Authored by:
Paul M. Kurowski

President
Tom P. Kurowski
Project Engineer
Design Generator Inc.
London, Ontario, Canada

Edited by Leslie Gordon
leslie.gordon@penton.com

Resources:
Design Generator Inc.
, www.designgenerator.com
Note: All examples were solved with SolidWorks Simulation 2010. You can download the models at www.designgenerator.com/downloads.htm.

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.

For an example of a linear buckling analysis, consider a model of a beam in compression (beam material is 1060 aluminum). The model is studied in two configurations: with a free loaded end, and with a sliding loaded end.

In the Notched beam figure, note the small notch — its importance will soon become clear. Also, try predicting which way the beam should buckle in each configuration.

In the Linear buckling, free end figure, why has the beam buckled in the wrong direction, and not toward the side where the notch is? Because the linear-buckling analysis only predicts the buckled shape and not the direction of buckling.

The Linear buckling, sliding end figure shows the beam buckling in the correct direction, but this is purely coincidental. The color legends with displacement magnitudes in both figures point out their meaninglessness. They show that linear buckling analysis cannot provide quantitative results for displacements or stresses.

In addition, linear-buckling analysis does not show what happens to a structure after buckling. Does the structure collapse, or will it retain the load-bearing ability in the buckled shape? How much will it deform when it buckles? To obtain more information than the BLF and the qualitative buckled shape, it is necessary to enter the domain of nonlinear buckling analysis.

Nonlinear-buckling analysis
As with any other nonlinear analysis, nonlinear-buckling analysis requires that a load be applied gradually in multiple steps rather than in one step as in a linear analysis. Each load increment changes the structure’s shape, and this, in turn, changes the structure’s stiffness. Therefore, the structure stiffness must be updated at each increment. In this approach, which is called the load control method, load steps are defined either by the user or automatically so the difference in displacement between the two consecutive steps is not too large.

Although the load-control method is used in most types of nonlinear analyses, it would be difficult to implement in a buckling analysis. When buckling happens, the structure undergoes a momentary loss of stiffness and the load control method would result in numerical instabilities. Nonlinear buckling analysis requires another way of controlling load application — the arc length control method. Here, points corresponding to consecutive load increments are evenly spaced along the load-displacement curve, which itself is constructed during load application.

In contrast to linear-buckling analysis, which only calculates the potential buckling shape with no quantitative values of importance, nonlinear analysis calculates actual displacements and stresses. To better understand the inner workings of nonlinear-buckling analysis, first consider what happens in running a nonlinear-buckling analysis on an idealized structure. Imagine a perfectly round and perfectly straight column under a perfectly aligned compressive load. Theoretically, buckling will never happen, but in actuality, buckling will take place because of imperfections in the geometry, loads, and supports.

When real-world imperfections are absent in the FEA model, buckling will still happen, but it will be initiated by imperfections introduced by discretization errors. Therefore, nonlinear buckling requires a model with some initial imperfection. When no such imperfections are present, they must be added to control the onset of buckling. In the example, the imperfection is the notch.

Consider what happens in a nonlinear buckling analysis of the notched beam.

Note that in the figure Nonlinear buckling, free end, the red line shows that buckling happens at about 550 N, close to what the linear-buckling analysis predicted. The beam buckles towards the notched side as it should. Then, the displacement grows at an almost constant load. The result implies that the beam is capable of balancing the 550 N acting at an increasing load offset, but this is clearly unrealistic because the buckling of the notched beam initiates a chain of events leading to structural collapse.

To model these events, it is necessary to account for another source of nonlinearity beside the already considered geometric nonlinearity. That other source is yielding. This is done by using an elastic perfectly plastic material, the simplest type of nonlinear material models.

In the figure Nonlinear buckling, free end, the blue line, which represents the load-displacement curve for this material, looks much different than the red line. The buckling still happens at 550 N, indicating that the onset of buckling took place with the material still in the elastic range. After a period of elastic buckling (the vertical portion of the blue curve that leads to its maximum load the beam can support), the load rapidly drops. This is because to maintain equilibrium, it would be necessary to reduce the applied force.

However, when the load stays the same, as is most often the case, the end of the vertical portion of the curve marks the beam’s structural collapse. Here, the beam is totally plasticized and can no longer support the load. If you examined the deformed shape you would have noted that most of the deformation takes place at the support where the plastic “hinge” develops.

In repeating the analysis for the same beam with the sliding restraint, the results, as shown in Nonlinear buckling, sliding end, reveal that the linear material model (red) predicts buckling at about 9,500 N, close to the linear BLF. However, a load this high causes yielding of the entire cross section in the notched area and this cannot be modeled with a linear elastic material model. The elastic-perfectly plastic model (blue) shows that buckling takes place at 2,500 N, when the notched cross section is already plastic.

The upshot: linear-buckling analysis can only provide limited information, often not enough to make a design decision, especially when designing lightweight components. Nonlinear-buckling analysis should, therefore, become more routine in design departments. Most often the nonlinear-buckling analysis should account both for geometry nonlinearity and for nonlinear material. The increased computing power of hardware combined with easier-to-use FEA software now make it practical to add nonlinear buckling analyses to almost any design engineer’s toolbox.

Where modal and buckling analyses meet
Linear-buckling analysis is also called eigenvalue buckling or Euler buckling analysis because it predicts the theoretical buckling strength of an elastic structure. Eigenvalues are values of load at which buckling takes place. Eigenvectors are buckling shapes associated with the corresponding eigenvalues. According to eigenvalue-buckling analysis, buckling takes place when — as a result of subtracting the stress stiffness induced by a compressive load from the elastic stiffness — the resultant structure stiffness drops to zero.

Both linear-buckling analysis and modal analysis (which is also linear) can predict a large number of modes. In a buckling analysis, the only mode of practical importance is the first one with a positive buckling load factor (BLF). In modal analysis we are usually interested in the few first modes.

The natural frequencies may significantly depend on the applied load. If this is the case, modal analysis should account for prestress. Tensile stresses boost natural frequencies, illustrated by tuning a guitar string or by the stress stiffening of a rotating component such as a turbine blade.

Rotating machinery typically requires that designers consider the effect of prestress. Compressive stresses reduce natural frequencies. For example, an analysis of natural frequencies of a compressed column demonstrates that the first natural frequency declines with the load. Note that load magnitude corresponding to zero frequency equals the buckling load because this is when the compressed column looses its stiffness.

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