Lab_assignment_3 _ Truss Buckling Analysis 186o3

MAE 456 FINITE ELEMENT ANALYSIS Truss Buckling Analysis – LAB INSTRUCTIONS LAB ASSIGNMENT 3 Lab Objectives • • • •

Use beam elements in an FEA Eigenvalue buckling analysis. Perform FEA buckling analyses of a simple column and a more complex structure. the buckling analyses with hand calculations. Write-up the finite element stress analysis in an Engineering Report.

Lab Tasks 1. The analysis of the truss structures from Lab Assignment 1 did not for buckling of 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 instead of rod elements. First, to test the capability of NX to perform a beam buckling analysis, a simple column buckling test case is evaluated. A pipe is cast into a concrete floor and used to 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 steel: E = 30x106 psi, ν=0.29. P

D = 1.8 in 10 ft t = 0.15 in Figure 1. Pipe column and cross section Perform an FEA Eigenvalue buckling analysis for this test case to that the FEA results agree closely with Euler’s critical buckling load as given by the equation:

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A non-linear FEA stress analysis with slightly off-center loading can also be used and may be more accurate.

Pcr =

π 2 EI

(0.7 L )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. For this assignment, enter the Pcr from the hand calculation as the magnitude of the load. This should give a BLF of one. 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. Note that the BLF may converge to a number that is slightly different from “1.00”. To perform an Eigenvalue buckling analysis in NX 7.5: a) Create a CAD model of the structure. This can be simple lines or a complete solid. If a complete solid is created, an idealized CAD model with lines must also be created as described in Assignment 1. b) Perform hand calculations to determine the approximate BLF for the member or section of the structure that is anticipated to buckle first. This is the BLF (and safety factor) for the entire structure. c) Create a new FEM file as in Assignment 1. d) In the FEM file, create Mesh Points at the ends of the lines where loads and constraints will be applied. From the menu select: Insert Model Preparation Mesh Points… e) Create the 1D mesh, define the cross-sections and materials, and re-orient any beam cross-sections as Assignment 2. Be sure to use CBEAM elements rather than CROD elements, as the CROD elements cannot represent the bending which occurs in buckling. f) Create a new Simulation. As before, select NX Nastran as the type of simulation, however, this time, in the “Create Solution” dialog box, be sure to select “SEBUCKL 105” as the “Solution Type.” g) Apply the boundary conditions to the mesh points and solve in the usual way. When you get to the results you will notice a second set of results. This second set of results shows the key buckling modes under which the structure would fail. a. 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 will apply.)

b. 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. h) Repeat the analysis with different numbers of elements to observe when the BLF converges. Create a graph to show the convergence. 2. Perform a buckling analysis of your spatial structure from Lab Assignment 1. Replace each rod member that was in compression with four or five beam elements. Redesign the cross-sections of the if necessary to keep the structure from buckling with the prescribed loads. •

When using beam elements, be aware that the nodes will carry moments from one beam element to the next, unlike rod elements. Realize also that, depending on how the details of the t would be designed, there may be stress concentrations where beam elements are ed to each other that are not ed for in this analysis.

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Where the truss structure meets the wall, restrain only translations. Do not restrain rotations, to more closely replicate the situation from Assignment 1.

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To do the check calculations, choose one of the more heavily, compressively loaded . Apply Euler’s buckling equation to that member. In the buckling equation, use the length constant that is the most correct for the given situation, realizing that where beam elements , the translation and rotation are restricted, but not completely, so the correct length constant will be somewhere in between. Study the FEA results to see how much translation and rotation took place. Calculate the Buckling Load Factor by dividing Pcr from Euler’s buckling equation by the axial compressive force expected in that member. The axial force may be from hand calculations or the stress analysis results from Assignment 1.

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that for an FEA buckling analysis: FS =

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Pcr = BLF = Eigenvalue P

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 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 requirements 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 actual physical boundary conditions (loads and restraints) and anticipated results.



Show the mesh and types of elements used



Show how the restraints and loading conditions were applied. Which degrees-of-freedom were restrained at each restraint?

5) Discussion of results of FE analysis

For the new design, for the first positive buckling mode, show how the structure would buckle. What is the factor of safety with respect to the given loads?



For the original design from Assignment 1, compare the factor of safety from (i) the stress analysis, and (ii) the buckling analysis? Which one is the true factor of safety? Which failure would happen first?



Compare the new design with that from Assignment 1. What is the weight of each one? What is the factor of safety for each one?

6) Description of verification of FE analysis

Based on the column buckling test case, how many elements should be used for each member? How many did you use?



Does the buckling mode seem reasonable? Is it as expected? Are the restraint 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 course web page has notes with this assignment on how to identify the most critical member or they can be identified by looking at which (or sub-structure) are most highly deflected in the first positive mode of the buckling FEA. The web page notes also show 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? How heavy is the structure?)

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Be sure there is no confusion about units.

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The report should be 8-10 pages long, having mostly figures and calculations. A template is provided on the course web page.

Submit the assignment electronically as a single document by emailing to [email protected]. Due date/time: Feb. 22.