module 2: thermal stresses in a 1d beam fixed at both ends

UCONN ANSYS –Module 2

Module 2: Thermal Stresses in a 1D Beam Fixed at Both Ends

Table of Contents Page Number

Problem Description 2

Theory 2

Preprocessor 3

Scalar Parameters 3

Real Constants and Material Properties 4

Geometry 6

Meshing 7

Loads 8

Solution 9

General Postprocessor 9

Results 11

Validation 11


Problem Description

Nomenclature:

L =250mm Length of beam

D =25mm Diameter of beam

T =175 C Uniform temperature of beam

= 25 C Room Temperature

E =205GPa Young’s Modulus of ANSI 1030 Steel at Room Temperature

=0.30 Poisson’s Ratio of Steel

= Thermal Expansion (Secant) Coefficient of Steel

In this module we will study the thermal stresses resulting from an elevated temperature on a

round beam fixed at both ends. We will model the beam using one dimensional BEAM 4

elements and ANSI 1030, a low carbon steel. The theory for this analysis is shown below:

Theory

Thermal Stress

When most materials are heated, they tend to expand. ANSI is isotropic thus the expansion is

uniform in all directions. The non-dimensionalized form of this expansion is called thermal

strain which is given in the form:

(2.1)

Where is the temperature where the reference length of the beam is considered and is a

material property known as the Thermal Expansion Coefficient. This constant is called the

Secant Coefficient in Mechanical ANSYS APDL. For our beam, when the material tries to expand

in the x direction, the fixed supports provide reaction forces to keep the beam at the initial

length. These reaction forces result in a net compressive stress on the beam. Using some

definitions of axial stress, we can say:

(2.2)

Where E is young’s modulus.

y

T

L

D

x

𝑻𝒓𝒆𝒇


Substituting equation 2.1 into equation 2.2, we can derive:

= 359.7 MPa (2.3)

The yield strength of ANSI 1030 is 441 MPa, so the beam is stressed within the linear elastic

limits. If we want to design against buckling, we can check that the load applied from the fixed

supports doesn’t exceed the critical load for buckling. We will simplify this analysis for the

purposes of this tutorial. For more in depth analysis of buckling, see module 3.

Buckling Considerations

First, we must check to see if the beam is a Euler or Johnson column. The criteria are as follows:

(

) (

)

(2.4)

(

)

(2.5)

{

(

)

(

)

(2.6)

Where c = 1 in a conservative evaluation and

for a circular beam. Evaluating equation

2.6 we find that

(

) . Thus the beam is a Johnson Column. Now that we have classified the

beam, we must check to see that the critical buckling load ( is greater than the load applied

by the fixed ends. In a Johnson Column:

( (

)

(

)) = 19.7kN (2.7)

For axial stress:

(2.8)

where F is the axial load and

for a circular cross section. Thus, in order for no buckling

to occur,

(2.9)

Evaluating the right hand side of equation 2.6, we get 17.7kN. Thus, no buckling occurs.


Preprocessor

Scalar Parameters

First, we will declare some variables in ANSYS that will be

used throughout the remainder of the tutorial.

1. Go to Utility Menu -> Parameters ->

Scalar Parameters…

2. Under Selection type PI=acos(-1). ANSYS has the

capability of solving trigonometric functions. After

the statement has been written, press ENTER.

3. Repeat step two for the following statements:

D = 0.025

L = 0.25

The screen should look as shown.

4. Click Close

These variables are stored and can be accessed at any time.

Real Constants and Material Properties

Element Selection

We will be using BEAM 4 in this tutorial. For more information on BEAM 4, see module 1.1.

1. Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete

2. Click Add…

3. Select Library of Element Types -> Structural Mass -> Beam -> 3D Elastic 3

4. Click OK

5. Click Close

2

4


Real Constants

Now we will enter the cross sectional properties of the beam.

1. Go to Real Constants -> Add/Edit/Delete

2. Click Add

3. Click OK

4. Under AREA enter PI*D*D/4

5. Under IZZ type PI*D*D*D*D/64

6. Click OK

7. Click Close

Material Properties

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

2. Go to Structural -> Linear -> Elastic -> Isotropic

3. Under EX enter 205E9

4. Under PRXY enter 0.3

5. Click OK

6. Go to Structural -> Thermal Expansion ->

Secant Coefficient -> Isotropic

7. Under ALPX enter 11.7E-6

8. Click OK

9. Go to Define Material Model Behavior -> Material -> Exit

4

5

6

3

4

5

7

8


Geometry

Keypoints

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

Create -> Keypoints -> on Working Plane

2. Enter 0,0,0

3. Click Apply

4. Repeat steps 3 and 4 for L,0,0

5. Click OK

6. To get rid of the triad go to the Command Prompt and enter:

/triad,off

/replot

The resulting graphic should look as follows:

Line

1. Go to Main Menu -> Preprocessor -> Modeling -> Create -> Lines -> Straight Line

2. Enter 1,2 this connects a line from keypoint 1 to keypoint 2

3. Click OK

The resulting graphic should look as follows:

2

25


Meshing

As we will see later in the results section, linear thermal stresses problems are very accurate in

ANSYS. To get the point across, we will mesh the beam with two elements.

1. Go to Main Menu -> Preprocessor -> Meshing -> MeshTool

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

3. Under NDIV enter 2

4. Click OK

5. Click Mesh

6. Click Pick All

7. Click Close

8. Go to Utility Menu -> PlotCtrls -> Numbering …

9. Check NODE Node numbers

10. Click OK

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

Your mesh should look as follows:

2

2

3

24

2

5

26

2


Loads

Now we will constrain the ends of the beam and select a uniform temperature across the beam.

Fixed Ends

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

Structural -> Displacement -> On Nodes

2. Enter 1,2 and click OK

3. Under Lab2 DOFs to be constrained click ALL DOF

4. Under Value enter 0 and click OK

The resulting picture should look as shown below:

Uniform Temperature

1. The default reference temperature in Mechanical ANSYS APDL is 0. If we are working

with metric units, the reference temperature is in . If we are in British units, the

reference temperature is in . To change the reference temperature to room temperature,

go to the Command Prompt and enter MP,REFT,1,25 This sets the reference

temperature (REFT) material property (MP) on object 1 (the beam) to You are

instructing ANSYS that, at this reference temperature, the object experiences no thermal

strain.

2. Go to Main Menu -> Preprocessor -> Loads -> Define Loads -> Apply -> Structural

-> Temperature -> On Lines

3. Click Pick All

4. Under VAL1 enter 175

5. Click OK

6. If an error message appears,

ignore it. ANSYS considers

temperature to be a degree of

freedom at each node. Thus,

temperature definitions will be

translated to the nodes. This is a

more important consideration

when a non-uniform temperature

distribution is defined, as an

interpolation algorithm would be

applied across the nodes.

4

25

2


Solution

1. Go to Main Menu -> Solution

2. In the Command Prompt type solve and press ENTER in your keyboard.

3. Ignore the warning

4. Almost instantly, the problem will be solved. Click Close in the Note menu.

General Postprocessor

As in module 1, APDL has trouble graphing contour plots of stress, so we will access the

postprocessor results in the list files.

To check for buckling, we will first look at the displacement log file.

1. Go to Main Menu -> Postprocessor failure to do so will not allow access to the log files

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

Displacement vector sum

3. Click OK

As you can see from the chart, there are no displacements in the beam. Thus, no buckling has

occurred as expected.

Now we will check the forces applied at each node to get the stress distribution in the beam.

Using equation 2.8, we can find the axial stress across the beam.

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

All Available Force Items

5. Click OK


The log file should appear as follows:

As we can see, the reaction force is uniform across the beam as expected. Using Equation 2.8,

the axial stress at each location is 359.7 MPa.


Results

Axial Stress Error

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

(

) = 0% (2.10)

Due to quadrature, beam element functions are fourth order accurate. Since thermal stress

(equation 2.3) is a first order function, the stresses derived will be 100% accurate every time.

Validation

module 2: thermal stresses in a 1d beam fixed at both ends

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