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h1. Eigenvalue analysis
Please note that Buckling is the [load-case|kb:Load case] type for eigenvalue analysis.
Eigenvalue buckling analysis predicts the theoretical buckling strength of a structure idealized as elastic. In the classic Eigenvalue method, structural eigenvalues are computed from the loading and constraint conditions of a given system. Tabulated solutions provide buckling loads for basic structural configurations. Each load has an associated buckled mode shape which represents the shape a structure assumes when buckled. For real structures, imperfections and nonlinear behavior keep systems from achieving this theoretical buckling strength. Since eigenvalue analysis over-predicts buckling load, nonlinear buckling analysis is recommended.
h1. Nonlinear buckling analysis
Please note that Static, Nonlinear with P-Delta and Large Displacement is the load-case type for nonlinear buckling analysis.
Nonlinear buckling analysis is a more accurate method which incrementally increases applied loading until a structure becomes unstable. This condition of instability is indicated by a small increase in load level causing a very large change in deflection. Nonlinear buckling analysis is a static method which accounts for material and geometric nonlinearities (P-Δ and P-δ), load perturbations, geometric imperfections, and gaps. Either a small destabilizing load or an initial imperfection is necessary to initiate the desired buckling mode.
h1. Important considerations
* The primary output from linear buckling analysis is a set of buckling factors. Applied loading is multiplied by these factors to scale the loading such that it causes buckling in a given structure. Please refer to the [_CSI Analysis Reference Manual_|doc:Analysis Reference Manual] (Load Cases > Linear Buckling Analysis) for additional information.
* Since the deflections, forces, and reactions of linear buckling analysis correspond to the normalized buckled shape of a structure, users must run nonlinear buckling analysis to obtain the actual displacements, forces, and reactions. Figure 1 illustrates the nonlinear buckling-analysis output of a column subjected to an initial imperfection where lateral load induces displacement equal to 0.6% of column height. Softening behavior indicates the onset of buckling.
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!Evaluate_buckling_by_nonlinear_analysis.png|align=center,border=0!
{center-text}Figure 1 - Nonlinear buckling analysis of a column{center-text}
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Users may download the analytical model for this system through the [P-Delta effect for fixed cantilever column|tp:P-Delta effect for fixed cantilever column] test problem.
h1. See Also
* [kb:Nonlinear buckling]
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Eigenvalue Analysis (load case type "Buckling")
Eigenvalue buckling analysis predicts the theoretical buckling strength of an ideal elastic structure. It computes the structural eigenvalues for the given system, its loading and constraints. This is known as classical Euler buckling analysis. Buckling loads for basic structural configurations are readily available from tabulated solutions. Each load has an associated buckled mode shape; this is the shape that the structure assumes in a buckled condition. However, in real-life, structural imperfections and non-linearities prevent most real-world structures from reaching their eigenvalue predicted buckling strength; in other words, the eigenvalue analysis over-predicts the expected buckling loads. A more realistic, nonlinear buckling analysis is therefore recommended to analyze real-world structures for buckling.
Nonlinear Buckling Analysis (load case type "Static", Nonlinear with P-Delta and Large Displacements)
Nonlinear buckling analysis is more accurate than eigenvalue analysis because it employs non-linear, large-deflection, static analysis to predict buckling loads. Its mode of operation is very simple: it gradually increases the applied load until a load level is found whereby the structure becomes unstable (ie. suddenly a very small increase in the load will cause very large deflections). The true non-linear nature of this analysis thus permits the modeling of geometric imperfections, load perturbations, material nonlinearities and gaps. For this type of analysis, that small destabilizing loads or initial imperfections are necessary to initiate the desired buckling mode.
Important Considerations
(1) The primary output from linear buckling analysis is a set of buckling factors. These buckling factors are scale factors that must multiply the applied loads to cause buckling in a given model. Please refer to CSI Analysis Reference Manual, chapter "Load Cases", section "Linear Buckling Analysis" for additional information.
(2) Since the deflections, forces and reactions for linear buckling analysis correspond to normalized buckled shape of the structure, you would need to run nonlinear buckling analysis to obtain the actual displacements, forces and reactions for your structure. The screenshot below illustrates output from nonlinear buckling analysis of a column with initial imperfections (that are modeled by lateral load that will cause a lateral deflection of the column this equal to 0.6% of the column height). You will notice a softening of the structure that would indicate an onset of buckling.
The model for which the above plot was obtained can be downloaded from the P-Delta effect for fixed cantilever column test problem page.
See also
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