Buckling Analysis with FEA

Linear finite-element analysis does not provide enough information about buckling to make correct design decisions, especially when designing lightweight components.

In many design projects, engineers must calculate the factor of safety (FOS) to ensure the design will withstand the expected loadings. Calculations require correctly recognizing the mechanisms of failure, and this is a difficult task. All too often we associate structural failure only with yielding and are satisfied when design analysis shows a sufficient FOS related to yield.

 

However, yielding is not the only mode of failure. For example, it is necessary to consider displacements to ensure the part or assembly does not deform too much. Also important is buckling, which is all-too-often forgotten and yet poses a dangerous mode of design failure. Buckling happens suddenly, without little if any prior warning, so there is almost no chance for corrective action.

Certain problems tend to arise in buckling analysis performed with finite-element analysis (FEA). These problems are best presented in the context of two other failure modes: excessive displacements and yielding, as summarized in the Failure modes table.

Linear-buckling analysis
First, consider a linear-buckling analysis (also called eigenvalue-based buckling analysis), which is in many ways similar to modal analysis. Linear buckling is the most common type of analysis and is easy to execute, but it is limited in the results it can provide.

Linear-buckling analysis calculates buckling load magnitudes that cause buckling and associated buckling modes. FEA programs provide calculations of a large number of buckling modes and the associated buckling-load factors (BLF). The BLF is expressed by a number which the applied load must be multiplied by (or divided — depending on the particular FEA package) to obtain the buckling-load magnitude.

The buckling mode presents the shape the structure assumes when it buckles in a particular mode, but says nothing about the numerical values of the displacements or stresses. The numerical values can be displayed, but are merely relative. This is in close analogy to modal analysis, which calculates the natural frequency and provides qualitative information on the modes of vibration (modal shapes), but not on the actual magnitude of displacements.

 

Theoretically, it is possible to calculate as many buckling modes as the number of degrees of freedom in the FEA model. Most often, though, only the first positive buckling mode and its associated BLF need be found. This is because higher buckling modes have no chance of taking place — buckling most often causes catastrophic failure or renders the structure unusable.

The nomenclature is “the first positive buckling mode” because buckling modes are reported in the ascending order according to their numerical values. A buckling mode with a negative BLF means the load direction must be reversed (in addition to multiplying by the BLF magnitude) for buckling to happen.

As a consequence of discretization error, linear buckling analysis overestimates the buckling load and provides nonconservative results. However, BLFs are also overestimated because of modeling errors. FE models most often represent geometry with no imperfections and loads and supports are applied with perfect accuracy with no offsets. In reality though, loads are always applied with offsets, faces are never perfectly flat, and supports are never perfectly rigid. Even if supports are modeled as flexible, their stiffness is never evenly distributed. Imperfections are always present in the real world. Considering the combined effect of discretization error (a minor effect) and modeling error (a major effect), designers should interpret the results of linear buckling analysis with caution.