module 5: axisymmetric buckling of a thin walled spherecassenti/ansystutorial/modules... · there...

UCONN ANSYS –Module 5: Axisymmetric Buckling of a Thin Walled Sphere

Module 5: Axisymmetric Buckling of a Thin Walled Sphere

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


Problem Description:

9) External Pressure buckling of a thin walled sphere

Nomenclature:

r = 2m Sphere Radius

P = 1 Pa External Pressure

t = .02m Sphere Thickness

E= 210* Youngs Modulus

= 0.3 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 (9.3)

t

r

P


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.

Preferences

1. Go to Main Menu -> Preferences

2. Check the box that says Structural

3. Click OK

3

1

2

1

2

3


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

Arc


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

2

3

4

5

6

2

8

9


Line


Lines -> Lines -> Straight Line

2. Select Pick


4. Type 1,4 for points previously generated.

5. Click Apply

6. Type 3,6

7. Click OK

Area

1. Go to Utility Menu -> Plot -> Lines

2. Go to Utility Menu -> Plot Controls -> Numbering…

3. Check LINE Line numbers to ON

4. Click OK


Areas -> Arbitrary -> By Lines

6. Select Pick


8. Type 4,1,3,2 for lines previously generated.

9. Click OK

2

3

7

5

4

3

4

8

9


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

4

2

3


Preprocessor

Element Type

1. Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete

2. Click Add

3. Click Solid -> Axi-har 4node 25

4. Click OK

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.

3

4


Real Constants and Material Properties

We will 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 21E10 for the Young’s Modulus (Steel) 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 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

3 4

5

2

6

2

3

4

5

7

6


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

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.

10

11


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.

2

3

5

6


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

With the node numbers turned off the resulting graphic should be as shown:

2

3

4

5

6


Loads

1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural ->

Force/Moment -> On Nodes


3. Click OK

4. Under Direction of force/mom select FX

5. Under VALUE Force/moment value enter -1

6. Click OK

7. Go to Main Menu -> Solution -> Define Loads ->Apply->Structural ->Pressure ->

On Lines


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

2

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.

10

11

4

5

6


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


2

3

5

6

6


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


2

3

5

6


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


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.


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.

module 5: axisymmetric buckling of a thin walled spherecassenti/ansystutorial/modules... · there...

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