1. Linear Buckling Analysis
Chapter Seven
Linear Buckling Analysis
Follower force effect…

2. Chapter Overview
In this chapter, performing linear buckling analyses in Design Simulation will be covered.
In Design Simulation, performing a linear buckling analysis is similar to a stress analysis.
It is assumed that the user has already covered Chapter 4 Linear Static Structural Analysis prior to this section.
The capabilities described in this section are generally applicable to ANSYS DesignSpace Entra licenses and above.
Some options discussed in this chapter may require more advanced licenses, but these are noted accordingly.
Harmonic and nonlinear static structural analyses are not discussed here but in their respective chapters.
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3. Background on Buckling
Many structures require an evaluation of their structural stability. Thin columns, compression members, and vacuum tanks are all examples of structures where stability considerations are important.
At the onset of instability (buckling) a structure will have a very large change in displacement {x} under essentially no change in the load (beyond a small load perturbation). F
Stable
Unstable
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4. … Background on Buckling
Eigenvalue or linear buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure.
The eigenvalue formulation determines the bifurcation points of a structure. This method corresponds to the textbook approach of linear elastic buckling analysis.
The eigenvalue buckling solution of a Euler column will match the classical Euler solution.
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5. … Background on Buckling
However, imperfections and nonlinear behavior prevent most real world structures from achieving their theoretical elastic buckling strength. Linear buckling generally yields unconservative results, and should be used with caution.
Consider the buckling of a soda can:
Material response is inelastic. Geometrically nonlinear effects need to be considered. Contact is also required. Hence, these type of nonlinear behavior are not considered.
There may be slight imperfects in the soda can, such as a small dent, which would influence the response and not make the model symmetric. However, these small imperfections are also not usually considered in a linear buckling analysis.
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6. … Background on Buckling
Although unconservative, linear buckling has various advantages:
It is computationally cheaper than a nonlinear buckling analysis, and should be run as a first step to estimate the critical load (load at the onset of buckling).
Relative comparisons can be made of the effect of differences in design to buckling
Linear buckling can be used as a design tool to determine what the possible buckling mode shapes may be.
The way in which a structure may buckle can be used as a possible guide in design
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7. A. Basics of Linear Buckling
The idea behind performing linear buckling is that a bifurcation point is sought. The bifurcation point is where two configurations – the initial geometry and a buckled mode – are both possible, signifying the onset of buckling.
A linear static analysis can include the stress stiffness matrix [S], which is a function of the stress state:
If we consider the analysis to be linear, we can multiply the load and the stress state by a constant l:
In a buckling mode, displacements can be large (x+y) without an increase in load, so the following is also true:
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8. … Basics of Linear Buckling
If the last two equations are subtracted from each other, the following is the result:
The above equation is what is solved for during a linear buckling analysis.
The buckling load multiplier l is multiplied to the applied loads to get the critical load for buckling
The buckling mode shape y expresses the shape of buckling. However, the magnitude is not known since y is indeterminate.
There are actually many buckling load multipliers and modes, although the user is usually interested in the first few modes since these would occur before any higher buckling modes.
Note the similarity of linear buckling equation with the free vibration equation (Chapter 5). Both are known as eigenvalue problems which are solved for with similar matrix methods.
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9. … Basics of Linear Buckling
For a linear buckling analysis, two solutions are automatically performed internally:
A linear static analysis is performed first:
Based on the stress state from the static analysis, a stress stiffness matrix [S] is calculated:
The aforementioned eigenvalue problem is then solved to get the buckling load multiplier li and buckling modes yi:
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10. … Basics of Linear Buckling
For a linear buckling analysis, the eigenvalue problem below is solved to get the buckling load multiplier li and buckling modes yi: This results in certain assumptions related to the analysis:
[K] and [S] are constant:
Linear elastic material behavior is assumed
Small deflection theory is used, and no nonlinearities included
The response based on loading {F} is a linear function of li
Some additional restrictions:
Nonzero displacement supports or thermal loads are not allowed
It is important to remember these assumptions related to performing linear buckling analyses in Design Simulation.
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11. B. Buckling Analysis Procedure
The linear buckling analysis procedure is very similar to performing a linear static analysis, so not all steps will be covered in detail. The steps in yellow italics are specific to buckling analyses.
Attach Geometry
Assign Material Properties
Define Contact Regions (if applicable)
Define Mesh Controls (optional)
Include Loads and Supports
Request Buckling Results
Solve the Model
Review Results
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12. … Geometry and Material Properties
Similar to linear static analyses, any type of geometry supported by Design Simulation may be used:
Solid bodies
Surface bodies (with appropriate thickness defined)
Line bodies (with appropriate cross-sections defined)
Only buckling modes and displacement results are available for line bodies.
For material properties, Young’s Modulus and Poisson’s Ratio are required as a minimum
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13. … Contact Regions
Contact regions are available in buckling analyses. However, since this is a purely linear analysis, contact behavior will differ for the nonlinear contact types:
It is important to note the following:
The pinball region will influence some types of contact
All nonlinear contact types are reduced to either “Bonded” or “No Separation” contact.
No Separation contact should be used with caution in buckling analyses, as it provides no stiffness in the tangential direction. This may produce some superfluous buckling modes. Consider using bonded contact instead, if appropriate.
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14. … Loads and Supports
At least one structural load, which causes buckling, should be applied to the model:
All structural loads will be multiplied by the load factor to determine the buckling load. Hence, non-proportional or constant loading is not directly supported (see next slide)
No Given Displacement supports are allowed
No Thermal loading is allowed
Compression-only supports are nonlinear, so they are not recommended for use in buckling analyses
The structure should also be fully constrained
No rigid-body motion should be present in the model. Be sure to constrain the model appropriately.
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15. … Loads and Supports
Special considerations must be given if constant and proportional loads are present.
The user may iterate on the buckling solution, adjusting the variable loads until the load multiplier becomes 1.0 or nearly 1.0.
Consider the example of a pole with self weight WO and an externally applied force A. You can iterate, adjusting the value of A until l = 1.0.
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16. … Requesting Results
Most of the options for buckling analyses are similar to that of static analysis. However, Design Simulation knows to perform a buckling analysis when the Buckling tool is selected under the Solutions Branch:
The Buckling tool adds another branch to the Solutions branch
The Details view of the Buckling branch allows the user to specify the number of buckling modes to find. The default is to find the first buckling mode. Increasing the number of modes to calculate will increase the solution time. However, usually only a few buckling modes are usually desired.
Although most users are only concerned with the first buckling mode, it is generally a good idea to request the first 2 or 3 buckling modes. There may be closely-space buckling modes, so this would tell the user if the model may be susceptible to more than one failure mode.
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17. … Requesting Results
Requested results are located under the Buckling branch:
The buckling modes are controlled by the number of modes to find under the Details view of the Buckling branch
Stress, strain, or directional displacement results can be requested additionally under the Buckling branch
The buckling mode is specified for each stress, strain, or displacement result requested
If stresses or strains are requested for a model already solved, another solution is required.
No result may be requested directly under the “Solution” branch.
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18. … Requesting Results
The corresponding ANSYS commands for the Buckling tool are as follows:
A static analysis with PSTRES,ON is performed first
A buckling analysis (ANTYPE,1) is then run with PSTRES,ON
The buckling modes is set with BUCOPT,LANB,nmodes
The eigenvalue extraction method is always set to Block Lanczos, regardless of the “Solver Type” setting in the Solutions branch
Output requests are limited to what is requested
If any stress or strain results are requested for any modes, the stress results are expanded with MXPAND,,,,YES. Otherwise, MXPAND is not used.
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19. … Solution Options
The solution branch provides details on the type of analysis being performed
For a buckling analysis, none of the options in the Details view of the Solution branch usually need to be changed.
In the majority of cases, “Solver Type” should be left on the default option of “Program Controlled”. It only controls the solver used in the initial static analysis but not the buckling solution method.
“Weak springs” is also meant for the initial static analysis.
“Large Deflection” is not supported for a buckling analysis.
The “Analysis Type” will display “Buckling” for the case of a linear buckling analysis.
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20. … Solution Options
For a linear buckling analysis, none of the solution options have effect. These affect the initial static analysis only.
“Solver Type” can be set to “Direct” or “Iterative,” but it only sets the equation solver for the static analysis (EQSLV), not the buckling eigenvalue extraction method (BUCOPT)
“Weak Springs” are meant for the initial static analysis
One can use ‘weak spring’ option to automatically add COMBIN14 elements for the initial static analysis, but keep in mind that these elements will also be present for the buckling analysis.
“Large Deflection” is not supported for a linear buckling analysis
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21. … Solving the Model
After setting up the model, one can solve the buckling analysis just like any other analysis by selecting the Solve button.
A linear buckling analysis is more computationally expensive than a static analysis on the same model. This is because a static analysis and a buckling analysis are both performed.
The Worksheet tab of the Solution branch provides detailed solution output, including the amount of memory being used and how many modes have been extracted already.
If stress or strain results or more buckling modes are requested after a solution is performed, a new solution is required.
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22. … Reviewing Results
After the solution, the buckling modes can be reviewed
The Load Multiplier for each buckling mode is shown in the Details view. The load multiplier times the actual loads represent the critical load.
The buckling modes represent relative values, not absolute magnitudes. However, these can be used to determine what the failure modes may look like.
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Model shown is from a sample Inventor assembly.

23. … Reviewing Results
The buckling load multipliers can be reviewed in the Worksheet tab of the Bucking branch.
All requested modes will be summarized in the table below
As mentioned earlier, it may be advisable to request more than just the first buckling mode. This allows the user to see if the structure may be able to buckle in more than one way under a given applied load.
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24. C. Workshop 7 Workshop 7 – Linear Buckling Goal:
Verify linear buckling results in ANSYS Workbench for the pipe model shown below. Results will be compared to closed form calculations from a handbook.
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