KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES BY A FINITE ELEMENT APPROACH (original) (raw)
Exact solution and stability of postbuckling configurations of beams
Nonlinear Dynamics, 2008
We present an exact solution for the postbuckling configurations of beams with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated
Post-buckling behavior and imperfection sensitivity of L-frames
2005
Beginning with the work of Koiter in 1945, valuable insights into the postbuckling behavior of structures have been gained by Lyapunov-Schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point. Here this methodology is generalized to include nonlinear prebuckling behavior, as well as multiple, not necessarily coincident buckling modes. The expansion of the reduced equilibrium equations is performed about a reference point (which need not coincide with any of the bifurcation points), and applies no matter whether the modes are coincident, closely spaced, or well separated. From a variety of possible decompositions of the admissible space of displacements, two are incorporated into a finite element program. Theoretical considerations, and numerical examples in which asymptotic results are compared to 'exact' results, indicate that one of the decompositions has some important advantages over the other. Examples include a shallow arch, and a beam on elastic foundation problem exhibiting symmetry-breaking modal interaction.
Elastic postbuckling analysis via finite element and perturbation techniques. Part 1: Formulation
International Journal for Numerical Methods in Engineering, 1992
The main equations for the equilibrium, stability and critical state analysis of discrete elastic systems are presented following the works of Thompson, but in such a way that the original set of generalized coordinates and loads are preserved in the Total Potential Energy. This introduces differences in the resulting equations in bifurcation analysis but does not introduce any new feature regarding the physics of the problem. The new formulation is approximated by means of a standard finite element approach based on interpolation of displacements, in which the derivatives of the potential energy are approximated. The terms retained are those of moderately large rotation theory. The energy analysis is finally related to the more conventional finite element notation in terms of stiffness matrices, and it is shown how in such a way it can be included in present day codes.Part 2 of the paper deals with applications to the analysis of shells of revolution using a semi-analytical approximation. Two cases are presented in detail: bifurcation in axisymmetric and in asymmetric modes, and the results show good correlation with those of other authors. The influence of load and geometric imperfections is evaluated.
State-based buckling analysis of beam-like structures
Archive of Applied Mechanics, 2017
Beam, column, plate, and any other structure, under full or partial compressive loading, are prone to failure by the buckling phenomenon. At the instant of failure, the structure may be in unpredictable elastic, elastic-plastic, full plastic, cracked, or other forms of deterioration state. Therefore, in spite of so much study, there is no definite solution to the problem. In this paper a unified, simple, and exact theory is proposed where buckling is considered as the change of state of structure between intact and collapsed states, and then the buckling capacity is innovatively expressed via states and phenomena functions, which are explicitly defined as functions of state variable. The state variable is determined by calibration of the structure slenderness ratio. The efficacy of the work is verified via concise mathematical logics, and comparison of the results with those of the others via seven examples. Keywords State functions • Phenomena functions • State-based philosophy • Buckling • Capacity • Beam • Column 1 Introduction Buckling is a mathematical instability, leading to a failure mode. The formal meaning of the notion is found in engineering and sciences, regarding stability of systems. Theoretically, for a structural system, buckling is caused by a bifurcation in the solution to the equations of static equilibrium. In practice, buckling is characterized by a sudden failure of a structural member under a compressive stress, which is less than the ultimate compressive stress that the material is capable of withstanding. Failure occurs in a distinct, most of the times unpredictable, direction compared to the direction of the applied load. A structural member under compression, at any level, is always prone to failure via buckling. Although the stability of bars was first studied over 250 years ago by Euler [1], adequate solutions are still not available for many problems in structural stability. So much has been and is being studied and written in the field of structural stability, a question arise that why, after such intellectual and financial efforts, there are no definite solutions to these problems. Determination of the collapse load of a structure, due to loss of
Int J Numer Method Eng, 1993
The general theory developed in Part I of this paper for the finite element stability analysis of structural systems, using perturbation expansions in the vicinity of a critical point, is applied here to the analysis of shells of revolution. The discretization of the shell is performed by means of a semianalytical approximation, and the matrices required for the evaluation of critical points and postcritical equilibrium paths are obtained. Two cases are presented: bifurcation in axisymmetric and in asymmetric buckling modes. The derivatives required for an imperfection analysis are also obtained. A technique of switching between two paths using continuation methods is also discussed, in which the switch is performed using derivatives of the perturbation expansion. Results are presented for bifurcation in axisymmetric and in non-axisymmetric modes, and compared with known solutions or with results from changing the path using continuation methods; good correlation is shown. For structures displaying unstable bifurcation, the influence of load and geometric imperfections is evaluated.
Buckling and Postbuckling Analyses of Structure using Absolute Nodal Coordinate Formulation
IOP Conference Series: Materials Science and Engineering, 2019
In this paper, the three dimensional higher order beam element based on the absolute nodal coordinate formulation (ANCF) is used to study both the classical buckling and the nonlinear postbuckling problem. The analyses are performed using the Newton-Raphson method and the arc length method. The Newton-Raphson method is used for the Euler buckling whereas the Crisfield's arc length method is applied to track the equilibrium path of the William's Toggle. The solutions agree well with the analytical one or that given by the commercial finite element software ANSYS. Hence, the validity of the analyses is demonstrated.
Stability Analysis of Stiffened Panels Analytical and Finite Element Methods
2002
The design of large parts of an airframe is driven by post buckling stability requirements. The aim of the present work is to compare stability predictions obtained both by analytical and numerical methods. A study on the sensitivity of this method to material law has also been conducted. This way, the extent to which the precise shape of an high performance aluminium alloy stress / strain curve has an impact on stability results has been evaluated. Compression studies lead to acceptable differences between methods. The trend is however depending on the model : numerical calculations predict lower critical loads as well as upper ones. The field of improvement is undoubtedly modelling methods such as the introduction of a perturbation in the mesh and boundary conditions. More work is needed to better understand these differences. The study of the influence of the material also shows differences between both methods. Little changes are found when using detailed material model with ana...
International journal for numerical …, 1994
Adopting an updated Lagrangian approach, the general framework for the fully non-linear analysis of thin-walled framed structures is developed using a simple, two-node, Co-model (HMB2). The governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified Hellinger/Reissner type with independent assumptions for displacement and strain fields. All coupled significant modes of deformations, i.e. stretching, bending, shear, torsion and warping, are accounted for in the generalized-beam theory employed. Emphasis is placed on devising effective solution procedures to deal withjnite rotations in space, particularly with regard to their effect on the derivations of load-correction matrices corresponding to conjguration-dependent externally applied forces/moments. The effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space.
FSplines: A Software for Linear Stability Analysis of Thin-Walled Structures, Version 2.0
Innovation and Research, 2021
FSplines is a (geometrically) linear stability analysis tool of thinwalled structures with open section (useful for cold-formed steel profiles), that enables obtaining the bifurcation stresses (critical stress, load and moment) and the respective buckling modes by the Finite Strip Method (FSM). The Finite Strip Method: (i) allows analyzing prismatic steel members (commercial structural profiles), (ii) is an alternative to the Finite Element Method (FEM), and (iii) has some important advantages over FEM. In the present article, two variants of the FSM are presented: (i) the Semi-Analytical Finite Strip Method (SAFSM), where use is made of trigonometric functions and (ii) the Splines Finite Strip Method (SFSM), employing spline functions. The SAFSM has the advantage of being less time consuming. Its main restriction is the fact that it only allows modelling simple supported members (pinned restrained). The SFSM most important advantage is the ability to model members with all kinds of boundary conditions. This method is, however, more time consuming. It is worth noting that the bifurcation analysis, performed by the computer application FSplines, is required for the design of cold formed members according to the specifications of international standards. FSplines 2.0 is the second version of the computer application here presented. In this second improved version more cross-sections are available, and more section properties are presented.