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Modeling a Simple 2D Cantilever Beam Using PLANE Elements
Exercise 4-2
Modeling a Simple 2D Cantilever Beam Using PLANE Elements
Overview This exercise is a continuation of the cantilever beam analysis that was started in exercise 4-1. In exercise 4-2, you will take advantage of symmetry in the beam cross section to create a 2D model using PLANE182 elements.
File Management Create a new folder in your “Intro-to-ANSYS” folder named “Exercise4-2” Open a new session of ANSYS using the Mechanical APDL Product Launcher Change the Working Directory to the new “Exercise4-2” folder Change the Jobname to “Exercise4-2” Click Run to start ANSYS
Step 1: Define Geometry 1-1.
Create an area to serve as the center plane of the beam
• • • • • • 1-2.
Preprocessor > Modeling > Create > Areas > Rectangle > By Dimensions Enter 0 for X1 Enter 1 for X2 Enter −0.05 for Y1 Enter 0.05 for Y2 Click OK
Save the model geometry
Step 2: Define Element Types 2-1.
Define the element type to use for this model
• Preprocessor > Element Type > Add/Edit/Delete • Choose PLANE182 as the element type for this analysis • Do not close the Element Types dialog box
ANSYS Mechanical APDL for Finite Element Analysis. DOI: http://dx.doi.org/10.1016/B978-0-12-812981-4.00019-8
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Copyright © 2017 Elsevier Inc. All rights reserved.
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ANSYS Mechanical APDL for Finite Element Analysis
2-2.
Specify plane stress behavior for your model
• • • • •
If you have closed the Element Types dialog box, reopen it Click the “Options. . .” button For “Element behavior K3” choose “Plane stress w/thk” (Figure 4-2-1) Click OK Close the Element Types dialog box
Figure 4-2-1 Element Options Dialog Box.
In general, thickness is not required for 2D analyses. There are two exceptions to this rule: (1) when force loads are applied and (2) when you want to plot the full 3D results using the element shapes option. As noted in exercise 2-1, pressure loads in ANSYS are applied per unit area. When pressure loads are applied to lines, ANSYS assumes a unit depth of 1 unless otherwise specified. Usually this default value is ok. (For example, it was used in exercise 2-1 with no problem.) However, if the default value for depth will not result in the correct value for the applied load, the model thickness must be specified using a real constant. When forces are applied, the magnitude of the force must be adjusted for the thickness. Element shapes allow you to visualize the full 3D results of a lower order model. You took advantage of this option in exercise 4-1 to visualize the results of the first cantilever beam. Element shapes do not affect the model results. They only affect postprocessing. The model in this exercise can be created with or without a specified thickness. The results will be the same assuming that the correct value of force is applied. However, without the thickness value, you will not be able to generate the 3D plot shown in Figure 4-2-5. 2-3.
Define the real constants for your model
• • • • • •
Preprocessor > Real Constants > Add/Edit/Delete Click the “Add. . .” button Click OK For “Thickness THK” enter 0.1 (Figure 4-2-2) Click OK Close the Real Constants dialog box
In this model, the beam thickness has not been specified via the solid model geometry. Thus, it must be defined using a real constant.
Exercise 4-2
133
Figure 4-2-2 Real Constant Dialog Box.
Step 3: Define Material Properties 3-1.
Create a linear elastic material model for 6061-T6 aluminum
• • • • 3-2.
Preprocessor > Material Props > Material Models Choose a structural, linear, elastic, isotropic material model Enter 7.310e10 as the value for Young’s modulus (EX) Enter 0.33 as the value for Poisson’s ratio (PRXY)
Save your progress
Step 4: Mesh 4-1.
Create the mesh for the finite element model
• • • • • • • •
Preprocessor > Meshing > MeshTool In the third area of the Mesh Tool, click the “Set” button associated with “Areas” Click “Pick All” Set the “SIZE Element edge length” to 0.0125 Click OK In the fourth area of the Mesh Tool, click the “Mesh” button Click “Pick All” Close the Mesh Tool
This procedure specifies that all elements in the model should be 0.0125x0.0125 m long. It results in a coarse mesh with 8 elements through the thickness of the beam and 80 elements along the length of the beam. 4-2.
Change to the isometric view
• Click the “Isometric View” button in the Pan Zoom Rotate menu The isometric view clearly shows that this is a 2D model and that the finite element mesh is only one element thick. 4-3.
Return to the front view
• Click the “Front View” button in the Pan Zoom Rotate menu
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Step 5: Apply Constraint Boundary Conditions 5-1.
Select the nodes at x = 0
• • • • • • • 5-2.
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Choose “By Location” as the selection method Ensure that “X coordinates” is set as the coordinate to select by Enter 0 in the “Min,Max” box Ensure that “From Full” is selected Click OK
Plot the selected nodes to confirm that the selection operation was successful
• Utility menu > Plot > Nodes The plot shows that a single vertical line of nodes has been selected. These nodes are generally indistinguishable from any other line of vertical nodes. However, the presence of the global triad (i.e., the origin) at the center of the line shows that this is the correct line of nodes. 5-3.
Constrain the fixed end of the beam in x
• • • • • 5-4.
Solution > Define Loads > Apply > Structural > Displacement > On Nodes Click “Pick All” For “Lab2 DOFs to be constrained” choose “UX” For “VALUE Displacement value” enter 0 Click OK
Select the node at (0,0)
• • • • • • •
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Ensure that “By Location” is the selection method Change the coordinate to select by to “Y coordinates” Enter 0 in the “Min,Max” box Change “From Full” to “Reselect” Click OK
Reselecting entities from the currently selected set creates a new subset. Selecting “From Full” creates a new set from the set of all entities defined in the model. “Also Select” adds to the current set or creates a union between the old set and the new set. “Unselect” removes entities from the current set. See chapter 7 for more details. 5-5.
List the nodes to ensure that the correct node was selected
• Utility menu > List > Nodes. . . • Click OK • Close the NLIST window when you are finished reviewing the nodes You can confirm that the correct node was selected according to its (x,y,z) coordinates.
Exercise 4-2
5-6.
135
Constrain the fixed end of the beam in y
• • • • • •
Solution > Define Loads > Apply > Structural > Displacement > On Nodes Click “Pick All” For “Lab2 DOFs to be constrained” unselect “UX” if it is still selected For “Lab2 DOFs to be constrained” choose “UY” For “VALUE Displacement value” enter 0 Click OK
Constraining only a single point in y allows the beam to expand and contract in the y-direction and thus does not overconstrain the model. 5-7.
Save your constraints
Step 6: Apply Load Boundary Conditions 6-1.
Select the node at (1,0)
• • • • • • •
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Ensure that “By Location” is the selection method Change the coordinate to select by to “X coordinates” Enter 1 in the “Min,Max” box Change “Reselect” to “From Full” Click Apply
This selects all of the nodes at x = 1. • • • • 6-2.
Change the coordinate to select by to “Y coordinates” Enter 0 in the “Min,Max” box Change “From Full” to “Reselect” Click OK
List the nodes to ensure that the correct node was selected
• Utility Menu > List > Nodes. . . • Click OK • Close the NLIST window when you are finished reviewing the nodes 6-3.
Apply a downward load to the center of the free end of the beam
• • • • • 6-4.
Solution > Define Loads > Apply > Structural > Force/Moment > On Nodes Click “Pick All” For “Lab Direction of force/mom” choose “FY” For “VALUE Force/moment value” enter −5000 Click OK
List the forces to ensure that the load was correctly applied
• Utility Menu > List > Loads > Forces > On All Nodes • Close the FLIST window when you are finished reviewing the loads 6-5.
Save your loads
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Step 7: Set the Solution Options The default solution options can be used for this analysis.
Step 8: Solve 8-1.
Select everything in the model
• Utility Menu > Select > Everything 8-2.
Solve
• Solution > Solve > Current LS 8-3.
Save your results
Step 9: Postprocess the Results 9-1.
Plot the vertical deformation of the beam
• • • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “DOF Solution” Choose “Y-Component of displacement” Change the “Scale Factor” to “True Scale” Click OK
Figure 4-2-3 Plot of Displacement in Y: Win32 Graphics (top) and 3D Graphics (bottom).
9-2.
Plot the x component of stress in the beam
• • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “Stress” Scroll down and choose “X-Component of stress” Click OK
Exercise 4-2
137
Figure 4-2-4 Plot of X-Component of Stress: Win32 Graphics (top) and 3D Graphics (bottom).
9-3.
Change to the isometric view
• Click the “Isometric View” button in the Pan Zoom Rotate menu 9-4.
Turn element shape display on
• Utility Menu > PlotCtrls > Style > Size and Shape. . . • Turn “[/ESHAPE] Display of element shapes based on real constant descriptions” on • Click OK 9-5.
Plot the equivalent stress in the beam
• • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “Stress” Scroll down and choose “von Mises stress” Click OK
Figure 4-2-5 Plot of von Mises Stress (Element Shapes On): Win32 Graphics (left) and 3D Graphics (right).
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Step 10: Compare and Verify the Results The displacement plot in Figure 4-2-3 shows zero displacement at the fixed end of the beam and a maximum displacement of 2.739E-3 m at the free end of the beam. This is within 1% of the result from exercise 4-1 (2.758E-3 m) and very similar to the value predicted by beam theory (2.736E-3 m). The plot of the longitudinal stress in Figure 4-2-4 shows that the maximum stress occurs at the fixed end of the beam and has a value of 30.3 MPa. This is 1% higher than the maximum stress predicted by the model in exercise 4-1 and by beam theory. The plot of the equivalent (von Mises) stress in Figure 4-2-5 is 29.8 MPa. This is 1% lower than the maximum equivalent stress predicted by the model in exercise 4-1 and by beam theory. Based on these comparisons, we can conclude that the model is in excellent agreement with the theory and can be used for engineering design and analysis. We can also conclude that there are no major differences between the solutions of the 1D and 2D models.
Close the Program • Utility Menu > File > Exit. . .
Sample Input File /PREP7 RECT,0,1,-0.05,0.05 ET,1,PLANE182 KEYOPT,1,3,3 R,1,0.1, MP,EX,1,7.31e10 MP,PRXY,1,0.33 ESIZE,0.0125 AMESH,ALL FINISH
! Enter the Preprocessor ! Create a rectangle from x(0:1), y(-0.5:0.5) ! Use PLANE183 Elements ! Use plane stress w/thickness ! Set plane stress thickness to 0.1 m ! Define Young's modulus for material #1 ! Define Poisson's ratio for material #1 ! Set element size to 0.0125 ! Mesh all areas ! Finish and Exit Preprocessor
/SOLU NSEL,S,LOC,X,0 D,ALL,UX,0 NSEL,R,LOC,Y,0 D,ALL,UY,0 NSEL,S,LOC,X,1 NSEL,R,LOC,Y,0 F,ALL,FY,-5000 ALLSEL SOLVE FINISH
! Enter the Solution Processor ! Select all nodes at x = 0 ! Constrain those nodes in x ! Reselect the node at y = 0 ! Constrain that node in y ! Select all nodes at x = 1 ! Reselect the node at y = 0 ! Apply a downward load of -5000 N ! Select everything ! Solve the model ! Finish and Exit Solution
/POST1 /DSCALE,ALL,1 /VIEW,1,,,1 PLNSOL,U,Y,0,1 PLNSOL,S,X,0,1 /VIEW,1,1,1,1 /ESHAPE,1 PLNSOL,S,EQV,0,1 FINISH SAVE !/EXIT
! Enter the General Postprocessor ! Plot using true scale ! Change to front view ! Plot displacement in y ! Plot stress in x ! Change to isometric view ! Display element shapes using section data ! Plot the equivalent stress ! Finish and Exit Postprocessor ! Save the database ! Exit ANSYS
Modeling a Simple 2D Cantilever Beam Using PLANE Elements
Overview This exercise is a continuation of the cantilever beam analysis that was started in exercise 4-1. In exercise 4-2, you will take advantage of symmetry in the beam cross section to create a 2D model using PLANE182 elements.
File Management Create a new folder in your “Intro-to-ANSYS” folder named “Exercise4-2” Open a new session of ANSYS using the Mechanical APDL Product Launcher Change the Working Directory to the new “Exercise4-2” folder Change the Jobname to “Exercise4-2” Click Run to start ANSYS
Step 1: Define Geometry 1-1.
Create an area to serve as the center plane of the beam
• • • • • • 1-2.
Preprocessor > Modeling > Create > Areas > Rectangle > By Dimensions Enter 0 for X1 Enter 1 for X2 Enter −0.05 for Y1 Enter 0.05 for Y2 Click OK
Save the model geometry
Step 2: Define Element Types 2-1.
Define the element type to use for this model
• Preprocessor > Element Type > Add/Edit/Delete • Choose PLANE182 as the element type for this analysis • Do not close the Element Types dialog box
ANSYS Mechanical APDL for Finite Element Analysis. DOI: http://dx.doi.org/10.1016/B978-0-12-812981-4.00019-8
131
Copyright © 2017 Elsevier Inc. All rights reserved.
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ANSYS Mechanical APDL for Finite Element Analysis
2-2.
Specify plane stress behavior for your model
• • • • •
If you have closed the Element Types dialog box, reopen it Click the “Options. . .” button For “Element behavior K3” choose “Plane stress w/thk” (Figure 4-2-1) Click OK Close the Element Types dialog box
Figure 4-2-1 Element Options Dialog Box.
In general, thickness is not required for 2D analyses. There are two exceptions to this rule: (1) when force loads are applied and (2) when you want to plot the full 3D results using the element shapes option. As noted in exercise 2-1, pressure loads in ANSYS are applied per unit area. When pressure loads are applied to lines, ANSYS assumes a unit depth of 1 unless otherwise specified. Usually this default value is ok. (For example, it was used in exercise 2-1 with no problem.) However, if the default value for depth will not result in the correct value for the applied load, the model thickness must be specified using a real constant. When forces are applied, the magnitude of the force must be adjusted for the thickness. Element shapes allow you to visualize the full 3D results of a lower order model. You took advantage of this option in exercise 4-1 to visualize the results of the first cantilever beam. Element shapes do not affect the model results. They only affect postprocessing. The model in this exercise can be created with or without a specified thickness. The results will be the same assuming that the correct value of force is applied. However, without the thickness value, you will not be able to generate the 3D plot shown in Figure 4-2-5. 2-3.
Define the real constants for your model
• • • • • •
Preprocessor > Real Constants > Add/Edit/Delete Click the “Add. . .” button Click OK For “Thickness THK” enter 0.1 (Figure 4-2-2) Click OK Close the Real Constants dialog box
In this model, the beam thickness has not been specified via the solid model geometry. Thus, it must be defined using a real constant.
Exercise 4-2
133
Figure 4-2-2 Real Constant Dialog Box.
Step 3: Define Material Properties 3-1.
Create a linear elastic material model for 6061-T6 aluminum
• • • • 3-2.
Preprocessor > Material Props > Material Models Choose a structural, linear, elastic, isotropic material model Enter 7.310e10 as the value for Young’s modulus (EX) Enter 0.33 as the value for Poisson’s ratio (PRXY)
Save your progress
Step 4: Mesh 4-1.
Create the mesh for the finite element model
• • • • • • • •
Preprocessor > Meshing > MeshTool In the third area of the Mesh Tool, click the “Set” button associated with “Areas” Click “Pick All” Set the “SIZE Element edge length” to 0.0125 Click OK In the fourth area of the Mesh Tool, click the “Mesh” button Click “Pick All” Close the Mesh Tool
This procedure specifies that all elements in the model should be 0.0125x0.0125 m long. It results in a coarse mesh with 8 elements through the thickness of the beam and 80 elements along the length of the beam. 4-2.
Change to the isometric view
• Click the “Isometric View” button in the Pan Zoom Rotate menu The isometric view clearly shows that this is a 2D model and that the finite element mesh is only one element thick. 4-3.
Return to the front view
• Click the “Front View” button in the Pan Zoom Rotate menu
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ANSYS Mechanical APDL for Finite Element Analysis
Step 5: Apply Constraint Boundary Conditions 5-1.
Select the nodes at x = 0
• • • • • • • 5-2.
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Choose “By Location” as the selection method Ensure that “X coordinates” is set as the coordinate to select by Enter 0 in the “Min,Max” box Ensure that “From Full” is selected Click OK
Plot the selected nodes to confirm that the selection operation was successful
• Utility menu > Plot > Nodes The plot shows that a single vertical line of nodes has been selected. These nodes are generally indistinguishable from any other line of vertical nodes. However, the presence of the global triad (i.e., the origin) at the center of the line shows that this is the correct line of nodes. 5-3.
Constrain the fixed end of the beam in x
• • • • • 5-4.
Solution > Define Loads > Apply > Structural > Displacement > On Nodes Click “Pick All” For “Lab2 DOFs to be constrained” choose “UX” For “VALUE Displacement value” enter 0 Click OK
Select the node at (0,0)
• • • • • • •
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Ensure that “By Location” is the selection method Change the coordinate to select by to “Y coordinates” Enter 0 in the “Min,Max” box Change “From Full” to “Reselect” Click OK
Reselecting entities from the currently selected set creates a new subset. Selecting “From Full” creates a new set from the set of all entities defined in the model. “Also Select” adds to the current set or creates a union between the old set and the new set. “Unselect” removes entities from the current set. See chapter 7 for more details. 5-5.
List the nodes to ensure that the correct node was selected
• Utility menu > List > Nodes. . . • Click OK • Close the NLIST window when you are finished reviewing the nodes You can confirm that the correct node was selected according to its (x,y,z) coordinates.
Exercise 4-2
5-6.
135
Constrain the fixed end of the beam in y
• • • • • •
Solution > Define Loads > Apply > Structural > Displacement > On Nodes Click “Pick All” For “Lab2 DOFs to be constrained” unselect “UX” if it is still selected For “Lab2 DOFs to be constrained” choose “UY” For “VALUE Displacement value” enter 0 Click OK
Constraining only a single point in y allows the beam to expand and contract in the y-direction and thus does not overconstrain the model. 5-7.
Save your constraints
Step 6: Apply Load Boundary Conditions 6-1.
Select the node at (1,0)
• • • • • • •
Utility menu > Select > Entities. . . Ensure that “Nodes” are the entity to be selected Ensure that “By Location” is the selection method Change the coordinate to select by to “X coordinates” Enter 1 in the “Min,Max” box Change “Reselect” to “From Full” Click Apply
This selects all of the nodes at x = 1. • • • • 6-2.
Change the coordinate to select by to “Y coordinates” Enter 0 in the “Min,Max” box Change “From Full” to “Reselect” Click OK
List the nodes to ensure that the correct node was selected
• Utility Menu > List > Nodes. . . • Click OK • Close the NLIST window when you are finished reviewing the nodes 6-3.
Apply a downward load to the center of the free end of the beam
• • • • • 6-4.
Solution > Define Loads > Apply > Structural > Force/Moment > On Nodes Click “Pick All” For “Lab Direction of force/mom” choose “FY” For “VALUE Force/moment value” enter −5000 Click OK
List the forces to ensure that the load was correctly applied
• Utility Menu > List > Loads > Forces > On All Nodes • Close the FLIST window when you are finished reviewing the loads 6-5.
Save your loads
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ANSYS Mechanical APDL for Finite Element Analysis
Step 7: Set the Solution Options The default solution options can be used for this analysis.
Step 8: Solve 8-1.
Select everything in the model
• Utility Menu > Select > Everything 8-2.
Solve
• Solution > Solve > Current LS 8-3.
Save your results
Step 9: Postprocess the Results 9-1.
Plot the vertical deformation of the beam
• • • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “DOF Solution” Choose “Y-Component of displacement” Change the “Scale Factor” to “True Scale” Click OK
Figure 4-2-3 Plot of Displacement in Y: Win32 Graphics (top) and 3D Graphics (bottom).
9-2.
Plot the x component of stress in the beam
• • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “Stress” Scroll down and choose “X-Component of stress” Click OK
Exercise 4-2
137
Figure 4-2-4 Plot of X-Component of Stress: Win32 Graphics (top) and 3D Graphics (bottom).
9-3.
Change to the isometric view
• Click the “Isometric View” button in the Pan Zoom Rotate menu 9-4.
Turn element shape display on
• Utility Menu > PlotCtrls > Style > Size and Shape. . . • Turn “[/ESHAPE] Display of element shapes based on real constant descriptions” on • Click OK 9-5.
Plot the equivalent stress in the beam
• • • •
General Postproc > Plot Results > Contour Plot > Nodal Solu Choose “Stress” Scroll down and choose “von Mises stress” Click OK
Figure 4-2-5 Plot of von Mises Stress (Element Shapes On): Win32 Graphics (left) and 3D Graphics (right).
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ANSYS Mechanical APDL for Finite Element Analysis
Step 10: Compare and Verify the Results The displacement plot in Figure 4-2-3 shows zero displacement at the fixed end of the beam and a maximum displacement of 2.739E-3 m at the free end of the beam. This is within 1% of the result from exercise 4-1 (2.758E-3 m) and very similar to the value predicted by beam theory (2.736E-3 m). The plot of the longitudinal stress in Figure 4-2-4 shows that the maximum stress occurs at the fixed end of the beam and has a value of 30.3 MPa. This is 1% higher than the maximum stress predicted by the model in exercise 4-1 and by beam theory. The plot of the equivalent (von Mises) stress in Figure 4-2-5 is 29.8 MPa. This is 1% lower than the maximum equivalent stress predicted by the model in exercise 4-1 and by beam theory. Based on these comparisons, we can conclude that the model is in excellent agreement with the theory and can be used for engineering design and analysis. We can also conclude that there are no major differences between the solutions of the 1D and 2D models.
Close the Program • Utility Menu > File > Exit. . .
Sample Input File /PREP7 RECT,0,1,-0.05,0.05 ET,1,PLANE182 KEYOPT,1,3,3 R,1,0.1, MP,EX,1,7.31e10 MP,PRXY,1,0.33 ESIZE,0.0125 AMESH,ALL FINISH
! Enter the Preprocessor ! Create a rectangle from x(0:1), y(-0.5:0.5) ! Use PLANE183 Elements ! Use plane stress w/thickness ! Set plane stress thickness to 0.1 m ! Define Young's modulus for material #1 ! Define Poisson's ratio for material #1 ! Set element size to 0.0125 ! Mesh all areas ! Finish and Exit Preprocessor
/SOLU NSEL,S,LOC,X,0 D,ALL,UX,0 NSEL,R,LOC,Y,0 D,ALL,UY,0 NSEL,S,LOC,X,1 NSEL,R,LOC,Y,0 F,ALL,FY,-5000 ALLSEL SOLVE FINISH
! Enter the Solution Processor ! Select all nodes at x = 0 ! Constrain those nodes in x ! Reselect the node at y = 0 ! Constrain that node in y ! Select all nodes at x = 1 ! Reselect the node at y = 0 ! Apply a downward load of -5000 N ! Select everything ! Solve the model ! Finish and Exit Solution
/POST1 /DSCALE,ALL,1 /VIEW,1,,,1 PLNSOL,U,Y,0,1 PLNSOL,S,X,0,1 /VIEW,1,1,1,1 /ESHAPE,1 PLNSOL,S,EQV,0,1 FINISH SAVE !/EXIT
! Enter the General Postprocessor ! Plot using true scale ! Change to front view ! Plot displacement in y ! Plot stress in x ! Change to isometric view ! Display element shapes using section data ! Plot the equivalent stress ! Finish and Exit Postprocessor ! Save the database ! Exit ANSYS