Module 3: Buckling of 1D Simply Supported Beamcassenti/AnsysTutorial/Modules_APDL/Module 3_1...UCONN ANSYS –Module 3 Page 1 Module 3: Buckling of 1D Simply Supported Beam Table of

UCONN ANSYS –Module 3 Page 1

Module 3: Buckling of 1D 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


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

In this module, we model a simply supported steel beam with compressive loads. This module

stresses the importance of buckling in designs involving linear compressive loading and serves

as a starting point for later modules which deal with more complex buckling considerations.

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.

Theory

Buckling load

Hooke’s Law equates stress as shown:

(3.1)

Deriving both sides of equation 3.1 it shows

(3.2)

By solving for equilibrium:

(3.3)

Equation 3.3 is a nonlinear equation, however this equation can be linearized using eigenvalues.

y x


Since:

(3.4)

Then:

(3.5)

Plugging in Equation 3.1 for stress we find:

(3.6)

Plugging Equation 3.6 into Equation 3.3, Equation 3.6 becomes

(3.7)

Which is simplifies to:

(3.8)

By integrating two times Equation 3.8 becomes

(3.9)

At the fixed end (x=0), v=0,

, thus 0

At the supported end (x=L), v=0,

, thus 0

Equation 3.9 becomes

(3.10)

Equation 3.6 represents the Differential Equation for a Sin Wave

√

√

(3.11)

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

√

(3.12)

But A cannot equal zero or this problem is trivial.

At the supported end (x=L), v=0 Equation 12 becomes

√

(3.13)

Since A cannot equal zero, ( ) must equal zero:

Sin(nπ)=0 for n=(0, 1, 2, 3, 4…..)


So:

√

n ≠ 0 or it is trivial (3.14)

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 ( ).

(3.15)

Where the lowest Buckling Load is at

(3.16)

This is an over estimate so there are certain correction factors (C) to account for this. (C) is

dependent on the beam constraints.

(3.17)

Where C=1 for a fixed-simply supported beam.

So the Critical Buckling Load is

= 41.124 N (3.18)

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

1

2


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

1

2

3

2


Key points

Since we will be using 1D Elements, our goal is to model the length 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 key point at the

origin.

4. Click Apply

5. Repeat Steps 3 and 4 for 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

Line

1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->

Lines -> Lines -> Straight Line

2. Select Pick

3. Select List of Items

4. Type 1,2 for points previously generated.

5. Click Ok

6

2

3

2

3

4

7

5


The resulting graphic should be as shown:

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 Beam -> 2D Elastic 3

4. Click OK

4

2

3

3

4


Beam3 is a uniaxial element with tension, compression, and bending capabilities. The element

has three degrees of freedom at each node: translations in the nodal x and y directions and

rotation about the nodal z-axis

Real Constants and Material Properties

Now we will dimension our beam.

1. Go to Main Menu -> Preprocessor ->

Real Constants -> Add/Edit/Delete

2. Click Add

3. Choose Type 1 Beam3

4. Click OK

5. Under Cross-sectional area AREA

enter 10

6. Under Area moment of inertia IZZ

Enter 10/12

7. Under Total beam height HEIGHT

enter 10

8. Click OK

9. Close out of the Real Constants window

2

3

4

56

9

7

8


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 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 10.

This will create a mesh of a total 10 elements

4. Click OK

5. Click Mesh

6. Click Pick All

3 4

5

2

6

2

3

4

5

6


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:

ANSYS numbers nodes from the left extreme to the right extreme and then numbers from left to

right.

9

10


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.

2

3

5

6


Displacement

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

Displacement -> On Nodes

2. Select Pick -> Single -> and click node 1

3. Click OK

4. Under Lab2 DOFs to be constrained select UX and UY

5. Under VALUE Displacement value enter 0

6. Click OK


Displacement -> On Nodes 8. Select Pick -> Single -> and click node 2

9. Click OK

10. Under Lab2 DOFs to be constrained select only UY

11. Under VALUE Displacement value enter 0

12. Click OK

2

3

4

5

6

WARNING: UX and UY might already be highlighted, if so, leave UY

highlighted and click UX to remove it from the selection. Failure to only

constrain UY will result in incorrect results.


The graphics area should look as below:

Loads


Force/Moment -> On Nodes

2. Select Pick -> Single -> and click node 2

3. Click OK

4. Under Direction of force/mom select FX

5. Under VALUE Force/moment value enter -1

6. Click OK


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.


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 OK

4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options

5. Under NMODE No. of modes to extract input 1

6. Click OK



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 1

6. Click OK



2

3

5

6


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= 41.124 N

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




Results

The percent error (%E) in our model can be defined as:

(

) = 0%

As one can see, eigenvalue problems are very accurate for one dimensional elements and are

solved quickly by the solver.

Validation

Theoretical 10 Elements 2 Elements

Critical Buckling Load 41.124 N 41.124 41.433

Percent Error 0% 0% 0.7513%

This table provides the critical buckling loads and corresponding error from the Theory (Euler),

and two different ANSYS results; one with 2 elements and one with 10 elements. This is to

prove mesh independence, showing with increasing mesh size, the answer approaches the

theoretical value. The results here show that even using a coarse mesh of 2 elements the error

baseline is minimal. 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 beam so the structure will never actually

reach the eigenvalue load that is calculated.

Documents

Module 3: Buckling of 1D Simply Supported Beamcassenti/AnsysTutorial/Modules_APDL/Module 3_1...UCONN ANSYS –Module 3 Page 1 Module 3: Buckling of 1D Simply Supported Beam Table of