linear static and dynamic structural analysis

7/29/2019 LINEAR STATIC AND DYNAMIC STRUCTURAL ANALYSIS

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Faculty of Mechanical Engineering of Kragujevac

Laboratory for Engineering Software

Miroslav ivkovi Rodoljub Vujanac Sneana Vulovi

LLIINNEEAARRSSTTAATTIICC AANNDD DDYYNNAAMMIICCSSTTRRUUCCTTUURRAALL AANNAALLYYSSIISS

PPRREE-- AANNDD PPOOSSTT-- PPRROOCCEESSSSIINNGG EEXXAAMMPPLLEESS IINN PPRROOGGRRAAMM FFEEMMAAPP

FFOORR PPRROOGGRRAAMM PPAACCKKAAGGEE PPAAKK

Kragujevac

SERBIA


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Faculty of Mechanical Engineering

ENGINEERING SOFTWARE LAB

Sestre Janjic 6, 34000 Kragujevac, SERBIA

Tel: +381 34 300790

Fax: +381 34 300791

Mob: +381 64 8288777

e-mail: [email protected]


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CONTENTS

EXAMPLE 1 (SE1_1) Spatial truss structure ...............................................1

EXAMPLE 2 Static analysis of 2D truss .....................................................17

EXAMPLE 3 Linear static analysis of roof truss ........................................39

EXAMPLE 4 (SE2_2) Cylinder under internal pressure ............................57

EXAMPLE 5 (SE2_3) Stress concentration in tension of plate with hole ..81

EXAMPLE 6 (SE2_5) Cook`s problem (cantiliver with variable cross-

section)................................................................................105

EXAMPLE 7 (SE3_1) Cantilever subjected to triangular distributed

loading.................................................................................125

EXAMPLE 8 (SE8_1) Clamped square plate loaded by concetrated

force.....................................................................................147

EXAMPLE 9 (SE8_4) Cantilever of Tshape cross-section loaded by

concentrated force ..............................................................163

EXAMPLE 10 (SE8_5) Cylindrical roof under gravity loading ...............185

EXAMPLE 11 - (SE8_6) Half of sphere with hole subjected to concentrated

forces ..................................................................................203


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EXAMPLE 12 (SE8_8) Cylindrical shell with rigid diaphragms, loaded by

concetrated forces ...............................................................225

EXAMPLE 13 (SE8_9) Cantilever with rotated cross-section .................245

EXAMPLE 14 (SE8_13) Simply supported circular plate subjected to

pressure.................................................................... ...........261

EXAMPLE 15 (SM3_3) Thermal loading of coaxial rings .......................287

LINEAR DYNAMIC ANALYSIS

EXAMPLE 1D (SD_1) Eigenvalue analysis of cantilever beam ..............313

EXAMPLE 2D (SD_3) Cylindrical pipe loaded by radial line impulse

load......................................................................................335


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PAK-S Examples Linear Structural Analysis

Example 1 1

EXAMPLE 1 (SE1_1) Spatial truss structure

Problem Description/Objective Symmetric three-dimensional structure consisting of four equaltrusses is loaded by vertical concentrated force along the axes of symmetry. Boundary conditions

are as follows: all nodes laying at the Oxy-plane (nodes 1, 2, 3 and 4) are fully restrained (in all

three directions), and only loading point (node 5), common for all trusses is free, Fig. 1. This

example illustrates use of the straight (two-node) truss elements with all necessary spatialtransformations in the PAK-S program.

Geometrical and boundary data for structure are given in Fig. 1. Trusses are placed at the edges of

four-sided pyramid, with a squared base a x a, and height is 2/2a . Cross-sectional area for trusses

is A, Young modulus for material is E. The magnitude of the force is F. Poisson` s ratio is .

Numerical values are given in Fig. 1.

For this example, you will display deformed shape of the model with the rod axial forces and

displacement of loading point.

Fig. 1

Suggested Exercise Steps:

I Defining Materials and Properties

II Generation of Nodes and Elements

III Constraining the ModelIV Loading the Model

V Prepare and Submit the Model for a Linear Static Analysis

VI Postprocessing and Review the Results

a=10cm

A=1cm2

E=1x105NIcm

2

F=100 N

=0.3


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Example 1 2

If you have not already done, start FEMAP by double-click the icon on your Desktop or in the

Windows Start menu, pickStart>Programs>Femap v8.1>Femap v8.1. All new models are named

Untitled. The commandFile>Save As...allows you to change the model filename. The WindowsCommon File Save As dialog box will be display. Using this box, you can navigate around your

local computer. Move to or create a directory where you wish to store your model, type in the field

File name: Example1, and press Save.

Throughout this example, all commands that you will select from the FEMAP menu will be shownin the following styleFile>Save As...Which means, first selectFile from the menu, and then move

to the Save As command.

File>Save As...


Model>Material...

AfterFEMAP displays again dialog box Define Isotropic Materialpress Cancel to end command.

Note: FEMAP automatically repeats any creation commands assuming that during the creation ofmost finite element models that each command will be required more than once. This behavior is

designed to minimaze the number of times you will need to go into the menus or use the toolbox to

acces commands. This feature can be turned off in the File>Preferences command.

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Example 1 3

Model>Property...

AfterFEMAP displays again dialog box Define Property - ROD Element Typepress Cancel.

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Example 1 4


Model>Node... (Short-cut key Ctrl+N)

Note: To create the required model you will enter manually coordinates of the nodes in the properfield in the dialog box.

AfterFEMAP displays again dialog box Locate Enter Coordinates or Select with Cursorpress Cancel.

Press Ctrl+A or choose following command to autoscale the view:

View>Autoscale>Visible

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Example 1 5

You can change the colors of model enteties (nodes) using next commands:

Modify>Color>Node...

Display labelling of nodes and elements by ID

View>Options... (short-cut keyF6)

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Example 1 6

Model>Element...( Short-cut key Ctrl+E)

Note: You can select the proper nodes directly by the mouse from the screen, or enter their ID inthe field as shown figures bellow.

After FEMAP displays again dialog box Define ROD Element - Enter Nodes or Select withCursorpress Cancel.

Modify>Color>Elements...

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Example 1 7

Let`s rotate the view so you can see the whole model.

View>Rotate>Model... ( short-cut keyF8)

You can turn off the workplane for the clarity of model picture.

Tools>Workplanel... ( short-cut keyF2)

Use the following command to regenerate the screen:

View>Regenerate ( short-cut key Ctrl+G)

On the Fig.2is shown finite element model(nodes and elements).

Fig. 2

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Example 1 8

III Constraining the Model

Note: Since Constraints are Set based, you must create an empty Constrain Set before defining anyactual constrains.

Model>Constraint>Set... (short-cut keyShift+F2)

Model>Constraint>Nodal...

Note: There are 5 nodes in model, and, as described above, nodes 1, 2, 3 and 4 are restrained in allthree directions, and node 5 is free. You have to graphically (using mouse) pick the location of the

nodes 1, 2, 3 and 4 on screen.

AfterFEMAP displays again dialog box EntetySelection - Enter Node(s) Select press Cancel.

Modify>Update Other>Perm Constraint

Note: Degrees of freedom that corresponding to the rotations of nodes are restrained globallybecause only translational displacements are allowed for trusses.

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Example 1 9

IV Loading the Model

Note: Like constrains, loads are grouped in sets. Before creating any loads we must create a set tohold them.

Model>Load>Set... (short-cut key Ctrl+F2)

Model>Load>Nodal

Note: We will now apply a load in the negative z direction to simulate vertical concentrated forcealong the axes of symmetry. Select the node 5 moving the mouse on that node on the screen or enterits ID in the field ID in Entity Selection - Enter Node(s) to Select dialog box.

AfterFEMAP displays again dialog boxEntity Selection - Enter Node(s) to SelectpressCancel.

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Example 1 10

The model is now complete and looks like one on the Fig. 3.

Fig. 3

We suggested you to save your work that you have done till now, using File>Save (short-cut F4)


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Example 1 11


You should export your model as a FEMAP neutral file (*.NEU file ver.4.4). Then, using the PAKTranslator you will translate your model into an input file for FEA program PAK-S that willanalyze this model.

File>Export>FEMAP Neutral

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Example 1 12

Run the PAK Translator by double-click the icon on your Desktop or in the directory Paktrrunthe filePaktr.exe.

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Example 1 13

You should save your *. dat file in the same directory as you put a filePaks.exe. You need to run

filePaks.exe and after start it, typeExample1.dat in a DOS prompt (see the window below). Three

files should be create during this processExample1. LST,Example1. UNV, Example1. NEU.


After solution, FEMAP can read the output results for post-processing. Reading output results is

similar like writing model for analysis.

File>Import>FEMAP Neutral

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Example 1 14

Viewing the rod axial forces in the truss as follows:

View>Select... (short-cut keyF5)

AfterFEMAP displays again dialog box Select PostProcessing DatapressOK.

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Example 1 15

AfterFEMAP displays again dialog box View SelectpressOK.Througout a few steps from Command Toolbar and View Toolbar shown bellow you can setupappropriate view of your results on the screen.

Look Command Toolbar

Look View Toolbar

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Example 1 16

On Fig. 4, FEMAP displays on the screen the deformed model with color representation of the rod

axial forces.

Fig. 4 Rod axsial force

Displacements of the loading point (node 5) you can view in the Messages and List Window using

command:

List>Output>Query...

To turn the 3 lines of the text of the Messages and List Windowinto a complete window simplydouble click in the area at the bottom of the Main Window.

Numerical results obtained by program are exactly the same as analytical solution.

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Example 2 17

EXAMPLE 2 Static analysis of 2D truss

Problem Description/Objective Determine the member forces and reactions of the supports of thetruss described in the figure bellow subjected to the concentrate forces at the nodes 3, 6 and 8.

Geometrical (coordinates of nodes) data; boundary conditions and magnitude of the forces are given

in Fig. 1. All rods have the same cross-sectional area, A, and the same Young modulus, E. Poisson`

s ratio is . Numerical values are given in Fig. 1.

Fig. 1








VII Propose of Infinitesimal Displacements (determine the reactions of supports)

x

y

1(0,0)

2(0,4)3(2,4)

5(4,4)

4(2,6)

6(4,6)

7(6,4)

8(6,6)

Material S355

A=0.01 [ m2]

E=2.1e11 [N/m2]

=0.3


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Example 2 19

Model>Property...


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Example 2 20


Model>Node... (Short-cut key Ctrl+N)

Note: To create the required model you will enter manually coordinates of the nodes, Fig. 1, in theproper field in the dialog box.

AfterFEMAP displays again dialog box Locate Enter Coordinates or Select with Cursorpress Cancel.

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Example 2 21

To autoscale the view press Ctrl+A or choose command:


Model>Element...(Short-cut key Ctrl+E)

Note: You can select the proper nodes directly by the mouse from the screen, see Fig. 1, or enter

their ID in the field Nodes as shown figures bellow.

...

Enter the following data (see Table 1.) into proper fieldsNodes of previous dialog box when pop

up:

Table 1. Elements

Element 1. node 2. node

3 2 34 2 45 3 46 3 5

7 4 68 4 59 6 8

10 5 611 5 812 5 713 7 8

After the last element13 between the nodes 7and8 has been created andFEMAP displays againdialog box Define ROD Element - Enter Nodes or Select with Cursorpress Cancel to endcommand.



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Example 2 22

On the Fig.2,finite element model(nodes and elements)is shown.

Fig. 2

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Example 2 23





Note: First, you have to graphically (using mouse) pick the location of the node 1 on screen and restrain it in x and y

direction.

Now, select the node 7 on screen and restrain it in and y direction as following:


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Example 2 24


Note: Degrees of freedom that corresponding to the rotations of nodes are restrained globallybecause only translational displacements are allowed for trusses. Also, because of 2D problem only

two translational displacements are allowed in x and y directions.

The model should look like the one below on the Fig.3.

Fig. 3

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Example 2 25



Model>Load>Set... (short-cut keyCtrl+F2)

Model>Load>Nodal...

Note: We will now apply a load to the model as shown on Fig. 1. First, select the node 3 movingthe mouse on that node on the screen or enter its ID in the field ID in Entity Selection - EnterNode(s) to Select dialog box. Next, applay concentrate forces at nodes 6 and 8.

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Example 2 26


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Example 2 27

On the Fig. 4 you will see the loads and constarints on your model.

Fig. 4



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Example 2 28


You should export your model as FEMAP neutral file (*.NEU file ver.4.4). Then, using the PAKTranslator you will translate your model into an input file for FEA program PAK-S that willanalyze this model.


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Example 2 29


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Example 2 30








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Example 2 31



AfterFEMAP displays again dialog box Select PostProcessing DatapressOK.

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Example 2 32

AfterFEMAP displays again dialog box View SelectpressOK.Througout a few steps from Command Toolbar and View Toolbar shown bellow, you can setupappropriate view of your results on the screen.


Look View Toolbar

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Example 2 33

FEMAP displays on the screen the deformed model with color representation of the rod axialforces, Fig. 5.

Fig. 5

To show reactions of the supports you have to change your model throughout next few steps:

First, you should delete previous results from your model using following commands:

Delete>OutputSet..


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Example 2 34

View>Regenerate...( Short-cut key Ctrl+G)

FEMAP can display all enteties using Quick options ofView Toolbar, see following picture:

Look View Toolbar

Now, your model is looking like one at the Fig. 4.

Note:To show reactions of supports in FEMAP you must remove restraints in the x and y

directions at node 1 and in the y direction at the node 7 and instead of them you have to proposeinfinitesimal displacements at these nodes (1e-10 m) in the equivalent directions.

Delete>Model>Constraint - Set...

View>Regenerate...(Short-cut key Ctrl+G)

Note:Your model now loses restraints in the x and y directions at node 1 and in the y directions atthe node 7 that is shown on screen.

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Example 2 35

VII Purpose of Infinitesimal Displacements (to the determine reactions ofsupports)

Model>Load>Nodal...

Note: You can graphically (using mouse) pick the location of the nodes 1 and 7 on screen or enter their ID.

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Example 2 36

AfterFEMAP displays again dialog boxEntity Selection - Enter Node(s) to SelectpressCancel.On the Fig. 6you can see the model with loads and displacements.

Fig. 6

We need to repeat steps V and VI on same way that we have done before (until page 18. before Delete> OutputSet...) and submit the model for linear static analysis and post procesing andreviewing the results in the FEMAP. Also, we suggest to save your model and all new files (*.NEU,

*.dat) using new names, for instance Example2a.

After you went through all necessary steps you should show reactions of supports as follows:

View>Select... (short-cut key F5)


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Example 2 37

AfterFEMAP displays again dialog box View SelectpressOK

FEMAP displays on the screen the deformed model with color representation of the reactions of

supports, Fig. 7.

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Example 2 38

Fig. 7Constraint force

Reactions of supports (at nodes 1 and 7) you can view in the Messages and List Window usingcommand:

List>Output>Query...

To turn the 3 lines of the text of the Messages and List Windowinto a complete window simplydouble click in the area at the bottom of the Main window.

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Example 3 39

EXAMPLE 3 Linear static analysis of roof truss

Problem Description/Objective Determine the rod axial forces for bellow roof truss. Because ofmodel and load symmetry we will use only one half of this part. Geometrical and boundary data for

structure is given in Fig. 1. Cross-sectional area for trusses is A, Young modulus for material is E.

The magnitude of the force is F. Poisson` s ratio is .

Table 1.Coordinates of nodes in the plane Oxy

Node x [m] y [m] z [m]

1 0 0 0

2 0.4 0 0

3 0.8 0 04 1.2 0 0

5 0.2 -0.3 0

6 0.4 -0.3 0

7 0.6 -0.3 0

8 0.8 -0.3 0

9 1.0 -0.3 0

10 1.2 -0.3 0


I Defining Materials and PropertiesII Creating the Geometry

III Generation of Nodes and Elements

IV Constraining the Model

V Loading the Model

VI Prepare and Submit the Model for a Linear Static Analysis

VII Postprocessing and Review the Results

y

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F=200N

x

Material S355

E=2.1e11 [N/m2]

=0.3

A=0.01 [m2]

A

B

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D

0.3

Fig. 1.


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Example 3 40


Windows Start menu, pick Start>Programs>Femap v8.1>Femap v8.1. All new models are named

Untitled. The command File>Save As... allows you to change the model filename. The Windows

Common File Save As dialog box will be display. Using this box, you can navigate around your

local computer. Move to or create a directory where you wish to store your FEMAP model, type in

the field File name Example3, and press Save.

Throughout this example, all commands that you will select from the FEMAP menu will be shown

in the following styleFile>Save As...Which means, first selectFile from the menu, and then moveto the Save As command.

File>Save As...

Tools>Advanced Geometry...


Model>Material...


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Example 3 41

Model>Property...


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Example 3 42

II Creating the Geometry

Create Points

Geometry>Point...

Note: To create the required geometry you should enter manually coordinates of the points in the

proper field in the following dialog box that are given in the Table 1.

...

AfterFEMAP displays again dialog box Locate - Enter Coordinates or Select with Cursor enterthe data points given in Table 1.

To autoscale the view after press short-cut keyCtrl+A or choose command:



Tools>Workplane... (short-cut keyF2)

Create Lines Between the Points

Geometry>Curve-Line>Points...

Note: To create the required geometry you should select the first point by moving the mouse on thefirst point in a line and then move to the point at the other end of the line.

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Be sure to create all 17 line segments.

There is a geometry of 17 lines in the Fig. 2.

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Example 3 43

Fig. 2

These lines will be copied in the z direction to create the other side of the truss. Since we will be

working in 3-D, let`s rotate the view so we can see the new lines that will be generated behined the

ones you just created.

View>Rotate>Model... (short-cut keyF8)

Geometry>Copy>Curve...

To autoscale the view use command:

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Example 3 44

View>Autoscale>Visible (short-cut key Ctrl+A)

On the Fig. 3 is shown part of 3-D geometry.

Fig. 3Connect the Two Sides

We will now create new lines that connect the corresponding points from the two sides of the truss.

You should create 10 lines.

Geometry>Curve - Line>Points...

Note: To create the required geometry you should select the first point by moving the mouse on thefirst point in a line and then move to the point at the other end of the line, see Fig. 1.

...

You have completed the geometry section of this example as shown on Fig. 4.

Fig. 4

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Example 3 45


Note: The first thing we need to do is to set the mesh sizes for the curves (actually lines in thisexample) to an appropriate value; appropriate in that it will keep the total number of nodes

generated small enough to work in the demo version ofFEMAP.

Mesh>Mesh Control>Size Along Curve...

After FEMAP displays again dialog box Entety Selection - Select Curve(s) to Set Mesh Sizepress Cancel.

Mesh>Geometry>Curve...

View>Regenerate (short-cut key Ctrl+G)

To reduce the amount of information displayed (turn off geometry), use the FEMAP Quick Options

(short-cut keyCtrl+Q):

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Example 3 46

Note: Each line in the model now has rod element along its length. Since they were generatedindependetly, there will be multiple nodes at the end of every line coincident with the nodesgenerated for the other lines that also ended in the same point. In the world of FEA, these structural

members will not be connected. To effectively link the model we will use FEMAP`s ability to

check and optionally merge coincident nodes of model.

Tools>Check>Coincident Nodes...

Since coincident nodes have been merged with connecting all rod elements of this exampletogether, you should performe their renumber because their number are less now.

Modify>Renumber>Node...

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Example 3 47

On the Fig.5finite element model(nodes and elements)is shown.

Fig. 5

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Example 3 48




To simulate the simmetry of this part, we will constrain the four nodes that are the halfway point of

our structure. In this model, we are defining symmetry across the x plane through these points. By

imposing this type of constrain condition, we are actually introducing a stiffness exactly equal to the

structure modeled, just mirrored about the x plane.


Note: You have to graphically (using mouse) pick the location of these 4 nodes (nodes E, F, G andH) on screen, Fig.5, and restrain them as following.

FEMAP automatically prompts you for more nodes to constrain. Select nodes A and B at the left

end of the truss and fix these nodes, Fig. 1.

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Example 3 49



Note: Degrees of freedom that corresponding to the rotations of nodes are restrained globallybecause only translational displacements are allowed for trusses. Since we will be loading this part

in the negative y direction this part will be also require constraint against movement in the z

direction.

V Loading the Model

Note: Wewill now apply a load in the negative y direction to simulate something hanging from thistruss.Like constrains, loads are grouped in sets. Before creating any loads we must create a set tohold them.

Model>Load>Set... (Short-cut key Ctrl+F2)

Model>Load>Nodal...

Note: Select two nodes E and F moving the mouse on those nodes on the screen, Fig. 5. Next,applay concentrate forces at these nodes.

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Example 3 50

Due to simmetry of model enter half value of force (-100) in the FY field.


The model shown on the Fig. 6is now complete and ready for analysis.

Fig. 5


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Example 3 51




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Example 3 52


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Example 3 53








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Example 3 54



AfterFEMAP displays again dialog box Select PostProcessing DatapressOK.AfterFEMAP displays again dialog box View SelectpressOK.

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Example 3 56

FEMAP displays on the screen the deformed model with color representation of the rod axial

forces, Fig. 7.

Fig. 7 Rod axial force


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Example 4 57

EXAMPLE 4 (SE2_2) Cylinder under internal pressure

Problem Description/Objective Cylinder of thick walls shown in Fig. 1 is loaded by internalpressure. It is considered that axial deformation is restrained. Geometrical and material data are

shown in Fig. 1, with cross-section of cylinder segment. Finite element model should consist of 4

finite elements with 8 nodes per element. Boundary conditions are such that only all radial

displacements are free. Pressure loading on axial axisymmetric surface is employed in PAK-S.

The objective is to find displacements, stresses and strains in an axisymmetric body loaded by

axisymmetric loading using 2D elements.

Fig. 1


I Defining Material and Properties




V Loading the Model




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Example 4 58





the field File name Example4, and press Save.Throughout this example, all commands that you will select from the FEMAP menu will be shown


File>Save As...



Model>Material...


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Example 4 59

Model>Property...

To define finite elements with 8 nodes per element you must pick options Parabolic Elements inthe following dialog box.

AfterFEMAP displays again dialog box Define Property AXISYMMETRIC Element Typepress Cancel.

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Example 4 60


Geometry>Surface>Corners

Note: Corners of surface have been defined in order to get 4 equal elements on created surface.

AfterFEMAP displays again dialog box Locate Enter First Corner of Surfacepress Cancel.

To autoscale the view press Ctrl+A or choose command:



Tools>Workplane... (short-cut keyF2)


View>Regenerate(short-cut keyCtrl+G)

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Example 4 61

Display labelling of curves and surfaces by ID


You have designed the geometry of surface as shown on Fig. 2.

Fig. 2

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Example 4 62


Note: The first thing we need to do is set the mesh size for the surface.

Mesh>Mesh Control>Mapped Divisions on Surface...

AfterFEMAP displays again dialog box Entity Selection Select Surface(s) to Set Mesh Size

press Cancel.Mesh>Geometry>Surface

Four finite elements with 9 nodes per element have been generated. You should solve the problem

by using 8-node 2D axisymmetric elements, so ninth node in the middle of the each element must

be deleted.

Delete>Model>Node...

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Example 4 63

You will see in the Messages and List Window following message:

To see effect of deleting use the following command to regenerate the screen:

View>Regenerate(short-cut keyCtrl+G)To reduce the amount of information displayed (turn off geometry), use the FEMAP Quick Options




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Example 4 64

Since 4 nodes were deleted you have to performe the renumber of nodes.

Modify>Renumber>Node

Finite element model is shown in the Fig 3.

Fig. 3

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Example 4 65




Note: since boundary conditions are such that only all radial displacements are free, you canconstrain the model as following:


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Example 4 67

The model shown on the Fig. 4 is now complete and ready for analysis.

Fig. 4



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Example 4 68




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Example 4 70








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Example 4 71

Viewing the total translation and stresses:



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Example 4 73

FEMAP displays on the screen the deformed model with color representation of the totaltranslation, Fig. 5.

Fig. 5 Total translation field


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Example 4 74

AfterFEMAP displays again dialog box View SelectpressOK.

FEMAP displays on the screen the deformed model with color representation of the equivalent

stress, Fig. 6

Fig. 6 Effective stress field

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Example 4 76

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Example 4 77

On the Fig. 7FEMAP displays radial displacements distribution.

Fig. 7


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Example 4 78

If you want to display distribution of radial stress xx and after that circular stress zz, you will

select in a dropdown menu Output Vector instead of 2. T1 TRANSLATION PLATE options7020..SOLID X NORMAL STRESS and 7022..SOLID Z NORMAL STRESS respectively in the

dialog box Select XY Curve Data. On the Fig. 8 and Fig. 9 radial stress distribution and circuralstress distribution are shown respectively.

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Example 4 79

Fig. 8

Fig. 9

Good agreeement between the numerical and the analytical solutions is observed.


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Example 4 80


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Example 5 81

EXAMPLE 5 (SE2_5) Stress concentration in tension of plate withhole

Description/Objective Plate of unit thickness with a hole shown in Fig. 1 is exposed to a uniformtension at two sides. Stress concentration develops in the hole region. Geometrical and material data

are shown in Fig. 1. Due to symmetry in geometry, boundary conditions and loading, only one-quarter is modelled using PAK 8-node 2D plane stress elements. Plate is loaded by distributed

loading (pressure). Boundary conditions take in account symmetry conditions: ux=0 along line AB,

uy=0 along line CD. Finer mesh is used around hole due to high gradients in displacement, strain

and stress fields in that region. This approach in meshing provides good results with a small number

of finite elements.

The objective is to find stress distribution around a hole using 2D plane stress elements and to show

deformed and undeformed model.

L

b

d A

C

D

B

p

p

L= 56 mmb= 20 mmd= 10 mm

E= 7.0 10 MPa

= 0.25

p= 25.0 Mpa

= 1 mm

4

X

Y

E

Fig. 1


I Defining Material and Properties




V Loading the Model




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Example 5 82








File>Save As...



Model>Material...


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Example 5 85

After FEMAP displays again dialog box Entety Selection Select Curve(s) to Breakpress

Cancel.


You will now create three curves that are necessary to get three surfaces.

Geometry>Curve-Line>Project Points

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Example 5 86

AfterFEMAP displays again dialog box Locate Enter First Location for Projected LinepressCancel.







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Example 5 87


You can see the created lines on the Fig. 2.

Fig. 2

Create three surfaces among cuves 1 and 3, 2 and 4, 4 and 5 that will represent quarter of model.

Geometry>Surface>Ruled

Note: You can select the proper curves directly graphically by the mouse from the screen, or entertheir ID in the field as shown figures bellow.

AfterFEMAP displays again dialog box Create Ruled Surfacepress Cancel.

Note: Since the surfaces were generated independetly, there will be multiple curves at the edge ofevery surface coincident with the curve generated for the other surface that also lie in the same

edge. In the world of FEA, these geometrical members will not be connected. To effectively link

the model we will use FEMAP`s ability to check and optionally merge coincident curves of model.

Tools>Check>Coincident Curves

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Example 5 88

To see effect of merging use the following command to regenerate the screen:



Fig. 3

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Example 5 89


Note: The first thing we need to do is to set the mesh size for the all three surfaces and set the densemesh in the hole vicinity using optionsBias as following.

Mesh>Mesh Control>Mapped Divisions on Surface...

Note: You may graphically pick the location of these 3 surfaces on screen, Fig.3.

AfterFEMAP displays again dialog box Entity Selection Select Surface(s) to Set Mesh Sizepress Cancel.

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Example 5 90

Mesh>Geometry>Surface

Finite elements with 9 nodes per element have been generated. You should solve the problem by

using 8-node 2D elements, so ninth node in the middle of the each element must be deleted.



To see an effect of deleting use the following command to regenerate the screen:


Note: Since the surfaces were generated independetly, there will be multiple nodes at the edgecurve of every surface coincident with the nodes generated for the other surface that also lie in the

same edge. In the world of FEA, these structural members will not be connected. To effectively link

the model we will use FEMAP`s ability to check and optionally merge coincident nodes of model.

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Example 5 91

Tools>Check>Coincident Nodes

Since 10 nodes were deleted (see Messages and List Window) you have to performe the renumber

of nodes.

Modify>Renumber>Node


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Example 5 92





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Example 5 94




Model>Constraint>Nodal

Note: You have to graphically (using mouse) pick the location of curve 7 on screen, Fig.3, and setsymmetry conditions, restrain it in x direction.

Note: Now, you can graphically (using mouse) pick the location of curves 8 and 10 on screen, Fig.3, and set symmetry conditions, restrain them in y direction.

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Example 5 95

After FEMAP displays again dialog box Entity Selection Enter Node(s) to Select pressCancel.

Note: Because of 2D plane stress elements only two translational displacements are allowed in xand y directions.


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Example 5 96

V Loading the Model



Model>Load>On Curve

Note: You have to graphically (using mouse) pick the location of curve 5, Fig.3.

AfterFEMAP displays again dialog box Entity Selection Enter Curve(s) to SelectpressCancel.

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Example 5 97

The model shown on the Fig. 6is now complete and ready for analysis.

Fig. 6



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Example 5 98




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Example 5 99


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Example 5 101

Display stress xx distribution:


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Example 5 102


Througout a few steps from Command Toolbar and View Toolbar shown bellow, you can setupappropriate view of your results on the screen.


Look View Toolbar

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Example 5 103

FEMAP displays on the screen the deformed model with color representation of the stressdistribution, Fig. 7.

Fig. 7 Stress xx field

Good agreement between the numerical and the analytical solutions for stress is observed even with

a coarse finite element mesh. In Newton-Cotes numerical integration nodes and integration points

on element boundary coincide

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Example 5 104

Deformed and undeformed configurations are shown on Fig. 8.



Fig. 8 Orginal and deformed configurations

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PAK-S Examples Liner Structural Analysis

Example 6 105

EXAMPLE 6 (SE2_5) Cook`s problem (cantiliver with variablecross-section)

Description/Objective Cantilever with variable cross section is loaded by distributed loading onthe right side as shown in Fig. 1. Geometrical and material data are given in Fig.1. Boundary

conditions are shown in the same figure. Loading is applied on the right side as distributedtangential load. The FE model use 16x16 mesh of 2D 4-node plane stress elements.

The objective is to show displacements in y direction of model.

Fig. 1






V Loading the Model



E=1 N/mm2;

=0.333333;

=1 mm;

p=1 N


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Example 6 106



Untitled. The commandFile>Save As...allows you to change the model filename. The WindowsCommonFile Save As dialog box will be display. Using this box, you can navigate around your



in the following styleFile>Save As...Which means, first selectFile from the menu, and then move


File>Save As...



Model>Material...

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Example 6 107

AfterFEMAP displays again dialog box Define Isotropic Material press Cancel to end command.Model>Property...

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Example 6 108

AfterFEMAPdisplays again dialog box Define Property - MEMBRANE Element Type pressCancel.II Creating the Geometry

Note: You can create two vertical lines of cantiliver and then surface among them, Fig. 1.

Geometry>Curve-Line>Project Points

AfterFEMAP displays again dialog boxLocate - Enter First Location for Projected Line pressCancel.

View>Autoscale>Visible... (short-cut key Ctrl+A)

Geometry>Surface>Ruled...

AfterFEMAPdisplays again dialog boxCreate Ruled Surface press Cancel.



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Example 6 109





You have completed the geometry as shown on Fig. 2.

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Example 6 110

Fig. 2


Note: The first thing we need to do is set the mesh size for surface.

Mesh>Mesch Control>Size Along Curve...

AfterFEMAP displays again dialog boxEntity Selection -Select Curve(s) to Set Mesh Size pressCancel.

Mesh>Geometry>Surface...

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Example 6 111



Display labelling of nodes by ID


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Example 6 112

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Example 6 113

Now, you should check elements direction (orientation of nodes must be in the positive

mathematical direction). Activate option Off>Element on the Status Line and then you may

graphically pick the location of every element on screen to check its direction. You will see pop-up

window with written orientation of nodes as you see on the figure bellow. After you finished, press

the button Offin the Status lineagain.

You can update element direction as following:

Modify>Update Elements>Reverse

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Example 6 114

Your model looks like as shown on Fig. 3.

Fig. 3

Turn off labelling of nodes by ID


A

B

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Example 6 115





Note: You have to graphically (using mouse) pick the location of curve 1 on screen, Fig.3, and setboundary conditions, restrain it in all direction and all rotation.

AfterFEMAP displays again dialog boxEntity Selection - Enter Nodes to Select (On Curves)pressCancel.

Note: Because of 2D plane stress elements only two translational displacements are allowed in xand y directions.

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Example 6 116

Modify>Update Other>Perm Constraint...

On the Fig. 4 is shown constraint model.

Fig. 4

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Example 6 117

V Loading the Model



Model>Load>Nodal...

Note: You have to graphically (using mouse) pick the location of the nodes A and B on curve 2(firs and last) on screen, Fig.3, and set vertical loads in positive y direction as following.

Note: Now,You will graphically (using mouse) pick all the other on curve 2 (between A and B) onscreen, Fig.3, and set vertical loads in positive y direction as following. You must pick 15 nodes.

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Example 6 119


You should export your model as FEMAP neutral file (*.NEU file ver.4.4). Then, using the PAKTranslator you will translate your model into an input file for FEA program PAK-S that willanalyze this model.


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Example 6 120


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Example 6 121








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Example 6 122

FEMAP can display displacements in y direction of model as following:



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Example 6 123



Look View Toolbar

On the Fig. 6displacements in the y direction are shown.

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PAK-S examples Linear Structural Analysis

Example 7 125

EXAMPLE 7 (SE3_1) Cantilever subjected to triangular distributedloading

Description/Objective cantilever is subjected to distributed loading of triangular shape, as shownin Fig. 1. Geometrical, material data and loading conditions are given in Fig. 1. Five 3D solid

elements, with 20 nodes per element, are used in FE model. Boundary conditions at the clampededge (plane yz) are as follows: ux=0 for all nodes; uy=0 for the mid-nodes on z-axis; uz=0 for mid-

nodes on y-axis. These conditions provide no rotation of the clamped edge, and no through-the-

thickness stresses. Loading is considered as surface-pressure loading.

Determine deflection at the free end of cantilever and distribution of deflection along the cantilever

axis (x axis).

Fig. 1






V Loading the Model

VI Prepare and Submit the Model for a Linear Static AnalysisVII Postprocessing and Review the Results


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Example 7 126








File>Save As...



Model>Material...

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Example 7 127

AfterFEMAPdisplays again dialog box Define Isotropic Materialpress Cancel to end command.Model>Property...

AfterFEMAP displays again dialog boxDefine Property - SOLID Element Typepress Cancel.

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Example 7 128


Note: First, you will create a surface in the yz plane.

Geometry>Surface>Corners...

AfterFEMAP displays again dialog boxLocate - Enter First Corner of Surfacepress Cancel.

Since we will be working in 3-D, let`s rotate the view so we can see the generated surface.

View>Rotate>Model... (short-cut keyF8)


Tools>Workplanel... (short-cut keyF2)

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Example 7 129

Use the short-cut key Ctrl+G to regenerate the screen.


On the Fig. 2 created surface is shown.

Fig. 2

Creating the 3D console.

Geometry>Volume>Extrude...

AfterFEMAP displays again dialog boxEntity Selection Select Surface(s) to ExtrudepressCancel.

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Example 7 130


Display labelling of curves by ID


You have completed the geometry as shown on Fig. 3.

Fig. 3

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Example 7 132



Five finite elements with 21 nodes per element have been generated. You should solve the problem

by using 20-node 3D solid elements, so 21st

node in the middle of the each element must be deleted.


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Example 7 133


To see effect of deleting use the following command to regenerate the screen:


Since 31 nodes were deleted you have to performe the renumber of nodes.

Modify>Renumber>Node...

Finite element model is shown in the Fig 4.

Fig. 4

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Example 7 134





Note: You have to graphically (using mouse) pick the location of nodes 1, 2, 3, 6, 7 and 8 on

screen, Fig.4, and restrain them in x direction.

Note: Now, you have to graphically (using mouse) pick the location of node 4 as a mid-node at zaxis on screen, too, Fig.4, and restrain it in x and y directions.

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Example 7 135

Note: Now, you have to graphically (using mouse) pick the location of node 5 as a mid-node at yand z axis on screen, too, Fig.4, and restrain it in x, y and z directions.

AfterFEMAP displays again dialog boxEntity Selection - Enter Node(s) to Selectpress Cancel.

Set the global constraints.

Modify>Update Other>Perm Constraint...

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Example 7 136

V Loading the Model


.

Model>Load>Elemental...

Note: Surface-pressure loadingis defined as variable value by formula: 0.2e6*Xel(!i)

AfterFEMAP displays again dialog box Entity Selection - Enter Elements to SelectpressCancel.

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Example 7 137

You can turn off the labelling of nodes and elements by ID and display load value for the clarity of

model picture.


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Example 7 138

The model shown on the Fig. 5 is now complete and ready for analysis.

Fig. 5



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Example 7 139




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Example 7 140


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Example 7 141








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Example 7 142

Viewing the deflection as following:



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Example 7 143



Look View Toolbar

FEMAP displays on the screen the deformed model with color representation of deflection of the

free end, Fig. 6.

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Example 7 144

Fig. 6 Field of displacement in y direction

Distribution of deflection along the cantilever axis could be displayed:


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Example 7 145

AfterFEMAP displays again dialog box View SelectpressOK.


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Example 7 146

On the Fig. 7FEMAP displays distribution of deflection along the cantilever axis.

Fig. 7

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Example 8 147

EXAMPLE 8 (SE8_1) Clamped square plate loaded by concetratedforce

Description/Objective A square plate is clamped along edges and subjected to concentrated forceat the plate centre. A view of the plate is shown in Fig. 1(a). Structural view, geometrical, material

and loading data are given in Fig. 1(b). One quarter of the structure is modelled by 4-node shellelements, due to symmetry in geometry, loading and boundary conditions. All displacements,

including rotations, are restrained along clamped edges AB and BD; while along lines AC and CD,

are applied symmetry boundary conditions: no displacement normal to the symmetry plane (v=0

along AC, u=0 along CD), and no rotation around symmetry line (1=0 for AC, and 2=0 for CD).

Five d.o.f. per node are used, with rotations 1 and 2 around local element axes V1 and V2, shown

in Fig. 1(b). One quarter of concentrated plate force F is taken to load on the part of the plate used

in the FE model. Display field of plate deflection and deflection of plate centre.

a)

b)

Fig. 1


I Defining Materials and PropertiesII Creating the Geometry



V Loading the Model




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Example 8 148







File>Save As...



Model>Material...

AfterFEMAP displays again dialog boxDefine Isotropic MaterialpressCancelto end command.

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Example 8 149

Model>Property...

AfterFEMAP displays again dialog box Define Property - PLATE Element Typepress Cancel.


Geometry>Surface>Corners..

AfterFEMAP displays again dialog boxLocate - Enter First Corner of Surfacepress Cancel.


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Example 8 150


Tools>Workplanel... (short-cut keyF2)

Use the short-cut key Ctrl+G to regenerate the screen.




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linear static and dynamic structural analysis

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