Module 5: Axisymmetric Buckling of a Thin Walled Sphere
Table of Contents
Page Number
Problem Description
2
Theory
2
Geometry
4
Preprocessor
Element Type
Real Constants and Material Properties
Meshing
7
7
8
9
Solution
Static Solution
Eigenvalue
Mode Shape
General Postprocessor
11
11
14
15
16
Results
18
Validation
18
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
Page 1
Problem Description:
9) External Pressure buckling of a thin walled sphere
t
P
r
Nomenclature:
r = 2m
P = 1 Pa
t = .02m
E= 210*
= 0.3
Sphere Radius
External Pressure
Sphere Thickness
Youngs Modulus
Poisson’s Ratio
This module takes a sphere and applies an external pressure field until buckling occurs. Taking
advantage of symmetry, this module will incorporate 2D PLANE elements along an axial cross
section swept across half the sphere. Buckling is inherently non-linear, but we will 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. This module will be compared against analytical results in an elasticity
textbook.
Theory
When a circular shell is under uniform axial compression, axisymmetric buckling is often the
lowest buckling mode. At the start of buckling, the strain energy is increased by midsurface
strain in the circumferential direction, bending, and axial compression. At this critical buckling
load, the increase in strain energy is equal to the work done by the uniform pressure owing to
axial straining and bending as the shell deflects. Thus:
(9.1)
Through a series of derivations using Hamilton’s Principle we find that the critical pressure is:
√
= 25.419556 MPa
(9.2)
We can also find the length of half-sine waves into which the shell buckles:
√
= 0.346m
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
(9.3)
Page 2
This can help us calculate the number of half-sine waves in our figure, since we modeled half of
a sphere we use simply pi multiplied by the radius:
= 18.36
(9.4)
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
Preferences
1
1. Go to Main Menu -> Preferences
2. Check the box that says Structural
3. Click OK
2
1
2
3
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Keypoints
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Keypoints -> On Working Plane
2. Click Global Cartesian
3. In the box underneath, write: 0,2.02,0
4. Click Apply
5. Repeat Steps 3 and 4 for the following points in order:
2.02,0,0
0,-2.02,0
0,2,0
2,0,0
0,-2,0
6. Click Ok
2
6
Arc
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Lines -> Arcs -> By End KPs & Rad
2. Select Pick
3. Select List of Items
4. Type 1,3 for the end points.
5. Click Ok
6. Window will pop up again, Type 2 for the midpoint
7. Click OK
8. Under RAD Radius of the arc type 2.02 for the outer radius
9. Click OK
10. Repeat Steps 1 through 9 for key points 4,6 and 5 with an
inner radius of 2
8
2
3
4
5
9
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Line
2
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Lines -> Lines -> Straight Line
2. Select Pick
3. Select List of Items
4. Type 1,4 for points previously generated.
5. Click Apply
6. Type 3,6
7. Click OK
Area
1.
2.
3.
4.
3
Go to Utility Menu -> Plot -> Lines
Go to Utility Menu -> Plot Controls -> Numbering…
Check LINE Line numbers to ON
Click OK
4
7
5
3
4
5. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Areas -> Arbitrary -> By Lines
6. Select Pick
7. Select List of Items
8. Type 4,1,3,2 for lines previously generated.
9. Click OK
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
8
9
Page 5
The resulting graphic should be as shown:
Saving Geometry
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
2
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
4
3
Page 6
Preprocessor
Element Type
1.
2.
3.
4.
Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete
Click Add
Click Solid -> Axi-har 4node 25
Click OK
3
4
PLANE25 is used for 2-D modeling of axisymmetric structures with nonaxisymmetric loading.
Examples of such loading are bending, shear, or torsion. The element is defined by four nodes
having three degrees of freedom per node: translations in the nodal x, y, and z direction. For
cross section nodal coordinates, these directions correspond to the radial, axial, and tangential
directions, respectively. Unless otherwise stated, the model must be defined in the Z = 0.0 plane.
The global Cartesian Y-axis is assumed to be the axis of symmetry. Further, the model is
developed only in the +X quadrants. Hence, the radial direction is in the +X direction.
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Real Constants and Material Properties
We will specify Young’s Modulus and Poisson’s Ratio
1.
2.
3.
4.
5.
6.
Go to Main Menu -> Preprocessor -> Material Props -> Material Models
Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic
Input 21E10 for the Young’s Modulus (Steel) in EX.
Input 0.3 for Poisson’s Ratio in PRXY
Click OK
of Define Material Model Behavior window
6
3
4
2
5
Meshing
1. Go to Main Menu -> Preprocessor ->
Meshing -> Mesh Tool
2. Go to Size Controls: -> Global -> Set
3. Under SIZE Element edge length put .02/4.
This will create a mesh of a total 4 elements through
the thickness.
4. Click OK
5. Check Mapped
6. Click Mesh
7. Click Pick All
2
3
4
5
6
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
7
Page 8
8. Go to Utility Menu -> Plot -> Nodes
9. Go to Utility Menu -> Plot Controls -> Numbering…
10. Check NODE Node Numbers to ON
11. Click OK
10
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.
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Static Solution
Analysis Type
1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis
2. Choose Static
3. Click OK
2
3
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.
5
6
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Displacement
1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural ->
Displacement -> On Nodes
2. Select Pick -> Single -> Type 641
This selects the node in the middle of sphere on the inside radius
3. Click OK
4. Under Lab2 DOFs to be constrained select UY and UZ
5. Under VALUE Displacement value enter 0
6. Click OK
4
2
3
5
6
With the node numbers turned off the resulting graphic should be as shown:
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Loads
1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural ->
Force/Moment -> On Nodes
2.
3.
4.
5.
6.
Select Pick -> Single -> Type 641
Click OK
Under Direction of force/mom select FX
Under VALUE Force/moment value enter -1
Click OK
4
5
2
6
3
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 not skew the eigenvalue answer.
7. Go to Main Menu -> Solution -> Define Loads ->Apply->Structural ->Pressure ->
On Lines
8. Select Pick -> Single -> Type 1
This selects the outside line to apply the external pressure
9. Click OK
10. Under VALUE Load PRES value
enter 1
11. Click OK
12. Go to Main Menu -> Solution ->
Solve -> Current LS
13. Go to Main Menu -> Finish
10
11
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Eigenvalue
1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis
2. Choose Eigen Buckling
2
3
3. Click OKGo to Main Menu -> Solution -> Analysis Type ->Analysis Options
4. Under NMODE No. of modes to extract input 4
5. Click OK
5
6
6
Go to Main Menu -> Solution -> Solve -> Current LS
6.
7. Go to Main Menu -> Finish
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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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
2
3
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
5
6
7. Go to Main Menu -> Solution -> Solve -> Current LS
8. Go to Main Menu -> Finish
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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General Postprocessor
Critical Pressure
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 Critical Pressure:
P= 24.5 MPa
Mode Shape
To view the deformed shape of the buckled beam vs. original beam:
1. Go to Main Menu -> General Postproc -> Read Results -> First Set
2. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape
3. Under KUND Items to be plotted select Def + undeformed
4. Click OK
3
4
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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The graphics area should look as below:
Results
The percent error (%E) in our model can be defined as:
(
)
= 3.618%
This shows that there is a very small error with 4 elements through the thickness.
As you can see in the figure there are 18 half-sine waves as well as predicted in the theory
section.
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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Validation
Theoretical
7935 Elements
5080 Elements
318 Elements
Critical Buckling
Load
25419556
24497000
24500000
24618000
Percent Error
0%
3.629%
3.618%
3.1533%
This table provides the critical buckling pressure and corresponding error from the theory, from
three different ANSYS results; one with 1 element, 4 elements and 5 elements through the
thickness. This is to prove mesh independence, showing with increasing mesh size, the answer
approaches a constant value. The results here show that even using a coarse mesh of 1 element
through the thickness, the error is minimal in comparison with the theoretical value. This
theoretical value uses approximations to linearize a problem which is inherently nonlinear, this
means this is not an exact answer. 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.
UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere
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