1. Appendix Seven Linear Buckling Analysis

2. 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: March 29, 2005 Inventory #002215 A7-2

3. … 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. March 29, 2005 Inventory #002215 A7-3

4. … 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: March 29, 2005 Inventory #002215 A7-4

5. … 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. March 29, 2005 Inventory #002215 A7-5

6. … 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 March 29, 2005 Inventory #002215 A7-6