynamic buckling of a cylindricaliy orthotropic truncated shallow conical cap with clamped edge. The cases of conical caps with a free central circular hole and with a hole plugged by a rigid central mass have been considered. The governing equations are formulated in terms of normal displacement w and stress function Ψ. The orthogonal point collection method is used for spatial discretisation and the Newmark-β scheme is used for time-marching. Analysis has been carried out for a uniformly distributed conservative load normal to the undeformed surface and a central axial ring load at the hole. Dynamic load is taken as a step function load. The influence of orthotropic parameter β and annular ratio on the buckling loads has been investigated. New results for static and dynamic buckling loads have been presented for the isotropic and orthotropic truncated conical caps. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.     

Plastic collapse of circular conical shells under uniform external pressure

The article reports on two theoretical investigations and an experimental investigation into the collapse of six circular conical shells under uniform external pressure. Four of the vessels collapsed through plastic non-symmetric bifurcation buckling and one vessel collapsed through plastic axisymmetric buckling. A sixth vessel failed in a mixed mode of plastic non-symmetric bifurcation buckling, combined with plastic axisymmetric buckling. The theoretical and experimental investigations appeared to indicate that there was a link between plastic non-symmetric bifurcation buckling and plastic axisymmetric buckling. The theoretical investigations were via the finite element method and were used to provide a design chart for these vessels.     

Buckling optimization of fiber-composite laminate shells considering in-plane shear nonlinearity

The buckling strengths of fiber-composite laminate shells with a given material system are maximized with respect to fiber orientations using a sequential linear programming method together with a simple move-limit strategy. While a modified Riks nonlinear solution algorithm is utilized to analyse the buckling and postbuckling behaviour of composite shells, both linear and nonlinear in-plane shear formulations are employed to form the finite-element constitutive matrix for fiber-composite laminae. Results of the optimization study for simply supported composite cylindrical shells using both linear and nonlinear in-plane shear formulations are presented.     

Displacement dependent pressure loads in nonlinear finite element analyses

Often pressure loading is falsely identified as a nonconservative load leading automatically to nonsymmetric load stiffness matrices. The present paper discusses in detail the conditions when a pressure load is conservative and when it is not. The essential part is a clear classification of the load definition. Here either body attached or space attached loads are considered. The load stiffness matrices are derived for a pressure loaded curved surface in space. Several numerical examples are given; among these are linear and nonlinear buckling analyses of beams, rings and shells.     

轴向载荷作用下圆柱壳的屈曲仿真

本文采用非线性有限元方法，在经典刚塑性理论的基础上，考虑了材料非线性、大变形效应对材料的影响，利用大型通用程序LS—DYNA对圆柱壳轴向冲击屈曲过程进行了计算机仿真分析。分析表明，利用非线性有限元数值仿真方法，可以对圆柱壳屈曲这一复杂的力学过程进行真实有效的模拟再现。     

Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole

Nonlinear axisymmetric static and dynamic buckling of clamped isotropic and cylindrically orthotropic elastic cap with central circular hole have been investigated. The governing equations are expressed in terms of normal displacement ω and stress function ψ. The orthogonal point collocation method has been used in the space domain and Newmark-β scheme in the time domain. The cases of shallow cap with free hole and with a hole plugged by rigid central mass are considered. Analysis has been carried out for uniformly distributed load and a ring load at the hole. Dynamic load is taken as a step function load. Detailed new results for static and dynamic buckling loads have been presented for the isotropic and orthotropic cases.     

Buckling of pressure loaded rings and shells by the finite element method

The correct formulations for solving nonlinear structural problems by the finite element method have now been established. Numerous investigators have given the derivation for the solution of problems by the incremental tangent stiffness method and total formulation methods. These derivations have been applied to many problems and the results have been shown to be quite accurate for the problems that have been selected. However there is one area of application that has received practically no attention. This is in the investigation of the buckling strength of pressure loaded rings and shells. The effect of pressure loading where the loading changes direction as the structure deforms has been included in several previous derivations, by what is known as the load stiffness matrix, but to the author's knowledge no one has investigated problems where this effect has been included in the solution procedure. For rings and some buckling modes of shells, the results can be in error by as much as 50%.This paper will describe an iterative process for solving the nonlinear equilibrium equations and correcting the loads to include the effect of changing geometry at each load level. This approach is different from the classical eigenvalue or bifurcation method. Several case studies will be described which were performed on ring and shell problems. The geometry of these example problems were axisymmetric and in order to apply a nonlinear collapse analysis, the structure had to be perturbed out of its axisymmetric pattern into a buckling pattern. Imperfect geometry and very small concentrated loads were used to cause this perturbation and this will be described in the paper. The sensitivity of the computed collapse pressure to the finite element mesh gradation will be discussed. A comparison will be made between results obtained by including the effect of following pressure load and those obtained by not including this effect.