Bifurcation and snap-through phenomena in asymmetric dynamic analysis of shallow spherical shells

The asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration is investigated. The solution of a linear eigenvalue problem yields the bifurcation paths and also the lower bound for the asymmetric dynamic snap-through buckling pressure. The asymmetric dynamic response of shells with a shape imperfection is studied. The asymmetric dynamic snap-through buckling load is defined to be the threshold value of the step pressure at which the asymmetric response shows significant growth rate. The snap-through buckling loads are obtained for a few shell parameters. The numerical results are compared with the available experimental results and they are in good agreement. Finally, a preliminary study of the phase planes is presented.     

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.     

On load interaction in the non linear buckling analysis of cylindrical shells

The elastic stability of shells or shell-like structures under two independent load parameters is considered. One of the loads is associated to a limit point form of buckling, whereas the second is a bifurcation. A simple one degree of freedom mechanical system is first investigated, for which an analytical solution is possible. Next, a cylindrical shell under the combined action of axial load and localised lateral pressure is studied via a non linear, two-dimensional, finite element discretization. It is shown that both problems display the same general behaviour, with a stability boundary in the load space which is convex towards the region of stability. The results show the need of performing a full non-linear analysis to evaluate the stability boundary for the class of interaction problems considered.     

Buckling analysis with the aska program system

The theoretical and practical considerations involved in extending the very large, general purpose finite element program system ASKA to include linear buckling mode analysis are presented. The main analysis features and their software implementation are discussed in the context of modular processors and data operands. A broader interpretation of buckling load factors and modes in relation to mode superposition is given, and typical applications are presented to illustrate both classical bifurcation buckling analyses and buckling mode superposition response.     

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.     

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.     

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.     

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $Q_1$ element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.     

Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion

This paper investigated the general instability of cylindrical shells in which the stiffeners formed spirals along the length and at an arbitrary angle with the axis. Two loading conditions were considered: uniform axial and lateral compressions and torsion. The stress-strain relations of the stiffeners were developed by rotation of the strain tensor. The buckling determinate was obtained by introducing into the equilibrium equations the admissible displacement functions consistent with the end constraints, thereby enforcing equilibrium by satisfying the characteristic equations.

The buclking equations were programmed for a computer which rearched through a finite set of stress resultants for assigned values of spiral angle and modes and printed out the buckling load. The optimum structure weight of the stiffened shell was determined by iterating the design parameters at the required spiral angle so that the buckling load approached the applied load as a limit until the difference between these loads was within the design allowance.     

Lateral-torsional buckling of singly symmetric web-tapered thin-walled I-beams: 1D model vs. shell FEA

This paper assesses the global performance and the underlying assumptions of a recently developed one-dimensional model characterising the elastic lateral-torsional buckling behaviour of singly symmetric tapered thin-walled open beams, which is able to account for the influence of the pre-buckling deflections. A comparative study involving the critical load factors and buckling modes yielded by (i) the one-dimensional model and (ii) two-dimensional shell finite element analyses (reference results) is presented and discussed. The results concern I-section cantilevers and simply supported beams (i) with uniform or linearly tapered webs, (ii) equal or unequal uniform flanges and (iii) acted by point loads applied at the free end or mid-span sections, respectively. In general, the one-dimensional predictions are found to agree well with the shell finite element results. Some significant discrepancies are also recorded (for the shorter beams), which are due to the occurrence of relevant cross-section distortion or localised buckling phenomena.     

Asymptotic study of the elastic postbuckling behavior of structures by the finite element method

Within the framework of finite element analysis, an asymptotic method is presented for the study of geometrically nonlinear static behavior of thin structures under one-parameter conservative loading. The method can be applied when the prebuckling behavior is moderately non-linear so that the bifurcation analysis is no longer accurate enough. An iterative process makes it possible to find suitable deformation modes which enable a good approximation of the structural behavior around the buckling point. As in the Rayleigh-Ritz approach, a reduced energy can be formulated with only a small number of generalized degrees of freedom. The influence of small initial imperfections on the buckling load can easily be analyzed as in Koiter's asymptotic method. A comprehensive treatment and several improvements of this asymptotic iterative method are given and selected examples illustrate the basic features of the method.     

Buckling analysis of ring-stiffened oval cylindrical shells

An energy principle is employed to derive the equations governing the stability of a simply-supported, eccentrically ring-stiffened, oval, orthotropic cylindrical shell. The kinematic relations used are those of Love-type shell theory and the effect of reinforcing rings is accounted for by a distributed stiffness approach. The cylinder is subjected to a combination of uniform axial and lateral pressures.

It is determined that the domain of stability of such a stiffened cylinder is bounded by two distinct solutions, herein denoted as corresponding to ‘long’ and ‘short’ axial wavelengths, with the extent of the short wavelength solution being dependent upon the degree of stiffening afforded by the rings.

The analysis of the effects of ring eccentricity shows that ovals are affected in a similar manner to circular cylinders in that outside rings provide the greatest capacity for sustaining axial compression, while inside rings are capable of supporting the greatest lateral pressure.

Finally, it is found that the buckling load of an oval cylinder under uniform lateral pressure slightly exceeds the corresponding value for an equivalent circular cylinder. As a further verification of this phenomenon, a Rayleigh-Ritz procedure is employed to determine the buckling load of an oval ring under uniform radial load. The results of this analysis corroborate those obtained for the cylinder.     

Analysis of pneumatic shells with or without cable net; general finite-element formulation

Utilising a curved quadrilateral element system derived from surface patching technique, a general finite element formulation is done to analyse the non-linear behaviour of cable-reinforced air supported shells taking consideration also of the action of the membrane. This formulation, though presently restricted to isotropic and linearly elastic materials, is valid for any arbitrary geometry and boundary of inflatable shells with or without prestressing of the cable nets. An incremental precedure is used to approximate the non-linear solution.Test of convergency is carried out on a hemispherical shell with internal pressure, whose exact solution is well known. Among the other sample problems analysed are pressurized hemisphere under wind load and a hyperbolic paraboloid prestressed cable net with distributed as well as concentrated loads. Comparison with earlier known solutions of these problems establishes the validity of the method.     

An iterative procedure for collapse analysis of reinforced concrete plates

An iterative procedure is proposed for evaluating the ultimate load of a laterally loaded plate discretized by finite elements. The procedure regards reinforced concrete plates, but it can be extended to metallic plates without any conceptual change. The stress and displacement fields are approximated by means of a finite element model with constant stress and linear displacement fields. Consequently, any load distribution is represented by the equivalent system of nodal forces for a given mesh. In the set of mechanisms compatible with the assumed discretization the best upper bound to the collapse multiplier of the actual load is obtained via linear programming. By dualization a sequence of linear programming problems is obtained which allows an evaluation of a lower bound of the collapse multiplier for the equivalent load system. When the mesh gets finer and finer, the value obtained does not change substantially anymore. This value can be regarded as an estimate of the collapse multiplier for the original load system. Some numerical examples of plates subjected to uniform pressure confirm the reliability of this approximate multiplier.     

Effects of random system properties on the thermal buckling analysis of laminated composite plates

This paper examines the effect of random system properties on thermal buckling load of laminated composite plates under uniform temperature rise having temperature dependent properties using HSDT. The system properties such as material properties, thermal expansion coefficients and thickness of the laminate are modeled as independent random variables. A C⁰ finite element is used for deriving the eigenvalue problem. A Taylor series based first-order perturbation technique is used to handle the randomness in the system properties. Second-order statistics of the thermal buckling load are obtained. The results are validated with those available in the literature and Monte Carlo simulation.     

基于向量式有限元的空间格构柱屈曲破坏

基于向量式有限元法对空间格构柱受载荷下的屈曲行为进行数值分析，编制屈曲分析的数值计算程序，通过屈曲平衡路径追踪获得失稳临界载荷，探讨不同加载控制方法对计算结果的影响。考虑材料非线性的影响，对5层空间格构柱的载荷屈曲过程进行分析，获得载荷 位移全过程曲线，并与格构柱的加载破坏试验进行对比，结果表明：向量式有限元采用位移控制法能够较好地分析空间格构柱的屈曲失稳破坏过程，获取准确的极限稳定承载力。     

Buckling load for nonuniform columns

A method to determine an approximate lower bound buckling load for a nonuniform column is presented. The method is applicable to both hinged-hinged and cantilever columns. A finite difference scheme is used to represent the equilibrium equation for the column. Recursion relations are developed for the coefficient of the characteristic equation from which an approximate lower bound buckling load may be calculated. One advantage of the present method is that it can be programmed on a programmable calculator with as few as 15 data storage locations and 200 program steps.     

A general finite element model for shells of arbitrary geometry

A finite element formulation is developed for the large displacement analysis of arbitrary shells. Formulation is based on a convected coordinate system and a tensorial approach is followed. The strain-displacement relationships used do not reflect the Kirchhoff hypothesis and Love's approximations. Isoparametric interpolation is used for the discretization of the problem, and the number of nodal points is variable. The numerical examples include the buckling analysis of cylindrical shells as well as two problems to test the convergence and accuracy of the algorithm.