KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES BY A FINITE ELEMENT APPROACH (original) (raw)
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Computer Methods in Applied Mechanics and Engineering, 1993
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.
Direct Methods for Limit States in Structures and Materials, 2014
The analysis of slender structures, characterized by complex buckling and postbuckling phenomena and by a strong imperfection sensitivity, is heavily penalized by the lack of adequate computational tools. Standard incremental iterative approaches are computationally expensive and unaffordable, while FEM implementation of the Koiter method is a convenient alternative. The analysis is very fast, its computational burden is of the same order as a linearized buckling load evaluation and the simulation of different imperfections costs only a fraction of that needed to characterize the perfect structure. In this respect it can be considered as a direct method for the evaluation of the critical and post-critical behaviour of geometrically nonlinear elastic structures. The main objective of the present work is to show that finite element implementations of the Koiter method can be both accurate and reliable and to highlight the aspects that require further investigation.
Initial postbuckling behavior of thin-walled frames under mode interaction
Thin-Walled Structures, 2013
An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means that the stiffer structure is also the less imperfection sensitive. Finally for each one of the cases studied, the effect of initial imperfection has been considered and the real load carrying capacity of the frames has been determined. Finally some results have been compared with those obtained using the FE code ABAQUS.
Mathematical Problems in Engineering, 2008
Structural systems liable to asymmetric bifurcation usually become unstable at static load levels lower than the linear buckling load of the perfect structure. This is mainly due to the imperfections present in real structures. The imperfection sensitivity of structures under static loading is well studied in literature, but little is know on the sensitivity of these structures under dynamic loads. The aim of the present work is to study the behavior of an archetypal model of a harmonically forced structure, which exhibits, under increasing static load, asymmetric bifurcation. First, the integrity of the system under static load is investigated in terms of the evolution of the safe basin of attraction. Then, the stability boundaries of the harmonically excited structure are obtained, considering different loading processes. The bifurcations connected with these boundaries are identified and their influence on the evolution of safe basins is investigated. Then, a parametric analysis ...
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.