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UCONN ANSYS –Module 3 Page 1
Module 4: Buckling of 2D Simply Supported Beam
Table of Contents Page Number
Problem Description 2
Theory 2
Geometry 4
Preprocessor 7
Element Type 7
Real Constants and Material Properties 8
Meshing 9
Solution 11
Static Solution 11
Eigenvalue 14
Mode Shape 15
General Postprocessor 16
Results 18
Validation 18
UCONN ANSYS –Module 3 Page 2
Problem Description:
Nomenclature:
L =200mm Length of beam
b =10mm Cross Section Base
h =1 mm Cross Section Height
P=1N Applied Force
E=200,000
Young’s Modulus of Steel at Room Temperature
=0.33 Poisson’s Ratio of Steel
Moment of Inertia
This module is a simply supported beam subject to two opposite edge compressions until the
material buckles. Buckling is inherently non-linear, but we linearize the problem through the
Eigenvalue method. This solution is an overestimate of the theoretical value since it does not
consider imperfections and nonlinearities in the structure such as warping and manufacturing
defects. We model the beam with 2D elements.
Theory
Buckling load
Hooke’s Law equates stress as shown:
(4.1)
Deriving both sides of equation 3.1 it shows
(4.2)
By solving for equilibrium:
(4.3)
Equation 3.3 is a nonlinear equation, however this equation can be linearized using eigenvalues.
y x
UCONN ANSYS –Module 3 Page 3
Since:
(4.4)
Then:
(4.5)
Plugging in Equation 3.1 for stress we find:
(4.6)
Plugging Equation 3.6 into Equation 3.3, Equation 3.6 becomes
(4.7)
Which is simplifies to:
(4.8)
By integrating two times Equation 3.8 becomes
(4.9)
At the fixed end (x=0), v=0,
, thus 0
At the supported end (x=L), v=0,
, thus 0
Equation 3.9 becomes
(4.10)
Equation 3.6 represents the Differential Equation for a Sin Wave
√
√
(4.11)
A and B are arbitrary constants which are calculated based on Boundary Conditions.
At the fixed end (x=0), v=0 proving B=0. Equation 3.11 becomes
√
(4.12)
But A cannot equal zero or this problem is trivial.
At the supported end (x=L), v=0 Equation 12 becomes
√
(4.13)
Since A cannot equal zero, ( ) must equal zero:
Sin(nπ)=0 for n=(0, 1, 2, 3, 4…..)
UCONN ANSYS –Module 3 Page 4
So:
√
n ≠ 0 or it is trivial (4.14)
We are interested in finding P which is the Critical Buckling Load. Since n can be any integer
greater than zero and a continuous beam has theoretically infinite degrees of freedom there are
infinite amount of eigenvalues ( ).
(4.15)
Where the lowest Buckling Load is at
(4.16)
This is an over estimate so there are certain correction factors (C) to account for this. (C) is
dependent on the beam constraints.
(4.17)
Where C=1 for a fixed-simply supported beam.
So the Critical Buckling Load is
= 41.124 N (4.18)
Geometry
Opening ANSYS Mechanical APDL
1. On your Windows 7 Desktop click the Start button
2. Under Search Programs and Files type “ANSYS”
3. Click on Mechanical APDL (ANSYS) to start
ANSYS. This step may take time.
3
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UCONN ANSYS –Module 3 Page 5
Preferences
1. Go to Main Menu -> Preferences
2. Check the box that says Structural
3. Click OK
Title:
To add a title
1. Utility Menu -> ANSYS Toolbar -> type /prep7 -> enter
2. Utility Menu -> ANSYS Toolbar -> type /Title, “ Title Name” -> enter
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UCONN ANSYS –Module 3 Page 6
Key points
Since we will be using 2D Elements, our goal is to model the length and width
of the beam.
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Keypoints -> On Working Plane
2. Click Global Cartesian
3. In the box underneath, write: 0,0,0 This will create a keypoint at the
Origin.
4. Click Apply
5. Repeat Steps 3 and 4 for the following points in order: 0,0,10
200,0,10
200,0,0
6. Click Ok
7. The Triad in the top left corner is blocking keypoint 1.
To get rid of the triad, type
/triad,off in Utility Menu -> Command Prompt
8. Go to Utility Menu -> Plot -> Replot
Areas
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Areas -> Arbitrary -> Through KPs
2. Select Pick
3. Select Min, Max, Inc
4. In the space below, type 1,4,1. If you are familiar with programming,
the ‘Min,Max, Inc’ acts as a FOR loop, connecting nodes 1 through 4
incrementing (Inc) by 1.
5. Click OK
6. Go to Plot -> Areas
7. Click the Top View tool
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UCONN ANSYS –Module 3 Page 7
Your beam should look as below:
Saving Geometry
We will be using the geometry we have just created for 3 modules. Thus it would be convenient
to save the geometry so that it does not have to be made again from scratch.
1. Go to File -> Save As …
2. Under Save Database to
pick a name for the Geometry.
For this tutorial, we will name
the file ‘Buckling simply
supported’
3. Under Directories: pick the
Folder you would like to save the
.db file to.
4. Click OK
Preprocessor
Element Type
1. Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete
2. Click Add
3. Click Shell -> 4node181
4. Click OK
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UCONN ANSYS –Module 3 Page 8
SHELL181 is suitable for analyzing thin to moderately-think shell structures. It is a 4-node
element with six degrees of freedom at each node: translations in the x, y, and z directions, and
rotations about the x, y, and z-axes. (If the membrane option is used, the element has
translational degrees of freedom only). The degenerate triangular option should only be used as
filler elements in mesh generation. This element is well-suited for linear, large rotation, and/or
large strain nonlinear applications. Change in shell thickness is accounted for in nonlinear
analyses. In the element domain, both full and reduced integration schemes are supported.
SHELL181 accounts for follower (load stiffness) effects of distributed pressures
Real Constants and Material Properties
Now we will add the thickness to our beam.
1. Go to Main Menu -> Preprocessor ->
Real Constants -> Add/Edit/Delete
2. Click Add
3. Click OK
4. Under Real Constants for SHELL18 ->
Shell thickness at node I TK(I) enter 1
for the thickness
5. Click OK
6. Click Close
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UCONN ANSYS –Module 3 Page 9
Now we must specify Young’s Modulus and Poisson’s Ratio
1. Go to Main Menu -> Preprocessor -> Material Props -> Material Models
2. Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic
3. Input 2E5 for the Young’s Modulus (Steel in mm) in EX.
4. Input 0.3 for Poisson’s Ratio in PRXY
5. Click OK
6. of Define Material Model Behavior window
Meshing
1. Go to Main Menu -> Preprocessor ->
Meshing -> Mesh Tool
2. Go to Size Controls: -> Global -> Set
3. Under NDIV No. of element divisions put 2.
This will create a mesh of 2 elements across the
smallest thickness, 4 in total.
4. Click OK
5. Click Mesh
6. Click Pick All
3 4
5
2
6
2
3
4
5
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UCONN ANSYS –Module 3 Page 10
7. Go to Utility Menu -> Plot -> Nodes
8. Go to Utility Menu -> Plot Controls -> Numbering…
9. Check NODE Node Numbers to ON
10. Click OK
The resulting graphic should be as shown using the Top View :
This is one of the main advantages of ANSYS Mechanical APDL vs ANSYS Workbench in that we
can visually extract the node numbering scheme. As shown, ANSYS numbers nodes at the left
corner, the right corner, followed by filling in the remaining nodes from left to right.
9
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UCONN ANSYS –Module 3 Page 11
Solution
There are two types of solution menus that ANSYS APDL provides; the Abridged solution menu
and the Unabridged solution menu. Before specifying the loads on the beam, it is crucial to be in
the correct menu.
Go to Main Menu -> Solution -> Unabridged menu
This is shown as the last tab in the Solution menu. If this reads “Abridged menu” you are
already in the Unabridged solution menu.
Static Solution
Analysis Type
1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis
2. Choose Static
3. Click OK
4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options
5. Under [SSTIF][PSTRES] Stress stiffness or prestress select Prestress ON
6. Click OK
Prestress is the only change necessary in this window and it is a crucial step in obtaining a final
result for eigenvalue buckling.
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UCONN ANSYS –Module 3 Page 12
Displacement
1. Go to Main Menu -> Solution ->Define Loads ->Apply ->
Structural ->Displacement -> On Nodes
2. Select Min, Max, Inc
3. In the space below, type 1,3,1. If you are familiar with programming,
the ‘Min,Max, Inc’ acts as a FOR loop, connecting nodes 1 through 3
incrementing by 1. This selects the nodes on the far left of the beam
4. Click OK
5. Under Lab2 DOFs to be constrained select UX, UY and ROTX
6. Under VALUE Displacement value enter 0
7. Click OK
8. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural ->
Displacement -> On Nodes 9. Select Pick -> Box
10. Box nodes: 4, 45 and 44 (far right nodes)
11. Click OK
12. Under Lab2 DOFs to be constrained highlight UY
and ROTX
13. Under VALUE Displacement value enter 0
14. Click OK
15. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural ->
Displacement -> On Nodes 16. Select Pick -> Single
17. Select nodes 2 and 4 (one node on the left of the beam and one node on the right)
18. Under Lab2 DOFs to be constrained highlight UZ
19. Under VALUE Displacement value enter 0
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UCONN ANSYS –Module 3 Page 13
20. Click OK
This creates the fixed end on the left and roller support on the right, while constraining
movement in the Z-direction
Loads
1. Go to Main Menu -> Solution -> Define Loads ->
Apply -> Structural ->Force/Moment -> On Nodes
2. Select Pick -> Single -> List of Items
3. In the space provided, type 45. Press OK.
4. Under Direction of force/mom select FX
5. Under VALUE Force/moment value enter -1
6. Click OK
WARNING: UX, UY and ROTX might already be highlighted, if so, leave
UY and ROTX highlighted and click UX to remove it from the selection.
Failure to only constrain UY and ROTX will result in incorrect results.
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USEFUL TIP: The force value is only a magnitude of 1 because
eigenvalues are calculated by a factor of the load applied, so having a
force of 1 will make the eigenvalue answer equal to the critical load.
UCONN ANSYS –Module 3 Page 14
Solve
1. Go to Main Menu -> Solution -> Solve -> Current LS
2. Go to Main Menu -> Finish
Eigenvalue
1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis
2. Choose Eigen Buckling
3. Click OKGo to Main Menu -> Solution -> Analysis Type ->Analysis Options
4. Under NMODE No. of modes to extract input 4
5. Click OK
6. Go to Main Menu -> Solution -> Solve -> Current LS
7. Go to Main Menu -> Finish
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UCONN ANSYS –Module 3 Page 15
Mode Shape
1. Go to Main Menu -> Solution -> Analysis Type -> ExpansionPass
2. Click [EXPASS] Expansion pass to ensure this is turned on
3. Click OK
4. Go to Main Menu -> Solution -> Load Step Opts -> ExpansionPass ->
Single Expand -> Expand Modes…
5. Under NMODE No. of modes to expand input 4
6. Click OK
7. Go to Main Menu -> Solution -> Solve -> Current LS
8. Go to Main Menu -> Finish
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UCONN ANSYS –Module 3 Page 16
General Postprocessor
Buckling Load
Now that ANSYS has solved these three analysis lets extract the lowest eigenvalue. This
represents the lowest force to cause buckling.
Go to Main Menu -> General Postproc -> List Results -> Detailed Summary
Results for Buckling Load:
P= 41.172 N
Mode Shape
To view the deformed shape of the buckled beam vs. original beam:
1. Go to Main Menu -> General Postproc -> Read Results -> By Pick
2. Select the lowest Eigenvalue: Set 1 -> Click Read
3
2
UCONN ANSYS –Module 3 Page 17
3. Click Close
4. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape
5. Under KUND Items to be plotted select Def + undeformed
6. Click OK
The graphics area should look as below using the Oblique view:
6
5
UCONN ANSYS –Module 3 Page 18
Results
The percent error (%E) in our model can be defined as:
(
) = 0.12%
This shows that there is no error baseline using one dimensional elements.
Validation
Theoretical 320 Elements 80 Elements 4 Elements
Critical Buckling
Load 41.124 N 41.142 41.172 67.195
Percent Error 0% .0438% 0.12% 63.4%
This table provides the critical buckling loads and corresponding error from the Theory (Euler),
and two different ANSYS results; one with 80 elements and one with 4 elements. This is to
prove mesh independence, showing with increasing mesh size, the answer approaches the
theoretical value. The results here show that using a coarse mesh of 4 elements creates an
unacceptable error. As mesh is refined it converges to a more accurate answer. The eigenvalue
buckling method over-estimates the “real life” buckling load. This is due to the assumption of a
perfect structure, disregarding flaws and nonlinearities in the material. There is no such thing as
a perfect structure so the structure will never actually reach the eigenvalue load that is calculated.
Thus, it is important to consider conservative factors of safety into your design for safe measure.
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