MAE 456 FINITE ELEMENT ANALYSIS
Truss Buckling Analysis – LAB INSTRUCTIONS
LAB ASSIGNMENT 3
Lab Objectives
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Use beam elements in an FEA Eigenvalue buckling analysis.
Perform FEA buckling analyses of a simple column and a more complex structure.
Verify the buckling analyses with hand calculations.
Write-up the finite element stress analysis in an Engineering Report.
Lab Tasks
The analysis of the truss structures from Lab Assignment 1 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 instead of rod elements.
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 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
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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:
Pcr =
π 2 EI
(0.7 L )2
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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 SolidWorks:
a) Create a CAD model of the structure by drawing lines and using a “Weldment” to
turn the lines into “Structural Members.” Be sure to orient the cross-section
correctly using the angle input at the bottom of the “Structural Member” dialog box.
Ensure that a material has been assigned to the CAD model.
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 Simulation Study. Choose “Buckling” as the type of study.
d) For the structural members of the study, select “Edit Definition” and ensure that:
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The type of element is “Beam”.
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The “End1 Connection” and “End2 Connection” are set to “Rigid”.
e) Apply the boundary conditions to the mesh points and solve 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 Factors of Safety”. We are only interested in the
first buckling mode.
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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.)
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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.
f) 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 1. For each structural member that was in compression, change from one
truss element to a sufficient number of beam elements. Record the results. If
necessary, resize the cross-sections of the members to keep the structure from buckling.
Record the new results and get screen captures.
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When using beam elements, be aware that the nodes will carry moments from one
beam element to the next, unlike truss elements. Realize also that, depending on
how the details of the joint would be designed, there may be stress concentrations
where beam elements are joined to each other that are not accounted 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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Where truss members meet each other, free the relative lateral rotations (i.e.,
create pinned joints as opposed to welded joints), to more closely replicate the
situation from Assignment 1. Right-click on the Structural Member and select “Edit
Definition.” Change the End Connections from “Rigid” to “Manual” and select
“Hinge 1st direction” and “Hinge 2nd direction.” Do this for one Structural Member
at a time, confirming each time that the Study still solves.
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To do the check calculations, choose one of the more heavily, compressively loaded
members. 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 join, 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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Remember that for an FEA buckling analysis:
FS =
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Pcr
= 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 should only
be discussed as part of the verification section, to verify that you used a reasonable number
of elements for each member.
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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 supports) and anticipated results.
Show the mesh and types of elements used
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 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 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 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 members (or
sub-structure) are most highly displaced 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? Will it fail first by yielding or buckling? 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.
4. Submit the report electronically as a single PDF document by emailing to
bpbettig@mix.wvu.edu. The PDF document file name must start with your name or mix
username and include the assignment number. The email subject line must indicate the
course and assignment number.
Due date/time: March 5.