UCONN ANSYS –Module 1.3 Page 1

Module 1.3: Distributed Loading of a 1D Cantilever Beam

Table of Contents Page Number

Problem Description 2

Theory 2

Geometry 4

Preprocessor 6

Loads

Solution 8

General Postprocessor 8

Results 11

Validation 12

1

1 33 44 55 66 77 88 99 1010 1111 2MN

MX

Y-Component of Deflection of a Cantilever Beam with Distributed Load

-.006275-.005578

-.00488-.004183

-.003486-.002789

-.002092-.001394

-.697E-030

NOV 13 2011

23:22:22

NODAL SOLUTION

STEP=1

SUB =1

TIME=1

UY (AVG)

RSYS=0

DMX =.006275

SMN =-.006275

UCONN ANSYS –Module 1.3 Page 2

Problem Description

Nomenclature:

L =110m Length of beam

b =10m Cross Section Base

h =1 m Cross Section Height

w=20N/m Distributed Load

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

=0.33 Poisson’s Ratio of Aluminum

In this module, we cover the static distributed loading of a cantilever beam using one

dimensional elements in ANSYS Mechanical APDL. The left side of the cantilever beam is fixed

while there is a distributed load of 20N/m across the top surface of the beam. 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:

(1.3.1).

Additionally, since the nodes of choice are located at the top surface of the beam, the shear stress

at this location is zero.

( . (1.3.2)

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

(1.3.3)

Bending Stress is given by:

(1.3.4)

Where

and

.

y

x

UCONN ANSYS –Module 1.3 Page 3

From statics, we can derive:

∑

(1.3.5)

Plugging into equation 1.3.4, we get:

(1.3.6)

= 72.6kPa (1.3.7)

Beam Deflection

As in module 1.1, the equation to be solved is:

(1.3.8)

Plugging in equation 1.3.5, we get:

(1.3.9)

Integrating once to get angular displacement we get:

(1.3.10)

At the fixed end (x=0),

, thus

(1.3.11)

Integrating again to get deflection:

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

(1.3.12)

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

(1.3.13)

UCONN ANSYS –Module 1.3 Page 4

Geometry

1. Go to Utility Menu -> Open ANSYS File and open 1D Cantilever.db that you

created in module 1.1. To refresh your memory, this was a 1x10x110m Aluminum

cantilever beam with two BEAM 4 elements through the thickness. The file that opens

should look as follows:

Mesh

We are going to re-mesh this beam to have ten elements through the length of the beam.

1. Go to Main Menu -> Preprocessor -> Meshing -> Mesh Tool ->

Size Controls : -> Global -> Set

2. Set Size Element Edge Length to 11. This will map 10 elements

through the length of the beam.

3. Click OK

4. Select Mesh: -> Lines

5. Click Mesh

6. Click Pick All

An error message should appear:

2

3

4

5

6

UCONN ANSYS –Module 1.3 Page 5

Unlike remeshing areas and solids, line meshes must be cleared manually before remeshing is

allowed.

7. Go to Main Menu -> Preprocessor -> Meshing -> Clear -> Lines

8. Click Pick All

9. Repeat steps 1-6 to update the mesh.

10. Go to Utility Menu -> Command Prompt and enter /triad, off

followed by /replot

If you go to Utility Menu -> Plot -> Nodes, use the Fit View followed by the Front

View and scroll in, the resulting mesh should look as follows:

As stated in previous tutorials, ANSYS numbers nodes from the left extreme to the right extreme

and then numbers from left to right.

7

UCONN ANSYS –Module 1.3 Page 6

Preprocessor

Loads

Displacement

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

Define Loads ->Apply ->Structural -> Displacement -> On Nodes

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

3. Click OK

4. Under Lab 2 DOFs to be constrained select All DOF

5. Under Value Displacement value enter 0

6. Click OK

The resulting graphic should look as shown below:

2

3

4

5

6

UCONN ANSYS –Module 1.3 Page 7

Line Pressure

Since our beam was made using 1D elements, we will use a line pressure as our distributed load.

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

Apply -> Structural -> On Beams

2. Click Pick All

3. Under LKEY, we are asked to specify a load key. A load key defines

the orientation of element loads. Enter 2.

The Load Keys for BEAM4 are as follows:

Face 1: -z normal

Face 2: -y normal

Face 3: +x tangential

Face 4: +x normal

Face 5: -x axial

A negative face value provides a loading in the opposite direction

For more information on Load Keys, search SFBEAM (the function being called)

In Utility Menu -> Help -> Help Topics

4. Under VALI enter 20. This enters a

uniform pressure of 20 N/m across

the length of the beam.

5. Click OK

The resulting picture should be as shown:

2

3

4

5

UCONN ANSYS –Module 1.3 Page 8

Solution

Now that our boundary conditions have been specified, it’s time to solve the problem.

1. Go to Main Menu -> Solution -> Solve -> Current LS

2. Click OK

General Postprocessor

Now that ANSYS has solved the load step, let’s create plots of Y Deflection and Equivalent Von

Mises Stress.

Y Component of Displacement

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

Solu

2. Go to Nodal Solution -> 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,Y-Component of Deflection of a Cantilever Beam with Distributed Load

followed by the return key and the command /replot

The resulting graph should look as shown below:

According to the list values, the max deflection is 6.27 mm

UCONN ANSYS –Module 1.3 Page 9

Equivalent (Von-Mises) Stress

Unfortunately, we cannot create a contour plot of Von-Mises stress for 1D elements unless more

complicated loading conditions are applied. We can, however, look up the moment reactions at

each element. If we plug this value into equation 1.3.4, 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 equivalent stress in the beam is 72.6 KPa, as expected.

UCONN ANSYS –Module 1.3 Page 10

This data can be exported to Excel for express computation of equivalent stress at all nodes.

4. Go to PRESOL Command -> File -> Save As …

5. Save the file as 1D_D_Equivalent Stress.lis to the

path of your choice

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

7. Open 1D_D_Equivalent Stress.lis in Excel

8. Click Fixed Width

9. Click Next >

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

1 Fx columns. This will cause Excel to separate these

columns into separate columns in the spreadsheet

11. Repeat for all columns of data to separate them.

12. Click Next >

13. Click Finish

This process can be repeated for Deflection data.

4

5

6

8

9

10

12

UCONN ANSYS –Module 1.3 Page 11

Results

Max Deflection Error

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

(

) = 0% (1.3.14)

This is a decent error baseline for the mesh considering equation 1.3.12 is quartic with respect to

displacement. Since the 1D Elements we are using are cubic functions between nodes, we can

expect a degree of truncation error in our model. As we will show in our validation section, our

model will converge to the expected solution as the mesh is refined.

Max Equivalent Stress Error

Using the same definition of error as before, we derive that our model has 0% error in the max

equivalent stress. Since we are using third order accurate beam elements and the stress

relationship is quadratic with respect to the length of the beam, there is no truncation error

regardless the number of elements used.

UCONN ANSYS –Module 1.3 Page 12