Linear Buckling Analysis

1. Chapter Seven Linear Buckling Analysis

2. Chapter Overview • In this chapter, performing linear buckling analyses in Simulation will be covered. • In 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. August 26, 2005 Inventory #002265 7-2

3. F F Stable Unstable A. 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). August 26, 2005 Inventory #002265 7-3

4. … Background on Buckling • Eigenvalue or linear buckling analysis predicts the theoretical buckling strength of an ideal linear elastic 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. August 26, 2005 Inventory #002265 7-4

5. … Background on Buckling • 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 imperfections 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. August 26, 2005 Inventory #002265 7-5

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 August 26, 2005 Inventory #002265 7-6

7. … 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:Assumptions: • [K] and [S] are constant: • Linear elastic material behavior is assumed • Small deflection theory is used, and no nonlinearities included • Some additional restrictions: • Nonzero displacement supports or thermal loads are not allowed • It is important to remember these assumptions related to performing linearbucklinganalyses in Simulation. August 26, 2005 Inventory #002265 7-7

8. 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 August 26, 2005 Inventory #002265 7-8

9. … Geometry and Material Properties • Similar to linear static analyses, any type of geometry supported by Simulation may be used in buckling analyses: • 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. • Although Point Masses may be included in the model, only inertial loads affect point masses, so the applicability of this feature may be limited in buckling analyses • For material properties, Young’s Modulus and Poisson’s Ratio are required as a minimum August 26, 2005 Inventory #002265 7-9

10. … 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: • 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. August 26, 2005 Inventory #002265 7-10

11. … 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 not recommended • The structure should be fully constrained, no rigid-body motion should be present in the model. August 26, 2005 Inventory #002265 7-11

12. … 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. August 26, 2005 Inventory #002265 7-12

13. 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. … Requesting Results • Most of the options for buckling analyses are similar to that of static analysis. • Simulation triggers a buckling analysis when the Buckling tool is inserted: • 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. August 26, 2005 Inventory #002265 7-13

14. … Requesting Results • Requested results are located under the Buckling branch: • Select the number of modes to find under the Details view of the Buckling branch • Stress, strain, or directional displacement results can be requested 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 additional results may be requested directly under the “Solution” branch. August 26, 2005 Inventory #002265 7-14

15. … 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. • “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 meant for theinitial static analysis. • “Large Deflection” is not supported fora buckling analysis. August 26, 2005 Inventory #002265 7-15

16. … Solving the Model • After setting up the model, solve the buckling analysis via the Solve button. • A linear buckling analysis is more computationally expensive than a static analysis on the same model. • If a “Solution Information” branchwas requested, detailed solutionoutput is available in the Worksheettab of that branch • If stress or strain results or morebuckling modes are requested aftera solution is performed, a newsolution is required. August 26, 2005 Inventory #002265 7-16

17. … 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 applied loads represent the critical load • The buckling modes can be used to determine what the failure modes may look like August 26, 2005 Inventory #002265 7-17 Model shown is from a sample Inventor part.

18. … Reviewing Results • Interpreting the Load Multiplier (): • The tower model below has been solved twice. In the first case a unit load is applied. In the second an expected load applied (see next page) August 26, 2005 Inventory #002265 7-18

19. … Reviewing Results • Interpreting the Load Multiplier (): Using the actual load: Using a unit load: August 26, 2005 Inventory #002265 7-19

20. … Reviewing Results • The buckling load multipliers can be reviewed in the Worksheet tab of the Bucking branch. • It is good practice to request more than one buckling mode to see if the structure may be able to buckle in more than one way under a given applied load. August 26, 2005 Inventory #002265 7-20

21. C. Workshop 7 – Linear Buckling • Workshop 7 – Linear Buckling • Goal: • Verify linear buckling results in Simulation for the pipe model shown below. Results will be compared to closed form calculations from a handbook. August 26, 2005 Inventory #002265 7-21