module 1.1: point loading of a 1d cantilever beam

UCONN ANSYS –Module 1.1

Module 1.1: Point Loading of a 1D Cantilever Beam

Table of Contents Page Number

Introduction 2

Problem Description 3

Theory 3

Geometry 4

Preprocessor 8

Element Type 8

Real Constants and Material Properties 9

Meshing 10

Loads 11

Solution 12

General Postprocessor 13

Results 15

Validation 17


Introduction

Welcome to the UCONN ANSYS Mechanical Training Suite! Modules 1.1-1.9 are designed to

be an introduction to the fundamental modeling considerations and features in ANSYS. Using

classical beam loadings, we will model fundamental structures in one two and three dimensions

in an environment where theoretical answers are known and can be compared against the created

models. We will study the tradeoffs and benefits of modeling in one two or three dimensions.

Also, we will investigate how different boundary conditions affect the number of mesh elements

required to achieve a converged solution. Modules 1.1-1.9 are also designed as an introduction to

Linear Static Structural problems, a general category of Finite Element problems which can be

solved in one load step and one iteration. These problems are generally quick to solve using the

software and are easier to set up. Completion of this first series of modules will help the user

gain proficiency in the layout of the APDL environment and draw attention to the modeling

process, common modeling mistakes and other modeling considerations. While most tutorials in

this suite use the ANSYS Mechanical APDL package, a small introduction to ANSYS Workbench

is explored in modules 1.3W, 1.5W and 1.7W.


Problem Description

Nomenclature:

L =110m Length of beam

b =10m Cross Section Base

h =1 m Cross Section Height

P=1000N Point Load

E=70GPa Young’s Modulus of Aluminum at Room Temperature

=0.33 Poisson’s Ratio of Aluminum

In this module, we will be modeling an Aluminum cantilever beam with a point load at the end

with one dimensional elements in ANSYS Mechanical APDL. We will be using beam theory and

mesh independence as our key validation requirements. The beam theory for this analysis is

shown below:

Theory

Von Mises Stress

Assuming plane stress, the Von Mises Equivalent Stress can be expressed as:

(1.1.1)

Since the nodes of choice are located at the top surface of the beam, the shear stress at this

location is zero.

( . (1.1.2)

Using these simplifications, the Von Mises Equivalent Stress from equation 1 reduces to:

(1.1.3)

Bending Stress is given by:

(1.1.4)

Where

and

. From statics, we can derive:

(1.1.5)

(1.1.6)

With Maximum Stress at:

= 66 KPa (1.1.7)

y

x


Beam Deflection

The governing equation of a beam in bending is given by the Euler-Bernoulli relationship:

(1.1.8)

Plugging in equation 1.7.5, we get:

(1.1.9)

Integrating once to get an angular displacement, we get:

(1.1.10)

At the fixed end (x=0),

, thus 0

(1.1.11)

Integrating again to get deflection:

(1.1.12)

At the fixed end.y(0)= 0 thus , so deflection ( is:

(

)

(1.1.13)

The maximum displacement occurs at the point load( x=L)

(1.1.14)

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.

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Preferences

1. Go to Main Menu -> Preferences

2. Check the box that says Structural

3. Click OK

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Keypoints

Since we will be using 1D Elements, our goal is to model the length of the beam.

Go to Main Menu -> Preprocessor -> Modeling -> Create ->Keypoints ->

On Working Plane

1. Click Global Cartesian

2. In the box underneath, write 0,0,0 creating a keypoint at the origin.

3. Click Apply

4. Repeat Steps 3 and 4 for the point 110,0,0

5. Click OK

6. 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

7. Go to Utility Menu -> Plot -> Replot

Your graphics window should look as shown:

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Line

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

Lines -> Lines -> Straight Line

2. Select Pick

3. Enter 1,2 for keypoints

4. Click OK

Go to Utility Menu -> Ansys Toolbar -> SAVE_DB

The resulting graphic should be as shown:

SAVE_DB

Since we have made considerable progress thus far, we will create a temporary save file for our

model. This temporary save will allow us to return to this stage of the tutorial if an error is made.

1. Go to Utility Menu -> ANSYS Toolbar ->SAVE_DB This creates a save checkpoint

2. If you ever wish to return to this checkpoint in your model generation, go to Utility

Menu -> RESUM_DB

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WARNING: It is VERY HARD to delete or modify inputs and commands to your model

once they have been entered. Thus it is recommended you use the SAVE_DB and

RESUM_DB functions frequently to create checkpoints in your work. If salvaging your

project is hopeless, going to Utility Menu -> File -> Clear & Start New -> Do not read file

->OK is recommended. This will start your model from scratch.


Preprocessor

Element Type

1. Go to Main Menu -> Preprocessor ->

Element Type -> Add/Edit/Delete

2. Click Add

3. Click beam -> 3D Elastic 4

4. Click OK

5. Click Close

6. Go to Utility Menu -> ANSYS

Toolbar -> SAVE_DB

* BEAM4 is a one dimensional linear element with 6 degrees

of freedom (UX,UY,UZ,ROTX,ROTY,ROTZ). It has

tension, compression, bending, and torsional capabilities.

For more information, consult the ANSYS HELP by

clicking HELP

ANSYS HELP

ANSYS Mechanical APDL at its

core is a command line driven

FEA code. Similar to the Java

APL or the Matlab HELP feature,

ANSYS has its own library of

internal functions known as

Commands that are used in the

backend from the GUI front end.

The ANSYS HELP library

also provides useful information

on the theory behind ANSYS

calculations and modeling best

practices. We encourage you to

explore the vast volumes of

ANSYS HELP to increase your

proficiency in ANSYS beyond

the scope of these tutorials

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Real Constants and Material Properties

1. Go to Main Menu -> Material Props -> Material Models

2. Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic

3. Enter 7E10 for Young’s Modulus (EX) and .33 for Poisson’s Ratio (PRXY)

4. Click OK

5. out of Define Material Model

Behavior

6. Go to Utility Menu -> SAVE_DB

Now we will add the thickness to our beam.


Real Constants -> Add/Edit/Delete

2. Click Add

3. Click OK

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4. Under Real Constants for BEAM4 ->Shell thickness

at node I TK(I) enter:

10 for cross sectional area

10/12 for moment of inertia IZZ

10 for thickness along Z axis

1 for thickness along Y axis

5. Click OK

6. Click Close

Meshing


Meshing -> Mesh Tool

2. Go to Size Controls: -> Global -> Set

3. Under SIZE Element edge length put 55.

4. Click OK

5. Click Mesh

6. Click Pick All

7. Click Close


Loads

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Saving Geometry

We will be using the geometry we have just created for the next 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 ‘1D Cantilever’

3. Under Directories: pick the

Folder you would like to save the

.db file to.

4. Click OK

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Displacements

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

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

3. Check NODE, Node Numbers to ON

4. Click OK

Your plot should look as shown:

5. Go to Main Menu -> Preprocessor -> Loads -> Define Loads ->

Apply -> Structural -> Displacement -> On Nodes

6. Click Pick -> Single and with your cursor, click on first node

7. Click OK

8. Click All DOF to secure all degrees of freedom

9. Under Value Displacement value put 0.

10. Click OK


The fixed end will look as shown below:

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Point Load

1. Go to Main Menu -> Preprocessor -> Loads -> Define Loads ->

Apply -> Structural ->Force/Moment -> On Nodes

2. Under List of Items enter 2 for node 2 and press OK

3. Under Lab Direction of Force/mom select FY

4. Under Value Force/moment value type -1000

5. Press OK


The load at the end face should look as below:

Solution

1. Go to Main Menu -> Solution ->Solve -> Current LS (solve). LS stands for Load Step.

This step may take some time depending on mesh size and the speed of your computer

(generally a minute or less). Ignore any warnings that may appear on your screen, as they

are irrelevant to the problem at hand.

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USEFUL TIP: If you wish to assign new force values, pick the nodes of

interest and replace that component of force with 0 before assigning new

values. This will delete the previous force assignment.

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General Postprocessor

We will now extract the Preliminary Displacement and Von-Mises Stress within our model.

Displacement

1. Go to Main Menu -> General Postprocessor -> Plot Results -> Contour Plot -> Nodal

Solution

2. Go to DOF Solution -> Y-Component of displacement

3. Click OK

4. To give the graph a title, go to

Utility Menu -> Command Prompt and type

/title, Deflection of a Cantilever Beam with a Point Load.

5. Press enter and write /replot to refresh the window.

6. Press enter

The Resulting Plot should look as shown below:

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Equivalent (Von-Mises) Stress

Unfortunately, we cannot create a contour plot of Von-Mises stress for 1D elements. We can,

however, look up the moment reactions at each element. If we plug this value into equation

1.1.4, we can readily calculate the bending stress in our model and by extension, the equivalent

stress.

1. Go to Utility Menu -> List -> Results -> Element Solution …

2. Go to Element Solution -> All Available force items

3. Click OK

This chart shows all reaction forces and moments at each node in the domain. Since we are

interested in reaction moments in the z direction, we will look to the last column in the chart:

According to the chart the maximum moment at the fixed end of the beam is .11E6 Nm.

Plugging into equation 1.1.4, we get the expected stress of 66 kPa.


Results

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

(

) = 0 % (1.1.15)

Max Deflection Error

Max Equivalent Stress Error

Using equation (1.1.15) above, the percent error for Max Deflection and Equivalent Stress in our

model is 0%. This is due to the fact that ANSYS uses Gaussian Quadrature to interpolate

between the integration points. This changes with respect to the element used. Beam4 used two-

point Gaussian Quadrature, a numerical technique which is fourth degree accurate. Since the

equations for deflection and stress are fourth order and second order respectively, the answer will

have no error because the Quadrature is accurate to the correct degree polynomial. Thus the one

dimensional method has zero percent error in deflection and stress.


Further Analysis

In addition to this baseline data, we can export both the deflection and Von-Mises data to Excel.

We will use the Y-deflection data as an example of how to do this.

1. Go to Utility Menu -> List -> Results -> Nodal Solution …

2. Select Nodal Solution -> DOF Solution -> Y-component of displacement

3. Click OK

4. The list file should populate. Go to

PRNSOL Command -> File -> Save As …

5. Save the file as 1D_P_YDeflection.lis to the

path of your choice

6. Go to PRNSOL Command -> File -> Close

7. Open 1D_P_YDeflection.lis in Excel

8. Click Fixed Width

9. Click Next >

10. Click a location on the ruler between the NODE and

UY columns. This will cause Excel to separate these

columns into separate columns in the spreadsheet

11. Click Next >

12. Click Finish

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Validation

module 1.1: point loading of a 1d cantilever beam

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