Buckling is a sudden, uncontrollable failure phenomenon occurring in thin structures subjected to compressive stresses. For the material to fail in buckling, the compressive stresses do not have to reach the ultimate compressive stresses that the material is capable of withstanding. This failure phenomenon can occur in different types of structural members like columns, beams, trusses, and thin-walled structures present in bridges, buildings and towers.
Importance Of FEA In Buckling Analysis
We can calculate the critical magnitude of compressive loads at which simple slender columns with different cross sections buckle using Euler's formula as discussed in the previous article. But it’s not always ideal to depend on mathematical and analytical methods to predict buckling in structural designs, especially when the problem involves complex geometry, loads, boundary conditions and nonlinear large deformations. Hence, FEA provides a powerful computational tool for analyzing buckling behavior, allowing the engineers to attain optimum designs that guarantee structural integrity under compressive loads.
We can perform two types of buckling analysis for a structure using FEA: Linear Eigenvalue Buckling Analysis and Nonlinear Buckling Analysis. Each type of these analyses offers unique insights into the stability and behavior of structures under different conditions. Linear analysis is a perturbation step used to determine the critical buckling mode shapes along with the corresponding load magnitudes for a mechanical structure. We can only provide elastic behavior for the material in linear analysis. Whereas in nonlinear analysis, geometric and material nonlinearities can be included for the structure. It extends the capability of FEA by accounting for large deformations, post-buckling behavior and material yielding.
In this article, let’s understand how a linear buckling analysis can be performed using some example problems.
Linear Eigenvalue Buckling Analysis
This type of analysis is used to determine the critical buckling mode shapes and their corresponding critical compressive loads of a structure. Only linear material behavior and small deformation theory can be assumed in this type of analysis. As this is a linear analysis, the physical model cannot have any substructure or contact definitions. All the parts have to be either connected using ties or direct nodal connections to form one single structure. A reference level of load is applied initially, ideally a unit load, to the structure.
The loads can be pressure, concentrated force, displacement, rotation or thermal loading. All the necessary boundary conditions are also applied, and a linear perturbation step is carried out. As everything is linear in this analysis, there will be no convergence issues and the computational time will be very small. Hence this can be used to perform fast model checking and determine parts of the structure that may lead to stability issues.
Theory
A linear static analysis is carried out to find stresses from the applied loads and boundary conditions to form geometric stiffness matrix defined as KG. The eigenvalue problem is defined as,
(K- λ KG) ν = 0
where, K is the stiffness matrix corresponding to the base state, λ is the eigen values or reference load multipliers and vector ν is the eigen vector corresponding to the eigen values. This problem is solved for λ and ν using Lanczos method.
The buckling load is associated with the first eigenvalue which is determined by,
Pcritical = λ1 Preference
Buckling in the structure occurs when λ1 < 1 and the structure is safe when λ1 > 1.
Limitations
Since linear buckling analysis can only account for linear material behavior, which is not always ideal in real life situations, the results are not always reliable. Because of this, even if the first eignenvalue is greater than 1 (λ1 > 1), this will not necessarily mean that stability failure won’t happen. We should always use further checks to increase confidence in the results.
The imperfections in the structural problem cannot be effectively modeled using this linear analysis. The imperfections will lead to a severe decrease in model stiffness and when these are introduced into the linear model, the decrease in stiffness may not be properly represented. The second order bending and material yielding at the imperfection will not be accurately captured at the imperfections, leading to over estimation of the critical loads.
Example Problem
Let’s perform the linear buckling analysis for the C channel made of 5 mm thick steel with compressive load on the top as shown below and explore the results.
The displacement, stress and strain magnitudes in these FEA results are meaningless and they only represent the buckling mode shape of the structure. The scale factor indicates that the first buckling mode occurs at 665,189 N.
Effect Of Support Conditions
In the previous article, we learned how different support conditions at the ends of a column affect the position of inflection points, thereby affecting the effective length of column and critical Euler buckling load. Now, let’s explore the this further by using linear buckling analysis for a column subjected to a unit compressive load as shown in the figure below.
Fixed - Fixed
Scale factor = 4,119
Pinned - Pinned
Scale factor = 1,034
Fixed – Pinned
Scale factor = 2,155
Fixed – Free Scale factor = 258
As can be seen above, the less rigidly the column is constrained, the lower the critical eigenvalue.
Final Thoughts
Hopefully, this article has summarized the theory, application, and limitations of linear buckling analysis. This type of analysis is used to calculate the magnitude of the critical compressive loads that cause sudden, catastrophic buckling failure in slender mechanical systems. This determines which parts of the structure can have possible stability issues, so that we can optimize the design in these critical regions. Since it is geometrically and materially linear, the results are not completely reliable and can only be used effectively for one-to-one design comparisons and fast failure checks.
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