MAE 456 FINITE ELEMENT ANALYSIS Truss Buckling Analysis – LAB INSTRUCTIONS LAB ASSIGNMENT 3
Lab Objectives
Perform a FEA Eigenvalue buckling analysis of a simple column. Perform a FEA Eigenvalue buckling analysis of a 3D truss structure. Verify the buckling analyses with hand calculations. Write‐up the FEA in an Engineering Report.
Lab Tasks
The analyses of the truss structure from Lab Assignments 1 and 2 did not account for buckling of members in compression. For slender shapes with heavy axial compressive loads, the structure will fail by buckling before it fails by yielding in axial compression. An “Eigenvalue” buckling analysis can be performed to determine the force level at which the buckling will occur.1 To perform a buckling FEA, beam elements must be used for members in compression.
1. First, a simple column buckling test case will be performed to test the capability of SolidWorks to perform a beam buckling analysis, as part of the verification. A pipe is cast into a concrete floor and used to support an axial load as shown in Figure 1. The top of the pipe may move vertically but may not move horizontally. (It may also rotate at the top as shown.) The cross section of the column has a 1.8 in. outside diameter and a wall thickness of 0.15 in. as shown. The pipe is made of AISI 1020 steel. Apply a force P = 1000 lbf.
P
D = 1.8 in 10 ft
t = 0.15 in
Figure 1. Pipe column and cross section
1 A non‐linear FEA stress analysis with slightly off‐center loading can also be used and may be more accurate. Perform an FEA Eigenvalue buckling analysis for this test case to verify that the FEA results agree closely with Euler’s critical buckling load as given by the equation: 2EI P . cr 0.7L 2
Here Pcr is the load at which buckling failure will occur if the load P is increased gradually. The “Buckling Load Factor” (BLF) is given by BLF = Pcr/P. It gives the factor by which the load(s) can be increased before buckling occurs. Thus the BLF also gives the Factor of Safety (FS) for a buckling analysis. Graph the BLF as a function of the number of elements. Determine the minimum number of beam elements that can accurately represent the solution to 2 decimal places.
To perform an Eigenvalue buckling analysis of a truss structure in SolidWorks: a) Create a CAD model of the structure by drawing lines and using a “Weldment” to define “Structural Members” based on the lines. Be sure to orient the cross‐section correctly using the angle input at the bottom of the “Structural Member” dialog box. Alternatively, use regular solid modeling operations like extrude or sweep to model truss members. Ensure that a material has been assigned to the CAD model. b) Perform hand calculations to determine the approximate BLF. c) Create a new Simulation Study. Choose “Buckling” as the type of study. d) For the structural members of the study, ensure that beam elements are being used: For members not‐created using “Structural Members,” in the Simulation Manager, right‐click on the solid () and select “Treat as Beam…” so that the member is treated as a 1‐D element. Ensure that the icon changes to . For members created using “Structural Members”, under the cut‐list (), under the list of structural members (), ensure that the member appears as a beam () and not as a truss element (). If necessary, right‐click on the icon, select “Edit Definition …” and select “Beam” as the Type. e) Set the analysis to give the first ten buckling modes. Right‐click on the buckling study () and select “Properties…” Set the number of buckling modes to 10 and press OK. f) Apply the boundary conditions to the joints and run the study in the usual way. You will notice that it gives a “Mode Shape” number and “Load Factor” in the information at the top left. This information can also be obtained by right‐clicking on the Results and selecting “List Buckling Factor of Safety”. The displayed mode shape can be changed by right‐clicking on the mode shape amplitude result () and clicking on “Edit Definition…” Each mode has an Eigenvalue which can be directly interpreted as the Buckling Load Factor. If the loads are increased gradually and proportionately, the structure will fail at the mode with the lowest positive BLF. (If the directions of all the loads are reversed and increased proportionately, then the lowest negative BLF may apply.) Note that the displacements and stresses from an Eigenvalue analysis are scaled arbitrarily. In real‐life the structure will just continue a sudden deflection according to the mode shape until the structure yields fully or fractures. g) Repeat the analysis with fewer and fewer elements to observe when the BLF converges. In SolidWorks, this can be accomplished by right‐clicking on the Mesh in the Simulation Manager and selecting “Apply Mesh Control.” Select the Structural Member and set the “Number of elements” as desired. Create a graph to show the convergence. Make a conclusion about the minimum number of elements needed to achieve the desired accuracy.
2. Perform a buckling analysis of your structure and boundary conditions from Lab Assignment 2. The only difference in setting up the buckling analysis is selecting “Buckling” instead of “Static” when creating a new study. For the check calculations, calculate the BLF for the member or section of the structure that is anticipated to buckle first. This will be the BLF (and safety factor) for the entire structure. See the Appendix for more detailed information. Remember that for an FEA buckling analysis: P FS cr BLF Eigenvalue P Remember that the displacements and stresses shown in an Eigenvalue analysis are scaled arbitrarily. Do not quote any actual stress values from the buckling FEA.
3. Complete a report on the truss buckling analysis. The column buckling analysis does not need to be written up. The report should include the following sections:
1) Summary (What is this report about? What structure did you analyze? What kind of failure did you study? What were the conclusions?)
2) Introduction (What were the objectives and constraints on the design? What kind of failures needed to be considered?)
3) Description of design (including geometric and material information, and mass) 4) Description of failure scenario and the FE model . Describe the overall situation. What types of failure are studied? What assumptions are made? . Describe the actual physical boundary conditions (loads and supports) for each load case. . Show the mesh and types of elements used. What simplifications did you make for your model? . Show how the supports and loading conditions were applied. Which degrees‐of‐freedom were restrained at each support? 5) Discussion of results of FE analysis . For each load case, for the first positive buckling mode, show how the structure would buckle. What is the factor of safety with respect to the given loads? . Compare the factor of safety with that from Assignment 2. Which one is the true factor of safety? Which failure would happen first? 6) Description of verification of FE analysis . Does the buckling mode seem reasonable? Is it as expected? Are the support conditions maintained as intended? Is it evident that rotation was restrained or allowed as intended?
. By hand, calculate Euler’s buckling load (Pcr) for the most critical member (or sub‐structure). The most critical member (or sub‐structure) is the one that experiences the buckling first. The Appendix has notes on how to identify the most critical member or they can be identified by looking at which members (or sub‐structure) are most highly displaced in the first positive mode of the buckling FEA. The Appendix also shows how to calculate Euler’s buckling load for different end conditions.
. Compute the Buckling Load Factor (BLF) for the critical member (BLF = Pcr / P), where P is the actual load that is expected for that member. The actual load can be obtained from the hand calculations from Assignment 1 or by multiplying the stress from the Assignment 1 FEA by the cross‐sectional area. . Compare the BLF for the critical member (which is also the BLF for the entire structure) with the BLF from the FEA.
7) Conclusions (Will the structure fail? What is the safety factor? Will it fail first by yielding or buckling? How heavy is the structure?)
Be sure there is no confusion about units. The report should be about 15 pages long, having mostly figures and calculations. A template is provided on the course web page.
4. Submit the report electronically via eCampus. The PDF document file name must start with your name and include the assignment number. Due date/time: April 5.
Appendix – Practical Aspects of Using Euler’s Buckling Formula