deflection of Beams and Shafts

8/12/2019 deflection of Beams and Shafts

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A finite element model of a steel overhanging beam will be built and analyzed. Due

to the simplicity of the example, beam elements will be used to model the structure.

This example problem was obtained from the Mechanics of Materials 7th EditionText R.C Hibbeler.

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Mesh Generation

Nodal

Displacements


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Example 12.12 - Hibbeler (Tutorial) Dassault Systmes, 2011 SIMULIA Learning Community

2

Creating the Model Geometry (Beams)

Go to the Start Menuand open Abaqus CAE

You may be prompted with an Abaqus/CAE 6.10 Student Editionbox (Figure

2). Close this box by clicking the Xin the top right hand corner.

Figure 2.Abaqus/CAE 6.10 Student Edition box.

Once the Student Editionbox is exited, the Abaqus CAE Viewport should look

similar to Figure 3. (Please note the model tr eeis the series of functions listed on

the left hand side of the viewport, while the moduleis the list of icons to the right

of the model tree)

Figure 3.Abaqus CAE Viewport

To create the beam model geometry of the steel overhanging beam, a line must be

generated.

Model Tree


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Using the left mouse button, double click Parts in the model tree and the Create

Part(Figure 4a) dialog box appears. Enter a new Name:for the part (BEAM),

and under the Modeling Spaceoption choose 2D Planar. In the Base Feature

option choose Wirefor shape (as in wireframe). Change the Approximate size:

option from the default 200 to 400. The Create Partdialog box should look

identical to Figure 4b.

Click Continue and the graphics window will change to a set of gridlines.

Figure 4a.Create Part Dialog Box Figure 4b.Create Part Dialog Box (BEAM)

Click the Create Lines: Connectedicon in the module, it is located directly

to the right of the Create Isolated Pointsicon.With the cursor, draw two connected horizontal lines of arbitrary length anywhere

in the viewport (Figure 5). (Please note, when the line is created, the line feature

is exited by clicking the Esckey on your computer keyboard.)


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Figure 5.Horizontal Lines

NOTE: If you accidentally create an unwanted line, you can select Edit > Delete

from the dropdown menu at the top of the screen and use the mouse to select a

line to delete.

Next, click the Add Constrainticon in the module and the Add Constraint

Dialogbox will appear. Under the list of constraints click Equal length. While

holding shift on the computer keyboard click both of the lines in the viewport

with the cursor. If this selection has been completed correctly, the lines will turn

from yellow to red.Click Done. Click the Xin the top right hand corner of the Add Constraint

dialog box to exit the tool.

Click the Add Dimension icon in the module. Click one of the lines and drag

the cursor away from the line and click again; this will reveal the arbitrary length

of the line in red.

At the bottom of the viewport enter a New dimension:for the line. Enter 144the

box since half of the total beam length is 12 ft. The correct length of both lines

will update in the viewport due to the equal length constraint.

Press Escon the computer keyboard to exit the dimensioning tool.

Click Done.

Sketch mode will automatically be exited, and the model geometry should look

identical to the beam shown in Figure 6.


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Figure 6.Final Model Geometry

Defining Material Properties

To define material properties for this model, double click on Materialsin the

model tree and the Edit Materialdialog box will appear (Figure 7a). Enter a

Name:for the material (STEEL), and click the Mechanicaltab, highlight

Elasticityand click Elastic. Enter values of Youngs Modulus= 29E06 psi, and

PoissonsRatio= 0.3. After the material properties have been entered, the Edit

Materialdialog box should look identical to Figure 7b.

Click OK.

Figure 7a.Edit Material Dialog Box Figure 7b.Edit Material Dialog Box (STEEL)


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Please note there is no dropdown menu or feature in Abaqus that sets specific

units. All of the dimensions have been input in inches; therefore the respective

Youngs Modulus units should be entered inpsi (pounds per square inch). The

units chosen for the definition of the material properties should be consistent and

dictate what units should be used for the dimensions of the structure.

At this point in preprocessing, the model should be saved. Click File then click

Save.Name the file Example 12.12 - Hibbeler. The file will save as a Model

Database (*.cae*) file. It may be of interest to save the file after each section of

this tutorial.

Creating Sections (and Profiles)

To create a beam section in Abaqus, double click Sectionsin the model tree and

the Create Sectiondialog box will appear (Figure 8a). Enter a Name:for the

section (BEAM), and choose Beamunder the Category option, and Beamunder

the Typeoption. Your Create Sectiondialog box should look identical to that in

Figure 8b.

Click Continue

Figure 8a. Create Section Dialog Box Figure 8b.Create Section Dialog Box (BEAM)

The Edit Beam Sectiondialog box will then appear where a Materialand

Profile can be prescribed for this section. Change the Section integration:to

Before analysis.

To the right of the Profile name:dropdown click Create The Create Profile

dialog box will immediately appear (Figure 9a). Create a Name:for the profile

(BEAM).Under the Shapeoption choose Generalized. The Create Profile

dialog box should look similar to that in Figure 9b.


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Figure 9a.Create Profile Dialog Box Figure 9b.Create Profile Dialog Box (BEAM)

Click Continue The Edit Profiledialog box will then appear, enter a value of

1for Area:and 125for I11: (all other values should have a 0value). The Edit

Profiledialog box should looks similar to Figure 10.

Figure 10.Edit Profile Dialog Box

Click OK.

The Edit Profiledialog box will disappear and in the Edit Beam Sectiondialog

box, the BEAMprofile will appear in the Profile name:dropdown menu. In the

Basic tab enter a Youngs Modulusof 29E06and a Shear modulusof 0. Finally

enter a Section Poissons ratio:of 0.3 (Figure 11).

Click OK.


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Figure 11.Edit Beam Section Dialog Box Final

Assigning Sections

Now that the beam section has been created, it can be assigned to the geometry. In

the model tree, click the +to the left of the Parts (1)icon, this will further expand

the model trees options. Next, click the +to the left of the part called BEAM,

further expanding the model tree (Figure 12).

Figure 12. Model Tree Expansion (Parts)

After the model tree has been expanded, double click Section Assignments. Use

the cursor to select the line by holding down the left mouse key and dragging the

cursor around the geometry to create a box around the whole model. If this drag

has been done correctly, the model will change color from grey to red.

Click Done.

The Edit Section Assignmentdialog box will appear (Figure 13). Because only

one section has been created, the dropdown menu defaults to the BEAMsection.


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If multiple sections had been created, the dropdown menu could be used to assign

different sections to different parts of the geometry.

Click OK. The model should now turn to a blue color.

Figure 13.Edit Section Assignment Dialog Box (BEAM)

Finally, a Beam Section Orientationmust be assigned. In the toolbar at the top

of the Viewport, there is a dropdown menu labeled Assign. Using the left mouse

button, click Assignand click Beam Section Orientation(Figure 14).

Figure 14. Beam Section Orientation Drop Down Menu

Using the cursor, hold the left mouse button while dragging the cursor around the

model to create a box around the whole geometry. If this drag is done correctly,

the model will change color from blue to red.

Click Done.

Using the computer keyboard, hit Enter. The model should look identical to

Figure 15.


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Figure 15.Beam Section Orientation

Hit OK.

Click Done. The model should turn back to a blue color.

This last step is used to define the orientation of a beam in space. Defining the

beam section orientation is critical when using beam elements because the local 1

and 2 coordinate axes must correspond correctly to the problem that is trying to be

solved.

Creating a MeshTo create a mesh for the model geometry, double click Mesh (Empty)in the

model tree. If this selection is done correctly, then the geometry should change

color to pink.

The first step in creating a mesh is to seed the part. Click the Seed Edgesicon

in the mesh module.

Using the cursor draw a box around the whole model, if this is done correctly the

model will turn from a pink to a red color.

Click Done.

The Local Seedsdialog box will appear (Figure 16a). Under the Basictab changethe Methodto By number, and under the Sizing Controlsoption change the

Number of elements:to 10. The Local Seedsdialog box should look similar to

Figure 16b.


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Figure 16a.Local Seeds Dialog Box Figure 16b.Local Seeds Dialog Box (20)

Click OK. The model will now appear to be seeded with evenly spaced pink

points along its length (Figure 17). Since the beam was sketched using twohorizontal lines, Abaqus treats each line as an edge. Each line was seeded to have

10 elements, thus giving a total element number of 20 for the complete model.

Figure 17.Seeded Geometry (20 Total Elements)

ClickDone.

The part is now ready to be meshed. In the mesh module, click the Mesh Part

icon . At the bottom of the viewport you will be prompted if it is OK to mesh

the part? Click Yes.

If this procedure was done correctly, the geometry will turn blue.


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Finally click the Assign Element Typeicon in the module. Using the cursor

draw a box around the whole model. If this has been done correctly the model will

turn to a red color.

Click Done. The Element Typedialog box will appear ensure that the Familyis

set to Beam, and under the Beam type:option Cubic formulationis chosen. TheElement Typedialog box should look similar to Figure 18.

Figure 18.Element Type Dialog Box

Click Ok.

Click Done.

Creating an Instance

Now that the part has been meshed, it can be brought into the assembly. To do

this task, click the+to the left of Assemblyin the model tree. The model tree will

expand and should look identical to Figure 19.

Figure 19. Model Tree Expansion (Assembly)

Double click on the Instancesicon in the expanded model tree. This feature will

allow multiple parts to be brought into the assembly. The Create Instancedialog

box will appear (Figure 20).


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Figure 20.Create Instance Dialog Box

The BEAMpart is selected by default because only one part has been created for

this tutorial. If multiple parts had been created, then this step would allow them to

be entered into the assembly.

Click OK. If this step was done correctly the model should turn a blue color(Figure 21).

Figure 21.Create Instance

Creating a StepA Step is where the user defines the type of loading, e.g. Static or Dynamic, and

defines the boundary conditions, e.g. support constraints and forces.

In the model tree, double click the Stepsicon. The Create Stepdialog box will

appear (Figure 22a). Create a Name:for the step (LOADINGSTEP). Under

Procedure typeensure that Generalis selected along with Static, General. The

Create Stepdialog box should look identical to Figure 22b.


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Figure 22a.Create Step Dialog Box Figure 22b.Create Step Dialog Box (LOADING)

Click Continue, and the Edit Stepdialog box will immediately appear. Acceptthe default values. The Edit Stepdialog box should look similar to that in Figure

23.

Figure 23.Edit Step Dialog Box

Click OK.

Apply Constraint Boundary Conditions

Boundary conditions will be defined which will simulate a pinned connection at

one end and a roller at the midpoint.

Double click BCsin the model tree and the Create Boundary Conditiondialog

box will appear (Figure 24a). Create a Name:for the boundary condition

(PINNED), and under the Stepdrop down menu make sure to choose

LOADINGSTEP. Under the Categoryoption choose Mechanical, and choose

Symmetry/Antisymmetry/Encastreunder the Types for Selected Stepoption.


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The Create Boundary Conditiondialog box should look identical to that in

Figure 24b.

Figure 24a.Create BC Figure 24b.Create BC (PINNED)

Click Continue

In the viewport the three visible yellow points will be located on the beam. Using

the cursor click the yellow point at the leftside of the beam. If this is donecorrectly the point will turn from a yellow to red color.

Click Done.

The Edit Boundary Condition dialogbox will immediately appear. Click

PINNED (U1=U2=U3=0). The Edit Boundary Conditiondialog box should

look identical to that in Figure 25.

Figure 25. Edit Boundary Condition Dialog Box (FIXED)

Click OK. If this procedure has been done correctly, the model should look

similar to that in Figure 26.


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Figure 26.Beam (PINNED)

Next, a roller boundary condition will be applied at the beams midpoint. Double

click BCsin the model tree and the Create Boundary Conditiondialog box will

appear (Figure 27a). Create a Name:for the boundary condition (ROLLER), and

under the Stepdrop down menu make sure to choose LOADINGSTEP. Under

the Categoryoption choose Mechanical, and choose Displacement/Rotation

under the Types for Selected Stepoption. The Create Boundary Condition

dialog box should look identical to that in Figure 27b.

Figure 27a.Create BC Figure 27b.Create BC (ROLLER)

Click Continue


the cursor click the yellow point at the midpoint of the beam. If this is done

correctly the point will turn from a yellow to red color.

Click Done.

The Edit Boundary Condition dialog box will immediately appear. Check the

box next to U2:this will add a constraint in the Ydirection while allowing


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translation in the Xdirection. The Edit Boundary Conditiondialog box should

look identical to that in Figure 28.

Figure 28. Edit Boundary Condition Dialog Box (ROLLER)

Click OK. If this procedure has been done correctly, the model should look

similar to that in Figure 29.

Figure 29.Constrained Beam

Applying LoadsThree point loads will be applied to the structure at points A, B, and C. Double

click Loadsin the model tree and the Create Loaddialog box will appear (Figure

30a). Enter a Name:for the load (LOADA) and ensure that the Step: dropdown is

set to LOADINGSTEP. Also that Mechanicalis selected under Categoryand

Concentrated forcefor Types for Selected Step. The Create Loaddialog box

should look similar to that in Figure 30b.


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Figure 30a. Create Load Dialog Box Figure 30b. Create Load Dialog Box (LOADA)

Click Continue


the cursor click the yellow point at the left end of the beam. If this is done


Click Done.

The Edit Load dialog box will immediately appear. In the CF2: option enter a

value of -5000. The Edit Loaddialog box should look identical to that in Figure

31.

Figure 31. Edit Load Dialog Box (LOADA)

Click OK. If this procedure has been done correctly, a small yellow arrow will

appear at point Aof the structure pointing in theYdirection.

Next, a load will be applied to point Bof the structure. Double click Loads (1)in

the model tree and the Create Loaddialog box will appear (Figure 32a). Enter aName:for the load (LOADB) and ensure that the Step: dropdown is set to

LOADINGSTEP. Also that Mechanicalis selected under Categoryand




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Figure 32a. Create Load Dialog Box Figure 32b. Create Load Dialog Box (LOADB)

Click Continue


the cursor click the yellow point at the midpointof the beam. If this is done


Click Done.


value of 10000. The Edit Loaddialog box should look identical to that in Figure

33.

Figure 33. Edit Load Dialog Box (LOADB)


appear at point Bof the structure pointing in the +Ydirection.

Finally, a load will be applied to point Cof the beam. Double click Loads (2)in

the model tree and the Create Loaddialog box will appear (Figure 34a). Enter aName:for the load (LOADC) and ensure that the Step: dropdown is set to

LOADINGSTEP. Also that Mechanicalis selected under Categoryand




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Figure 34a. Create Load Dialog Box Figure 34b. Create Load Dialog Box (LOADC)

Click Continue


the cursor click the yellow point at the right end of the beam. If this is done


Click Done.


value of -5000. The Edit Loaddialog box should look identical to that in Figure

35.

Figure 35. Edit Load Dialog Box (LOADC)


appear at point Cof the structure pointing in the -Ydirection.

If all loads have been applied correctly the model should look similar to Figure

36.


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Figure 36.Beam Loading

Creating a Job

To create a job for this model, double click the Jobs icon in the model tree. Up to

this point, you have been preprocessing the model. A job will take the input file

created by the preprocessor and process the model, i.e. perform the analysis.

In the Create Jobdialog box, create a Name:for this job (EXAMPLE_12_12).

Blank spaces are not allowed in a job name. Thus the use of the underscore in the

name. The Create Jobdialog box should look identical to that in Figure 37.

Figure 37. Create Job Dialog Box (EXAMPLE_12_12)

Click Continue

The Edit Jobdialog box will immediately appear (Figure 38).


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Figure 38.Edit Job Dialog Box

Accept the default values and click OK.

Setting the Work Directory

To ensure that the input files write to the correct folder, setting the work directory

must be accomplished. At the top of the screen, click File and in the dropdown

menu click Set Work Directory(Figure 39).

Figure 39. Set Work Directory

The Set Work Directoryscreen will immediately appear (Figure 40). Click

Select and use standard Windows practice to select (and possibly create) a

subdirectory.


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Figure 40. Set Work Directory (FOLDERS)

Click OK.

Click OK.

Writing the Input File (.inp)

To write the input file of the job that was created, first click the +next to the

Jobs(1)icon in the model tree.Right click the job called EXAMPLE_12_12and click the Write Inputoption.

This choice will write an input file (.inp) of this model to the work directory.

It may be helpful to go to the folder on the computer to which the work directory

is set to ensure that the input file was written there.

Model Analysis (Abaqus Command)

Method #1

Go to theStart Menu and openAbaqus Command

Abaqus is set to a default directory (Example C:\>). To change directories in the

Abaqus Command type the directory of choice followed by a colon (D:)then hitEnter.

To access a specific directory within that drive type cd followed by the specific

folder name in that directory (e.g., cd users) then hit Enter.

Now that the correct directory has been sourced in the command window type

Abaqus inter j=EXAMPLE_12_12 and then hit enter.

If the job has completed successfully the Abaqus prompt should look similar to

Figure 41.


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Figure 41. Abaqus Command Prompt (COMPLETED)

Method #2

An alternative method for submitting an *.inp file for processing by Abaqus can

be accomplished with Abaqus CAE

Right click the job called EXAMPLE_12_12and click the Submitoption. If you

see a warning (Figure 42) Click OK. The intent of this warning is to prevent the

user from accidentally overwriting a previously completed analysis with the same

name.

Figure 42.Warning Message

The model will now be submitted for analysis by Abaqus and the progress can be

viewed in the status window at the bottom of the screen (Figure 43).

Figure 43.Abaqus Progress

Postprocessing using Abaqus CAE

After the analysis has successfully completed in the Abaqus Command window

using Method #1 or using Method #2, return to view the Abaqus CAE viewport.

Because the last step of creating the model was to create a job/write (and possibly

submit) an input file, the EXAMPLE_12_12job should still be highlighted in


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Abaqus CAE model tree. Right clickthe EXAMPLE_12_12and then click

Results.

If this selection was done correctly, the model should turn to a green color

Visualizationmodule will be entered (Figure 44).

Figure 44. Analysis Results XY View

To rotate the model into different views, click Viewin the toolbar at the top of the

screen. Next, Click Toolbarsand make sure the option Viewshas a check mark

to the left of it. If not, then click it.

The Viewstoolbar will appear (Figure 45), and the Apply Front Viewbutton can

be clicked to view the model in the X Y plane.

Figure 45.Views Toolbar

To view the deformed shape of the model, click the Plot Contours on Deformed

Shapeicon in the Visualizationmodule. The model should look similar to

that in Figure 46.


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Figure 46.Deformed Shape

Obtaining Nodal Displacements

Upon viewing the deformed contour of the geometry, Abaqus CAE defaults to

view CF(Concentrated forces and moments). To view the spatial displacements

click CFin the Field Output dialog box at the top of the viewport and change the

dropdown to U. To the right of the Udropdown menu, change Magnitudeto U2

(Figure 47).

Figure 47. Field Output Dialog Box

At the top toolbar click Tools then click Query The Querydialog box will

appear, under the General Queriesoption click Node. (Figure 48).

Figure 48. Query Dialog Box (Node)


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Click on any of the individual nodes on the structure and the respective

displacement values will be listed in the message area. Since the example is to

solve for the displacement at point Cof the structure click the node at the right

end of the beam and the displacement will appear in the message area (Figure 49).

Figure 49.Message Area (Nodal Displacement C)

Note that the three columns of values correspond to the X, Y, and Z nodal

displacements. Point Chas displaced 2.745inches in theYdirection (U2).

Conclusion

Save the file by doing either File > Saveor clicking the disk icon (Figure 50).

Figure 50.Disk Icon (Saving)

Close Abaqus CAE: File > Exitor Ctrl+Q

This completes the Deflection of Beams and Shafts Tutorial (Example 12.12

Hibbeler).

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Content Provided by:

R.C. Hibbeler, Mechanics of Materials, 7th ed., Pearson Prentice Hall, New Jersey, 2008

Tutorial Created by:

Dimitri Soteropoulos

SIMULIA
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