Module 2: Thermal Stresses in a 1D Beam Fixed at Both Ends Table of Contents 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 Page Number UCONN ANSYS Module 2 Page 1
Problem Description T ref T D y L x 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: 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: Where E is young s modulus. (2.1) (2.2) UCONN ANSYS Module 2 Page 2
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. UCONN ANSYS Module 2 Page 3
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 4 2 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 UCONN ANSYS Module 2 Page 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 4 5 6 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 3 4 5 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 7 8 UCONN ANSYS Module 2 Page 5
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: 5 2 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: UCONN ANSYS Module 2 Page 6
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 2 4 8. Go to Utility Menu -> PlotCtrls -> Numbering 9. Check NODE Node numbers 10. Click OK 11. Go to Utility Menu -> Plot -> Nodes 3 5 6 Your mesh should look as follows: UCONN ANSYS Module 2 Page 7
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, 4 ignore it. ANSYS considers temperature to be a degree of 5 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. UCONN ANSYS Module 2 Page 8
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 UCONN ANSYS Module 2 Page 9
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. UCONN ANSYS Module 2 Page 10
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 UCONN ANSYS Module 2 Page 11