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ANSYS TUTORIAL Optimal Design of a Stiffened BeamANSYS Release 7.0

Dr. A.-V. Phan, University of South Alabama

1 Problem Description

Use ANSYS to perform the minimum weight design of a stiffened beam as shown. Structural steel ofcircular and rectangular cross-sections will be selected for the bar and beam members, respectively.The allowable stress in the elements is assumed to be 0.5Sy. The tip displacement is constrainedto be no greater than 104L. Let the yield strength and length of the beam be Sy = 250 MPa andL = 3 m, respectively.

45

2

1

1

3

o

2

X

Y

L

500 kN

Y

Z

W

H

(a) (b)

Figure 1: A stiffened beam

2 Data Preparation

Units to be used: Length (m), Force (N) Design variables: R (radius of the circular cross-section of the bar), W and H. Volume of the structure (the objective variable):

V = L(WH+

2R2)

Allowable stress (state variable):

a = 0.5Sy = (0.5)(250 106) = 125 106 Pa

Allowable tip displacement (state variable):

ua = 104L = (104)(3) = 3 104 m

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3 Problem Formulation

Minimize V(R,H,W) = L(WH+

2R2) [0.01 m3]Sub ject to 0 ||max 125 106 [100 Pa]

0u1

3

104 [

0 m]

0.01 R 0.25 [0.0025 m]0.02 W 0.5 [0.005 m]0.02 H 0.5 [0.005 m]

where ||max is the maximum absolute value of stress.

4 Analysis Preprocessing

1. Give the Job a Name

Utility Menu>File>Change Jobname ...

Enter a name in the Change Jobname window, then click on OK.

2. Initialize the Design VariablesUtility Menu>Parameters >Scalar Parameters ...

This brings up the Scalar Parameters window. Enter R=0.025 in the Selection field,then click on Accept.

Repeat the above step for W=0.05 and H=0.05. The Scalar Parameters windowshould look like this,

Click on the Close button.

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3. Define Element Types

(a) Bar/Truss ElementMain Menu>Preprocessor>Element Type>Add/Edit/Delete

This brings up the Element Types window. Click on the Add... button. The Library of Element Types window appears. Highlight Link and 2D spar 1.

After clicking on Apply you should see Type 1 LINK1 in the Element Typeswindow.

(b) Beam Element

Highlight Beam and 2D elastic 3 in the Library of Element Types window,then click on OK.

Type 2 BEAM3 should now be added to the Element Types window. Click onthe Close button.

4. Define Real Constants

(a) Bar/Truss ElementMain Menu>Preprocessor>Real Constants>Add/Edit/Delete

This brings up the Real Constants window. Click on the Add... button. The Element Type for Real Constants window appears. Highlight Type 1 LINK1,

then click on OK.

In the Real Constant Set Number 1, for LINK1 window that opens, enter 3.1416*R*Rfor the AREA field as shown,

Click on OK. You should see Set 1 appears in the Real Constants window.(b) Beam Element

In the Real Constants window, click on Add .... This brings up the Element Type for Real Constants window. Highlight Type 2 BEAM3

and click OK.

In the Real Constants for BEAM3 window that appears, enter W*H, W*H*H*H/12and H for AREA, IZZ and HEIGHT, respectively as shown. Click on OK.

Click on the Close button in the Real Constants window.

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5. Define Material PropertiesMain Menu>Preprocessor>Material Props>Material Models

In the right side of the Define Material Model Behavior window that opens, doubleclick on Structural, then Linear, then Elastic, then finally Isotropic.

The Linear Isotropic Material Properties for Material Number 1 window comes up.Enter in value for the Youngs modulus of 210 109 Pa (EX = 210E9), then click OK.

Click on OK in the Note window that appears. Then close the Define Material ModelBehavior window.

6. Define NodesWe are going to create 3 nodes given in the following table:

Node # X Y

1 0 02 3 03 3 3

Main Menu>Preprocessor>Modeling>Create>Nodes> In Active CS

To create node #1, enter its node number, X and Y coordinates in the Create Nodesin Active Coordinate System window that comes up. Click on Apply.

Repeat the above step for nodes #2 and #3. Note that you must click on OK insteadof Apply after entering data for the final node.

7. Define Elements

(a) Define Bar ElementMain Menu>Preprocessor>Modeling>Create>Elements>Elem Attributes

This brings up the Element Attributes window. Click on OK.

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(b) Create Bar ElementMain Menu>Preprocessor>Modeling>Create>Elements>Auto Numbered>Thru Nodes

The Element from Nodes window opens. Select node 1, then node 3. Click on OK in the Element from Nodes window.

(c) Define Beam Element

Main Menu>Preprocessor>Modeling>Create>Elements>Elem Attributes In the Element Attributes window that opens, select 2 BEAM3 for Element type number

and 2 for Real constant set number.

Click on OK.(d) Create Beam Element

Main Menu>Preprocessor>Modeling>Create>Elements>Auto Numbered>Thru Nodes

Select node 1, then node 2. Click on OK.

8. Mesh the Model

No need because we have defined the model using nodes and elements.

9. Apply Boundary ConditionsMain Menu>Preprocessor>Loads>Define Loads>Apply>Structural>Displacement>On Nodes

Pick nodes 2 and 3, then click OK in the picking window. In the Apply U,ROT on Nodes window that comes up, select All DOF for

DOFs to be constrained, Constant value for Apply as and enter 0 forDisplacement value, then click OK.

10. Apply Loads

Main Menu>Preprocessor>Loads>Define Loads>Apply>Structural>Force/MomentOn Nodes

Pick node 1, then click OK in the picking window. In the Apply F/M on Nodes window that opens, select FY for Direction of force/mom,

Constant value for Apply as and enter -5E5 for Force/moment value, (a forceof 500,000 N in the negative direction of the Y axis), then click on OK.

5 Analysis Processing (Solving)

Main Menu>Solution >Analysis Type>New Analysis

Make sure that Static is selected. Click OK.Main Menu>Solution >Solve>Current LS

Check your solution options listed in the /STATUS Command window. Click the OK button in the Solve Current Load Step window. You should see the message Solution is done! in the Note window that comes up. Close

the Note and /STATUS Command windows.

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6 Analysis Postprocessing

1. Parameterize the objective variable (Volume)

(a) Define the volumeMain Menu>General Postproc>Element Table>Define Table>Add ...

This brings up the Define Additional Element Table Items window. Select Geometry

and Elem volume VOLU as shown,

Click on OK, then click Close in the Element Table Data window.(b) Sum the element volumes to obtain the structure volume

Main Menu>General Postproc>Element Table>Sum of Each Item

The Tabular Sum of Each Element Table Item appears. Click on OK This opens the SSUM Command window showing that the total volume is 0.0158304

m3.

Close the SSUM Command window.(c) Obtain the structure volume

Utility Menu>Parameters >Get Scalar Data ...

In the Get Scalar Data window that opens, select Results data andElem table sums as shown,

Clicking on OK will bring up the Get Element Table Sum Results window. Com-plete the window as shown,

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Click on OK.2. Parameterize the state variables

(a) Obtain the tip displacementUtility Menu>Parameters >Get Scalar Data ...

In the Get Scalar Data window that appears, select Results data andNodal results. Click on OK.

Complete the Get Nodal Results Data window that opens as shown,

Click on OK.(b) Define the normal stresses in the elements

Main Menu>General Postproc>Element Table>Define Table

Click on Add ... in the Element Table Data window. In the Define AdditionalElement Table Items window that comes up, enter sequence number LS, 1.

Click on Apply, then enter sequence number NMISC, 1.

Repeat the previous step for the sequence numbers NMISC, 2, NMISC, 3 andNMISC, 4.

Click on OK in the Define Additional Element Table Items window. Click on Close in the Element Table Data.

(c) Determine the maximum normal stress in the elementsMain Menu>General Postproc>Element Table>Abs Value OptionTo determine the maximum stress in terms of absolute value, check the box in the

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window that appears as shown, then click on OK

Main Menu>General Postproc>Element Table>Find Maximum

To find the maximum of items LS1 and NMISC1, then assign the maximum as itemM1, complete the window that opens as shown and click on Apply.

Repeat the previous step to find M2 = max(M1, NMISC2), M3 = max(M2, NMISC3)and M4 = max(M3, NMISC4).

Click on OK.Main Menu>General Postproc>List Results>Sorted Listing>Sort ElemsTo find the maximum stress, complete the window that opens as shown and click OK.

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Utility Menu>Parameters >Get Scalar Data ...

In the Get Scalar Data window that appears, select Results data andOther operations. Click on OK.

Complete the Get Data from Other POST1 Operations window that opens asshown,

Click on OK.Utility Menu>Parameters >Scalar Parameters ...

This function is used to view the values of the objective and state variables. TheScalar Parameters window appears as shown,

Click on Close

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(d) Write the Log FileUtility Menu>File>Write DB log file ...This opens the Write Database Log to window. Assign a name to this file (the defaultextension is lgw). Click on OK. The log file should be similar to the following:

/BATCH

! /COM,ANSYS RELEASE 7.0 UP20021010 15:56:58 10/03/2004

/input,menust,tmp,,,,,,,,,,,,,,,,,1! /GRA,POWER

! /GST,ON

! /PLO,INFO,3

! /GRO,CURL,ON

! /REPLOT,RESIZE

/FILNAME,OptDesign,0

*SET,R,0.025

*SET,W,0.05

*SET,H,0.05

/PREP7

!*ET,1,LINK1

!*

ET,2,BEAM3

!*

!*

R,1,3.1416*R*R, ,

!*

R,2,W*H,W*H*H*H/12,H, , , ,

!*

!*

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,EX,1,,210E9

MPDATA,PRXY,1,,

N,1,0,0,,,,,

N,2,3,0,,,,,

N,3,3,3,,,,,

TYPE, 1

MAT, 1

REAL, 1

ESYS, 0

SECNUM,

TSHAP,LINE

!*

FLST,2,2,1

FITEM,2,1

FITEM,2,3

E,P51X

TYPE, 2

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MAT, 1

REAL, 2

ESYS, 0

SECNUM,

TSHAP,LINE

!*

FLST,2,2,1FITEM,2,1

FITEM,2,2

E,P51X

FLST,2,2,1,ORDE,2

FITEM,2,2

FITEM,2,-3

!*

/GO

D,P51X, ,0, , , ,ALL, , , , ,

FLST,2,1,1,ORDE,1

FITEM,2,1

!*

/GO

F,P51X,FY,-5E5

FINISH

/SOL

!*

ANTYPE,0

! /STATUS,SOLU

SOLVE

FINISH

/POST1

AVPRIN,0,0,ETABLE, ,VOLU,

!*

SSUM

!*

*GET,Volume,SSUM, ,ITEM,VOLU

!*

*GET,U1,NODE,1,U,SUM

!*

AVPRIN,0,0,

ETABLE, ,LS, 1

!*AVPRIN,0,0,

ETABLE, ,NMISC, 1

!*

AVPRIN,0,0,

ETABLE, ,NMISC, 2

!*

AVPRIN,0,0,

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ETABLE, ,NMISC, 3

!*

AVPRIN,0,0,

ETABLE, ,NMISC, 4

!*

SABS,1

SMAX,M1,LS1,NMIS1,1,1,SMAX,M2,NMIS1,NMIS2,1,1,

SMAX,M3,NMIS2,NMIS3,1,1,

SMAX,M4,NMIS3,NMIS4,1,1,

ESORT,ETAB,M4,0,0, ,

!*

*GET,SMax,SORT,,MAX

! LGWRITE,OptDesign,lgw,C:\DOCUME~1\DR8BDD~1.PHA\,COMMENT

7 Optimization Preprocessing

1. Assign the Log FileMain Menu>Design Opt>Analysis File>AssignIn the Assign Analysis File windows that pops up, specify the log file by typing its name orbrowsing for its location. Click OK

2. Define the Design VariablesMain Menu>Design Opt>Design Variables>Add...

Complete the window that opens as shown for the design variable H using the limits andtolerance specified in the problem formulation. Click on Apply.

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Repeat the previous step for the design variables R and W. Click on OK instead ofApplyafter entering data for W.

3. Define the State VariablesMain Menu>Design Opt>State Variables>Add...

Complete the pop-up window as shown for the state variable SMAX using the limits andtolerance specified in the problem formulation. Click on Apply.

Repeat the previous step for the state variables U1, then click on OK instead ofApply.4. Define the Objective Variable

Main Menu>Design Opt>ObjectiveSpecify VOLUME as the objective variable and enter the convergence tolerance defined in theproblem formulation as shown,

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8 Optimization Processing

1. Specify the Optimization MethodMain Menu>Design Opt>Method/Tool

In the Specify Optimization Method window that shows up, select First-Order. Clickon OK.

Complete the following window that opens as shown, then click on OK.

2. Execute the Optimization ProcessMain Menu>Design Opt>Run

Click on OK in the Begin Execution of Run window to start the optimization. The process converges after 11 iterations. Click on OK to close the Note window.

9 Optimization Postprocessing1. View the Final Results

Utility Menu>Parameters >Scalar Parameters ...This opens the Scalar Parameters window where the following final results are found:

R = 0.17972667 m

W = 0.279317374 m

H = 0.281996234 m

|

|max = 9, 143, 336.07 Pa

u1 = 3.015004667 104 mV = 0.279317374 m3

Note that displacement constraints at the tip is violated. However, the final result for u1 fallswithin the default tolerance set by ANSYS.

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2. Plot the Optimization HistoriesMain Menu>Design Opt>Design Sets>Graphs/Tables

History of the Design VariablesTo obtain a history graph of the design variables, in the following window that pops up se-lect Set number as X-variable parameter, and H, R and W as Y-variable parameter,

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Click on OK and your ANSYS screen should look like this,

History of the State VariablesThe following history graphs of the state and objective variables can be obtained in asimilar manner:

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History of the Objective Variable

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