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September 15, 2017 | Author: Chakravarthy Svsn | Category: Finite Element Method, Stress (Mechanics), Bending, Calculus Of Variations, Materials Science
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ANSYS 12 - Tensile Bar - Problem Specification Problem Specification A steel bar is mounted in a rigid wall and axially loaded at the end by a force P = 2 kN as shown in the figure below. The bar dimensions are indicated in the figure. The bar is so thin that there is no significant stress variation through the thickness. Neglect gravity.

In this exercise, you are presented with the numerical solution to the above problem obtained from finite-element analysis (FEA) using ANSYS software. Compare FEA results for the stress distribution presented to you with the corresponding analytical solution. Justify agreements and discrepancies between the two approaches (FEA vs. Analytical). Note that you will be using the ANSYS solution presented to you to explore the physics of the problem. You will be downloading the ANSYS solution prepared for you. The objective is to help you learn important fundamentals of mechanics through the interactive, visual interface provided by ANSYS. You will not be obtaining the FEA solution using ANSYS;

ANSYS 12 - Tensile Bar - Pre-analysis and Start-up

Pre-Analysis and Start-Up Since we don't expect significant variation of stresses in the z direction, it is reasonable to assume plane stress: The deformed structure will be in equilibrium. Thus, the 2D stress components should satisfy the 2D equilibrium equations:

We need to solve these equations in our rectangular domain and impose the appropriate boundary conditions: imposed displacement constraints at the left end and applied force at the right end. In effect, we have to solve a boundary value problem (BVP). Recall that the elements of a BVP are: 

Governing differential equations



Domain



Boundary conditions

You probably have solved simple BVPs before in your math classes. We will first review the analytical approach to solving this BVP. We'll then look at the FEA approach.

Analytical Solution Since we are ignoring the effects of gravity; there are no body forces per unit volume. Since the length is much larger than the width, we ignore end effects and neglect variations in the y direction. Plugging and chugging into the equilibrium equations yields Then the equilibrium equation in the x-direction becomes: Therefore,

Apply Boundary Conditions: If we make a vertical cut in the geometry, then the stress must be P/A. Therefore, This is of course a well-known result.

Numerical Solution using FEA In the numerical solution using FEA, we solve the 2D BVP directly by dividing the structure into small elements and approximating the solution for these small elements. Unlike the analytical approach, we do not assume that there is no variation in the y direction. Also, end effects are not neglected. The FEA solution is an approximate solution to the 2D BVP. The approximation gets better as the elements become smaller. In contrast, the analytical solution presented above is the exact solution to the 1D BVP obtained by making approximations to the 2D BVP. In other words, in the analytic solution, we have swapped the actual 2D BVP problem for a 1D BVP problem that we can solve in closed form. Both approaches have value in engineering and complement each other. We have checked that the FEA solution presented to you is reasonably accurate

te. The following figure summarizes the contrasts between the analytical and numerical approaches. Load FEA Solution obtained using ANSYS

As mentioned before, we are providing the FEA solution obtained using ANSYS so that you can focus on comparing the analytical and numerical solutions (which is the goal of this exercise). Without further ado, let's download the ANSYS solution and load it into ANSYS. 1. Download "Tensile Bar Demo.zip" by clicking here The zip should contain a Tensile Bar Demo folder with the following contents: - Tensile Bar Demo_files folder - Tensile Bar Demo.wbpj Please make sure both these are in the folder, otherwise the solution will not load into ANSYS properly. (Note: The solution provided was created using ANSYS workbench 13.0 release, there may be compatibility issues when attempting to open with other versions). Be sure to extract before use. 2. Double click "Tensile Bar Demo.wbpj" - This should automatically open ANSYS Workbench (you have to twiddle your thumbs a bit before it opens up). You will then be presented with the ANSYS solution in the project page.

A tick mark against each step indicates that that step has been completed. 3. To look at the results, double click on Results - This should bring up a new window (again you have to twiddle your thumbs a bit before it opens up). 4. On the left-hand side there should be an Outline toolbar. Look for Solution (A6).

We'll investigate the items listed under Solution in the next step of this tutorial.

ANSYS 12 - Tensile Bar - Results

Results Before we explore the ANSYS results, let's take a peek at the mesh.

Mesh Click on Mesh (above Solution) in the tree outline. This shows the mesh used to generate the ANSYS solution. The domain is a rectangle. This domain is discretized into a number of small "elements". For each element, ANSYS approximates how the structure responds to the forces acting on the element. A finer mesh is used in areas of greater stress concentration. We have checked that the solution presented to you is reasonably independent of the mesh.

Displacement To view the deformed structure, click on Solution > Displacement in the tree outline. The black rectangle shows the undeformed structure. The deformed structure is colored by the magnitude of the displacement. Red areas have deformed more and blue areas less. You can see that the left end has not moved as specified in the problem statement. This means this boundary condition has been applied correctly. The displacement increases from left to right as we intuitively expect. There is also not much variation in the y-direction. Note the extremely high deformation near the point load. This extremum is unrealistic and should be ignored (there are no point loads in reality).

sigma_x Next, let's take a look at the stress components starting with sigma_x. Click on Solution > sigma_x in the tree outline. The stress is uniform away from the ends. To check what the value is in the uniform region, click on Probe in the toolbar (see snapshot below) at the top and move the cursor on the structure; Probe values in the middle as well as at the ends.

The value of sigma_x away from the ends is nearly 200 MPa (the unit is indicated above the plot). This matches with the P/A value expected from the analytical solution of the equilibrium equations.

In the sigma_x plot, we see that there is deviation from the analytical value in two regions: 

Around the point load (again the extremely high values very close to the point load are unrealistic).



At the fixed end.

The analytical solution breaks down in these regions. In fact, as the mesh is refined further, the stress at the point load will approach infinity.

sigma_y Next, let's take a look at sigma_y. Click on Solution > sigma_y in the tree outline. Again, probe values in the middle as well as at the ends. The value in the middle is close to zero as expected from the analytical solution. There is significant deviation from the analytical solution at both ends. The analytical method assumed a long bar, therefore by definition, the stresses in the y direction are assumed to be zero. Since this bar does have width, the stresses in the y direction are symmetrical about the middle axis. Note that there are areas where sigma_y is negative. This is a consequence of deformation along the x-axis. Since no extra material is being added, stretching the bar in the x direction would cause a contraction of the bar in the y-direction, and therefore compressive stresses in the y direction.

tau_xy We expect tau_xy to be zero in the middle. Near the ends, since sigma_x and sigma_y are non-zero, we expect

Plot tau_xy, look at the range of values and use Probe to check actual values. Are the above statements valid?

Equivalent Stress (Von Mises): The Equivalent or Von Mises stress is used to predict yielding of the material. We can consider the maximum and minimum equivalent stresses as the critical design points. We can see that the analytical solution under-predicts the maximum equivalent stress. Thus, one would need to use a large factor of safety if using the analytical result while designing such a structure. One would use a factor of safety with the FEA result also but it does not have to be as large.

ANSYS 12 - Tensile Bar - Homework Exercise

Homework Exercise A steel plate of width, w =0.25m , of length L =1m , and of specific weight ρ=7.9x10³kg/m³ is hung vertically from the ceiling. A force, F =100N is applied as shown in the figure below. The thickness is

1mm. The plate is so thin that there is no significant stress variation through the thickness.

You can download our ANSYS FEA solution to this problem by clicking here. Load this into ANSYS Workbench as in the tutorial. Please determine if this ANSYS solution is correct. If the solution is not right, use paper-and-pencil calculations to predict what the right values should look like. Justify your answer through quantitative reasoning.

Plate with a Hole Tutorial - Problem Specification Consider the classic example of a circular hole in a rectangular plate of constant thickness with a symmetric loading.

In this exercise, you are presented with the numerical solution to the above problem obtained from finite-element analysis (FEA) using ANSYS software.Compare FEA results for the stress distribution presented to you with the corresponding analytical solution. Justify agreements and discrepancies between the two approaches (FEA vs. Analytical). Note that you will be using the ANSYS solution presented to you to explore the physics of the problem. You will be downloading the ANSYS solution prepared for you. The objective is to help you learn important fundamentals of mechanics through the interactive, visual interface provided by ANSYS. You will not be obtaining the FEA solution using ANSYS;

Plate with a Hole Tutorial - Pre-Analysis and Start-Up Pre-Analysis Calculations ANSYS Simulation Now, let's load the problem into ANSYS and see how a computer simulation will compare. First, start by downloading the files here The zip file should contain the following contents: 

Plate With a Hole_files folder



Plate With a Hole.wbpj

Please make sure to extract both of these files from the zip folder, the program will not work otherwise. (Note: The solution was created using ANSYS workbench 12.1 release, there may be compatibility issues when attempting to open with other versions). 2. Double click "Plate With a Hole.wbpj" - This should automatically open ANSYS workbench (you have to twiddle your thumbs a bit before it opens up). You will be presented with the ANSYS solution.

A tick mark against each step indicates that that step has been completed. 3. To look at the results, double click on "Results" - This should bring up a new window (again you have to twiddle your thumbs a bit before it opens up).

4. On the left-hand side there should be an "Outline" toolbar. Look for "Solution (A6)".

We'll investigate the items listed under Solution in the next step in this tutorial.

Results Displacement

Okay! Now we can check our solution. Let's start by examining how the plate deformed under the load. Before you start, make sure the software is working in the same units you are by looking to the menu bar and selecting Units > US Customary (in, lbm, lbf, F, s, V,

A). Now, look at the Outline window, and select Solution > Total Deformation.

The colored section refers to the magnitude of the deformation (in inches) while the black outline is the undeformed geometry superimposed over the deformed model. The more red a section is, the more it has deformed while the more blue a section is, the less it has deformed. Notice that the deformation is at its highest where the load is applied, and there is no a lot of variation in the y-direction, as one intuitively expect.

Sigma-x Now lets examine the stress in the x-direction. Look to the Outline window, then click

Solution > Sigma-x

From this, you can see that most of the plate is in constant stress, and there is a stress concentration around the hole. The more red areas correspond to a high, tensile (positive) stress and the bluer areas correspond to areas of compressive (negative) stress. Let's use the probe tool to compare the ANSYS simulation to what we expected from calculation. In the

menu bar, click the Probe button; this will allow you to hover over the model and it will display the stress at each point. Start by hovering over the area far from the hole. The stress is about 1e6 psi, which is the value we would expect for a plate with a constant traction. If you click the max tag (located next to the probe tool in the menu bar), it will locate and display the maximum stress, which is shown as 3.0335e6 psi. This is about a 0.0055% difference from the calculation we did in the Pre-Analysis, which shows that the FEA is very, very accurate!

Sigma-r Now let's look at the radial stresses in the plate. Look to the outline window and click

Solution > Sigma-r. This will display the radial stresses.

Does this match what we expect? First, let's examine the hole at r = a. From our precalculations, we found that the stress at the hole in the radial direction should be 0. Using our probe tool, we find that the stress in this area ranges from -450 to 450 psi. Although the simulation does not approach exactly zero, keep in mind that 450 psi is less than 1% of the average stress, so it can be thought of as approximately zero. Now, let's first look at the case when r >> a. As we found in the pre-calculations, when r >> a, the radial stress is a function of the angle theta only. This matches the behavior seen in the simulation. From our Pre-Calculations, we also found that

Using the probe tool, we find that indeed at this location, the stress is equal to 1e6 psi, which is the value we calculated in our Pre-Analysis. Also from our Pre-Analysis, we found that when

Checking the simulation with our trusty probe tool, we find that the ANSYS simulation matches up quite nicely with our calculation.

Sigma-Theta Now, let's compare the simulation to our pre-calculations for the theta stress. Look to the Outline window, then click Solution > Sigma-theta

First, let's compare the case when r = a. From the pre-calculations, we found that the stress at the hole acts as a function of theta. Specifically:

From this equation, we find that at zero degrees, we expect the stress to be -1e6 psi at zero degrees and at 90 degrees, we expect the stress to be 3e6 psi. From the simulation we find that the stress at 0 degrees is -1.0285e6 psi (a 2.85% deviation), and the stress at 90 degrees is 3.0323e6 psi (a 1.06 % deviation) Now, let's look at the case when r >> a. From our pre-calculations, we found that the theta stress is a function of theta only. This behavior is represented in the simulation. Also, at the points such that r >> a and

the stress is equal to

Using a probe tool and hovering over this area, we see that the stress is indeed about F/A in the simulation. However, looking at the area when

we find that the stress from the simulation is between 1000 psi and 2000 psi. Although this seems large compared to zero, one must keep in mind that the stress at this location is 1% of the average stress. We expect that the stress here will get closer to zero on refining the mesh since the finite-element error becomes smaller.

Tau-r-theta Now let's look at how the simulation match our predictions for the shear stress. Look to the Outline window, then click Solution > Tau-r-theta

In our pre-calculations, we determined that at r = a the shear stress should be 0 psi. Using our probe tool, we find that the stress ranges between -5 and -500 psi at the hole. Because 500 psi is .05% of the average stress, we can say this result does represent what we expect to happen very well. In our pre-calculations, we determined that far from the hole the shear stress should be a function of theta only. This can be shown by using the probe tool a hovering over a radial line from the hole. The colors (representing higher and lower stresses) only change only as the angle changes, but not as the move away from the hole. We also found that far from the hole at

the stress is zero. Using the probe tool, we can see that this is indeed the case for the simulation as well.

Homework Exercise

Consider the bar above. It is .4 inches thick, 3 inches wide (w), the groove radius r is .5 inches, and the diameter d of the hole is .3 inches. The traction is 10,000 psi. You can download our ANSYS solution by clicking here. Unzip the file and load the solution into ANSYS Workbench as in the tutorial. Double-click on Results in Workbench to bring up the FEA results that have been calculated already. Note that our ANSYS solution makes use of symmetry and models half the bar. You might need to change the units for the results display by selecting Units > US Customary (in, lbm, lbf, F, s, V, A). 1. Look at the deformation plot. From this plot, what can you tell about the boundary conditions (displacement constraints and traction) that have been applied to this model? 2. Look at the sigma_xx plot. There are three distinct regions in this geometry: a. Away from the hole and symmetric grooves b. Adjacent to the symmetric grooves c. Adjacent to the hole What are the sigma_xx values you expect in these three regions? What is the % deviation of the corresponding ANSYS result from these values?

Problem Specification

A curved beam with a rectangular cross section is subjected to a moment of 300 inchpounds. The curved beam has an inner radius of 10 inches and outer radius of 12 inches. The beam is .25 inches thick.

Calculate the stresses at r = 11.5 inches. In this exercise, you are presented with the numerical solution to the above problem obtained from finite-element analysis (FEA) using ANSYS software. Compare FEA results for the stress distribution presented to you with the corresponding analytical solution. Justify agreements and discrepancies between the two approaches (FEA vs. Analytical). Note that you will be using the ANSYS solution presented to you to explore the physics of the problem. You will be downloading the ANSYS solution prepared for you. The objective is to help you learn important fundamentals of mechanics through the interactive, visual interface provided by ANSYS. You will not be obtaining the FEA solution using ANSYS;

Pre-Analysis & Start-Up Pre-Analysis There are three difference theories for finding the solution for the bending of a curved beam. There is elasticity theory, where

There is Winkler Bach Theory, where

And there is the straight beam theory, where

ANSYS Simulation Now, let's load the problem into ANSYS and see how a computer simulation will compare. First, start by downloading the files here The zip file should contain the following contents: 

Curved Beam Solution_files folder



Curved Beam Solution.wbpj

Please make sure to extract both of these files from the zip folder, the program will not work otherwise. (Note: The solution was created using ANSYS workbench 12.1 release, there may be compatibility issues when attempting to open with other versions). 2. Double click "Curved Beam Solution.wbpj" - This should automatically open ANSYS workbench (you have to twiddle your thumbs a bit before it opens up). You will be presented with the ANSYS solution.

A tick mark against each step indicates that that step has been completed. 3. To look at the results, double click on "Results" - This should bring up a new window (again you have to twiddle your thumbs a bit before it opens up). 4. On the left-hand side there should be an "Outline" toolbar. Look for "Solution (A6)".

We'll investigate the items listed under Solution in the next step in this tutorial.

Results

Now we will examine the simulation results from ANSYS.

Mesh Before we dive in to the solution, let's take a look at the mesh used for the simulation. In the outline window, click Mesh to bring up the meshed geometry in the geometry window.

Only one-half of the geometry is modeled using symmetry constraints, which reduces the problem size. Look to the outline window under "Mesh". Notice that there are two types of meshing entities: a "mapped face meshing" and a "face sizing". The "mapped face meshing" is used to generate a regular mesh of quadrilaterals. The face sizing controls the size of the element edges in the 2D "face".

Displacement Okay! Now we can check our solution. Let's start by examining how the beam deformed under the load. Before you start, make sure the software is working in the same units you are by looking to the menu bar and selecting Units > US Customary (in, lbm, lbf, F, s, V,

A). Now, look at the Outline window, and select Solution > Total Deformation.

The colored section refers to the magnitude of the deformation (in inches) while the black outline is the undeformed geometry superimposed over the deformed model. The more red a section is, the more it has deformed while the more blue a section is, the less it has deformed. For this geometry, the bar is bending inward and the largest deformation occurs where the moment is applied , as one would intuitively expect.

Sigma-theta Click Solution > Sigma-theta in the outline window. This will bring up the distribution for the normal stress in the theta direction.

Sigma-theta, the bending stress, is a function of r only as expected from theory. It is tensile (positive) in the top part of the beam and compressive (negative) in the bottom part. There is a neutral axis that separates the tensile and compressive regions. The bending stress, Sigma-theta, is zero on the neutral surface. We will use the probe to locate the region

where the bending stress changes from tensile to compressive. In order to find the neutral axis, let's first enlarge the geometry. Do this by clicking the Box Zoom tool

then click

and drag a rectangle around the area you want to magnify. Now, click the probe tool in the menu bar

This will allow you to hover the cursor over the geometry to see the

stress at that point. Hover the cursor over the geometry until you have a good understanding of where the neutral axis on the beam is. To zoom out, click "Zoom to Fit" We will now look at Sigma-theta along the symmetry line. Click Solution > Sigma-theta

along symmetry in the outline window to bring up the stress distribution at the middle of the bar.

Look at the color bar to see the maximum and minimum stresses. The maximum thetastress is 1697.63 psi and the minimum theta-stress is -1916.2 psi.

Sigma-r In the outline window, click Solution > Sigma-r. This will bring up the distribution for the normal stress in the r-direction.

Looking at the distribution, we can see that the stress varies only as a function of r as expected. The magnitude of Sigma-r is quite a bit lower than Sigma-theta (this is why Winkler-Bach theory assumes Sigma-r =0). Also, we can see that there is a stress concentration in the area where the moment is applied. In the theory, this effect is ignored. In order to further examine the Sigma-r, let's look at the variation along the symmetry line. Click on Solution > Sigma-r along symmetry. This solution is the normal stress in the rdirection at the midsection of the beam.

Looking at the color bar again, we can see that the maximum r-stress is -.110 psi, and the minimum r-stress is -82.302 psi. At r=a and r=b, Sigma-r ~ 0 as one would expect for a free surface.

Tau-r-theta In the details window, click Solution > Tau-r-theta to bring up the stress distribution for shear stress.

Hover the probe tool over points on the geometry far from the moment. You will notice that the stress is on the order of 10e-7. For a beam in pure bending, we assume that the shear stress is zero. However, ANSYS does not make this assumption: it calculates a value for shear stress at every point on the beam. Therefore, it is reassuring that the shear stress is almost negligible, which reinforces our assumption that it is zero.

Solution at r = 11.5 Inches Now that we have a good idea about the stress distribution, we will look specifically at solving the problem in the problem specification. First, we will look at the stress in the rdirection at r = 11.5 inches. In the outline window, click Solution > Sigma-r at r =11.5. This will bring up the stress in the r-direction along the path at r = 11.5 inches (from the center of curvature of the bar).

In the window below, there is a table of the stress values along the path. To find the value of sigma-r at r = 11.5 in, we want to look far away from the stress concentration region due to the moment. The path is defined in a counter-clockwise direction, so looking at the last

value of the table should tell us the stress at r = 11.5 inches at the midsection of the bar. This value of sigma-r is -57.042 psi. Now, we will do the same for the stress in theta direction to determine sigma-theta at r = 11.5 inches. In the outline window, click Solution > Sigma-theta at r =11.5. This will bring up the stress in the theta-direction along the path at r 11.5 inches.

Look again at the table containing the stresses along the path. Look to the bottom of the table to find the stress in the theta-direction at the midpoint of the bar. We find that sigmatheta at this point is 910.950 psi. Compare this to what you would expect from curved beam theory. Finally, we will examine the shear stress at r = 11.5 in. In the outline window, click Solution

> Tau-r-theta at r =11.5.

Again, look at the bottom of the table. You will find that the shear stress is very small at this point as we mentioned above.

Comparison. Now that we have our results from the ANSYS simulation, let's compare them to the theory calculations. Below is chart comparing the values found in ANSYS, and through calculations using the Elasticity Theory, Winkler-Bach Theory, and Straight Beam Theory (Note: all stress values are in psi)

Now, let's see how the stress distributions vary along the beam for each theory. First, let's see how the Elastic Theory compared to th

e ANSYS solution:

From what we can see from the able graphs, the Elastic Theory matched the ANSYS solution very well. The same can be said for the Winkler-Bach theory:

When we approximate the beam as a straight beam, the analytical solution deviates from the ANSYS solution.

Now that we have gone through a simulation for bending of a curved beam, it is time to see if you can do the same on your own!

Homework Look at Problem 5.17 in Deformable Bodies and their Material Behavior. Complete parts (a), (b), and (c) by hand. Next, download our ANSYS solution here and compare the simulation's results to your Elasticity calculations, Winkler-Bach Theory calculations, and a straight beam relation calculation from parts (a), (b) and (c).

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