Buckling Analysis
Linearized Buckling Analysis
A linearized buckling analysis can be used for estimating the critical load at which a structure becomes unstable. This is a predefined study type that consists of two study steps: An initial step in which a unit load is applied to the structure, and a second step in which an eigenvalue problem is solved for the critical buckling load.
The idea behind this type of analysis can be described in the following way:
Consider the equation system to be solved for a stationary load f, in case of geometric nonlinearity;
Here the total stiffness matrix, K, depends in the solution, since the problem is nonlinear. It has been split into a linear part, KL, and a nonlinear contribution, KNL.
In a first-order approximation, KNL is proportional to the stress in the structure and thus to the external load. If the linear problem is solved first for an arbitrary initial load level f0,
(2-40)
then the nonlinear problem can be approximated as
where λ is called the load multiplier.
An instability is reached when this system of equations becomes singular, so that the displacements tend to infinity. The value of the load at which this instability occurs can be determined by, in a second study step, solving an eigenvalue problem for the load multiplier λ.
COMSOL reports a critical load factor, which is the value of λ at which the structure becomes unstable. The corresponding deformation is the shape of the structure in its buckled state.
The level of the initial load used is immaterial since a linear problem is solved. If the initial load actually was larger than the buckling load, then the critical value of λ is smaller than 1. It is also possible that the computed value of λ is negative. This signifies that a reversed load will give the critical case.
Live and Dead Loads
In some situations, not all loads that act on the structure can vary, and you may want to compute the critical load factor only with respect to the uncertain (‘live’) load. For example, gravity is often treated as a ‘dead’ load. The dead loads do, however, affect the stress distribution in the structure, so they cannot be completely ignored. Using the same terminology as above,
Here, the superscripts ‘d’ and ‘l’, stands for live and dead loads, respectively. By solving two problems, one with the live loads, and one with the dead loads, it is possible to separate their effects:
Now, the following eigenvalue problem is solved to determine the critical load factor:
For any load feature, you can specify that is a dead load by selecting the Treat as dead load check box in the Linear Buckling section.
To perform a buckling analysis including both live and dead loads, you need three study steps: two stationary steps, and the linear buckling study step.
1
2
3
Finally, in settings for the Linear Buckling study step, you need to point to these two solutions as Values at linearization point and Live loads solution, respectively.
Figure 2-32: Selecting the solutions for a buckling analysis with live and dead loads.
Geometric Nonlinearity
Sometimes, the preload case requires a geometrically nonlinear analysis in order to produce the correct state. This means that it is no longer solved using the linear set of equations given by Equation 2-40, but rather
(2-41)
The assumption for the buckling analysis is still that KNL is proportional to the external load, even though this may be disputable for a strongly nonlinear case. KNL is based on the stresses, which must be computed in the same way for both cases, that is, under the same assumption about geometric nonlinearity. The effect is that the stiffness matrix at the linearization point includes the nonlinear part from Equation 2-41, and the eigenvalue problem is reformulated as
Follower Loads
Loads that depend on the deformation are called follower loads. An example of this is a pressure load, since the orientation of the load will depend on surface deformation. Such loads contribute to the stiffness matrix, and can thus affect the buckling load. As a default, all loads in the structural mechanics interfaces are multiplied by the load factor λ in a linear buckling study step.
Be aware that for some structures, the true buckling load can be significantly smaller than what is computed using a linearized analysis. This phenomenon is sometimes called imperfection sensitivity. Small deviations from the theoretical geometrical shape can then have a large impact on the actual buckling load. This is especially important for curved shells.
Studies and Solvers and Linear Buckling in the COMSOL Multiphysics Reference Manual
Linear Buckling in the theory section of the Structural Mechanics Module User’s Guide
Bracket — Linear Buckling Analysis: Application Library path Structural_Mechanics_Module/Tutorials/bracket_linear_buckling
Buckling Analysis of a Truss Tower: Application Library path Structural_Mechanics_Module/Buckling_and_Wrinkling/truss_tower_buckling