1. Buckling of Arch Workshop Twelve REFERENCE: Training Manual Geometric Instability (9-56)

2. Workshop Supplement September 30, 2001 Inventory #001492 W12-2 Workshop Twelve: Buckling of Arch Geometric Instability (Buckling) Purpose Run both linear and nonlinear buckling analyses of an arch. Goal Become familiar with the procedure for performing eigenvalue buckling, adding a geometric imperfection, and running nonlinear buckling analysis.

3. Workshop Supplement September 30, 2001 Inventory #001492 W12-3 1.Read in the input file named “ANL_W12_ArchBuck.inp” Use the GUI menu method: –Utility Menu > File > Read Input From… Select the file “ANL_W12_ArchBuck.inp” Click on [OK] or Command Input method: –/INPUT,ANL_W12_ArchBuck,inp Notes: This will read in an input file which will generate the geometry, loads, and boundary conditions for the current exercise. We will have an arch meshed with SHELL181 elements. The current database will be erased, and the jobname will be changed to “Exercise_12”. The graphics and other settings will also be set to be consistent with these slides. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

4. Workshop Supplement September 30, 2001 Inventory #001492 W12-4 Before continuing with the exercise, you can also examine the mesh and the boundary conditions on the model to become more familiar with it. The model is comprised of SHELL181 elements. A pressure of 1 unit is applied on the top of the arch. The ends are fixed, and symmetry boundary conditions are applied along the edges. A linear eigenvalue buckling analysis will be performed first to determine the critical load, as well as using the first mode as an initial geometric imperfection to be applied to the model. A nonlinear static buckling analysis will then be performed on the model. External Pressure Fixed B.C. at ends Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

5. Workshop Supplement September 30, 2001 Inventory #001492 W12-5 2.Verify the element options for SHELL181 Use the GUI menu method: –Main Menu > Preprocessor > Element Type > Add/Edit/Delete … Select “Type 1 SHELL181” Click on [Options] Verify element options, then click on [OK] Select [Close] or Command Input method: –/PREP7 –ETLIST Notes: Element type 1 will use SHELL181. The default element options will be used. If in-plane response dominates, use of “Full integration w/ incompatible modes” for “Integration option” (KEYOPT(1)) is recommended. Since this model does not exhibit any significant in-plane response, “Reduced integration” is fine. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

6. Workshop Supplement September 30, 2001 Inventory #001492 W12-6 3.Specify prestressed static analysis Use the GUI menu method: –Main Menu > Preprocessor > Loads > Analysis Options … Select “Prestress ON” from “Stress stiffness or prestress” menu. Select [OK] or Command Input method: –PSTRES,ON Notes: In order to perform linear eigenvalue buckling analysis, recall that a static linear analysis with prestress effects turned on must be done first. The prestress flag tells ANSYS to save the stress- stiffness matrix which is required for the eigenvalue buckling problem. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

7. Workshop Supplement September 30, 2001 Inventory #001492 W12-7 4.Solve the linear static analysis Use the GUI menu method: –Main Menu > Solution > -Solve- Current LS After reviewing the status window, close it by selecting “File > Close” To start the analysis, click on [OK] or Command Input method: –/SOLU –SOLVE Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

8. Workshop Supplement September 30, 2001 Inventory #001492 W12-8 5.Specify eigenvalue buckling analysis Use the GUI menu method: –Main Menu > Finish –Main Menu > Solution > -Analysis Type- New Analysis … Select “Eigen Buckling” Click on [OK] to close the dialog box or Command Input method: –FINISH –/SOLU –ANTYPE,BUCKLE Notes: After performing the linear static analysis, we will now perform the linear buckling analysis. Note that we must exit the solution processor first when specifying a new analysis type. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

9. Workshop Supplement September 30, 2001 Inventory #001492 W12-9 6.Specify the mode extraction method Use the GUI menu method: –Main Menu > Solution > Analysis Options … Specify “Block Lanczos” as the “Mode extraction method” Enter “3” for “No. of modes to extract” Click on [OK] to close the dialog box or Command Input method: –BUCOPT,LANB,3 Notes: The Block Lanczos method is the preferred eigenvalue extraction method due to efficiency. Although we are usually interested only in the first buckling mode, it is a good idea to extract a few modes. This is to verify that there are not many modes close to each other. If this is the case, the structure may be imperfection-sensitive, so a few nonlinear runs may be required to determine the critical load. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

10. Workshop Supplement September 30, 2001 Inventory #001492 W12-10 7.Specify number of modes to expand Use the GUI menu method: –Main Menu > Solution > -Load Step Opts- Expansion Pass > Expand Modes … Enter “3” for “No. of modes to expand” Select “Yes” for “Calculate elem results” if you want to view relative stress results. Click on [OK] to close the dialog box or Command Input method: –MXPAND,3,,,YES Notes: In the previous slide, we specified the number of modes to extract (eigenvalues). In this slide, we are specifying the number of modes to expand (eigenvectors). Usually, these two values will be the same. However, in some cases, users may be interested in doing a separate expansion pass to reduce results file sizes, for example. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

11. Workshop Supplement September 30, 2001 Inventory #001492 W12-11 8.Solve the linear buckling analysis Use the GUI menu method: –Main Menu > Solution > -Solve- Current LS After reviewing the status window, close it by selecting “File > Close” To start the analysis, click on [OK] or Command Input method: –/SOLU –SOLVE Notes: In the status window, we can verify that a eigenvalue buckling analysis will be performed using the Block Lanczos extraction method. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

12. Workshop Supplement September 30, 2001 Inventory #001492 W12-12 9.Review first buckling mode Use the GUI menu method: –Main Menu > General Postproc > Results Summary... Note that the three load factors are 11.7, 19.1, and 34.7. Select the first buckling mode and click on [Read]. Select [Close] –Main Menu > General Postproc > Plot Results > Deformed Shape... Select “Def + undef edge” Click on [OK] to plot the first mode, as shown on the right or Command Input method: –/POST1 –SET,LIST,2 –SET,FIRST –PLDISP,2 Notes: The first buckling mode is an asymmetric one. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

13. Workshop Supplement September 30, 2001 Inventory #001492 W12-13 10.Retrieve the first load factor as a parameter MYLOADF Use the GUI menu method: –Utility Menu > Parameters > Get Scalar Data... Select “Current Settings” from the left and “Current settings” from the right. Click on [OK] Type “MYLOADF” for “Name of parameter to be defined”. Select “Postprocessing” from the left and “Freq of ld step” from the right Click on [OK] to close the dialog box or Command Input method: –*GET,myloadf,active,,set,freq Notes: Review the Output Window/File to verify that the parameter MYLOADF is assigned the value of the first load factor (around 11.7). Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

14. Workshop Supplement September 30, 2001 Inventory #001492 W12-14 11.Add a small imperfection, based on the first buckling mode Use the GUI menu method: –Main Menu > Preprocessor > -Modeling- Update Geom... Enter “0.005” for the “Scaling Factor” Enter “1” for “Load step” Enter “1” for “Substep” Click on “Browse” and select the results file, “Exercise_12.rst” Click on [OK] to close the dialog box or Command Input method: –UPGEOM,0.005,1,1,Exercise_12,rst Notes: The buckling modes are normalized to max displacement of 1.0. Since the shell thickness is 0.5, we will add an imperfection of 1/100 the thickness (0.005). Although this is a very small value, it will suffice for this example. Additional Notes: The “load step” value should always be “1” for buckling analyses. The “substep” value is the buckling mode number to be used for the imperfection. Since we are using the first buckling mode, enter “1”. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

15. Workshop Supplement September 30, 2001 Inventory #001492 W12-15 12.Reenter Solution processor and specify a static analysis. Use the GUI menu method: –Main Menu > Solution > -Analysis Type- New Analysis … Select “Static” Click on [OK] to close the dialog box or Command Input method: –/SOLU –ANTYPE,STATIC Notes: After performing the linear static analysis, we will now perform the linear buckling analysis. Note that we must exit the solution processor first when specifying a new analysis type. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

16. Workshop Supplement September 30, 2001 Inventory #001492 W12-16 13.Specify nonlinear solution options. Use the GUI menu method: –Main Menu > Solution > -Analysis Type- Sol’n Control … Select “Large Displacement Static” for “Analysis Options” Enter “MYLOADF*1.1” for “Time at end of loadstep” Enter “20” for “Number of substeps”, “1e5” for “Max no. of substeps”, and “20” for “Min no. of substeps” Select “Write every substep” for “Frequency” Do NOT click on [OK] yet. Go to next slide. or Command Input method: –NLGEOM,ON –TIME,myloadf*1.1 –NUSBST,20,1e5,20 –OUTRES,ALL,ALL Notes: “Time” has been entered as “MYLOADF*1.1”, so we can use “TIME” to represent the actual load value. All intermediate results will be saved for postprocessing purposes. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

17. Workshop Supplement September 30, 2001 Inventory #001492 W12-17 14.Turn on line search Use the GUI menu method: –Solution Control Dialog Box > Nonlinear tab Select “On” for “Line search” Click on [OK] to close the dialog box or Command Input method: –LNSRCH,ON Notes: Line search helps for nonlinear buckling analyses. As will be noted later, the MAX DOF INC oscillates, and line search helps aid convergence in these situations. By default, line search is only turned on for contact analyses, so this must be manually done if contact elements are not present, as in this situation. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

18. Workshop Supplement September 30, 2001 Inventory #001492 W12-18 15.Scale pressure loads by 1.1*MYLOADF Use the GUI menu method: –Main Menu > Solution > -Loads- Operate > -Scale FE Loads- Surface Loads... Enter “1.1*MYLOAD” for “Scale factor” Click on [OK] to close the dialog box or Command Input method: –SFSCALE,ALL,1.1*myloadf Notes: We used a pressure of 1.0 unit. We need to scale the loads to cause buckling. We use a factor of 1.1 to ensure that the critical load will be reached. Earlier, ending time was also set to 1.1*MYLOADF. This way, “time” will also represent the current pressure load applied to the model. Only FE loads can be scaled, so please keep this in mind when using this command. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

19. Workshop Supplement September 30, 2001 Inventory #001492 W12-19 16.Solve the nonlinear buckling analysis Use the GUI menu method: –Main Menu > Solution > -Solve- Current LS After reviewing the status window, close it by selecting “File > Close” To start the analysis, click on [OK] or Command Input method: –/SOLU –SOLVE Notes: In the status window, we see that nonlinear geometric effects are turned on. Line search is on, and all intermediate results will be saved to the results file. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

20. Workshop Supplement September 30, 2001 Inventory #001492 W12-20 The solution will not converge, as expected, because at the point of buckling, the stiffness will be zero. If we wanted to examine post-buckling behavior, we need to use other methods discussed later in Chapter 9 (also see Workshop 13). An error and warning will be displayed, similar to the ones on the right. These messages indicate that the solution has not converged, and the unconverged results are labeled as substep 999999. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

21. Workshop Supplement September 30, 2001 Inventory #001492 W12-21 17.Plot the force-deflection curve at Node #69 Use the GUI menu method: –Main Menu > General Postproc > Variable Viewer... Select the results file, “Exercise_12.rst” Click on the “+” icon (Add Data) Select “Nodal Solution > DOF Solution > X- Component of displacement”. Click on [OK] Select a node 1/4 the distance (for example, Node #69 as shown on right). Click on [OK] or Command Input method: –/POST26 –NSOL,2,69,U,X Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

22. Workshop Supplement September 30, 2001 Inventory #001492 W12-22 Plot the force-deflection curve (cont’d) Use the GUI menu method: –Variable Viewer Click on “X-Axis” for “UX_2” row Select “TIME” Click on the “Graph” icon (Graph Data) or Command Input method: –XVAR,2 –PLVAR,1 Notes: The plot is reversed because the deflection is in the negative x-direction. In the next slide, we will reverse the x-axis to plot the force- deflection curve in a more familiar manner. Also, the author finds putting markers on the graph very useful in determining where the converged substeps lie. This helps to determine if the non-convergence is due to numerical or structural instability (if slope is near zero). Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

23. Workshop Supplement September 30, 2001 Inventory #001492 W12-23 Plot the force-deflection curve (cont’d) Use the GUI menu method: –Utility Menu > PlotCtrls > Style > Graphs > Modify Curve... Select “Single” for “Thickness of curves” Select “Triangles” for “KEY to define marker type”. Click on [OK] –Utility Menu > PlotCtrls > Style > Graphs > Modify Axes... Select “Yes” for “Reverse order X-axis values” Click on [OK] –Utility Menu > Plot > Replot or Command Input method: –/GTHK,CURVE,1 –/GMARKER,1,1,1 –/GROPT,REVX,1 –/REPLOT Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

24. Workshop Supplement September 30, 2001 Inventory #001492 W12-24 The resulting force-deflection curve should look similar to the one below. The limit load is near 11.2. This is a little less than the value of 11.7 determined by eigenvalue buckling done earlier (see MYLOADF parameter). Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)

25. Workshop Supplement September 30, 2001 Inventory #001492 W12-25 If time permits, rerun the nonlinear analysis with line search off. You may note that the resulting behavior is quite oscillatory, as shown in the graph below. Line search helps convergence and stabilize the solution. Workshop Twelve: Buckling of Arch … Geometric Instability (Buckling)