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.

Look to the outline window under "Mesh". Notice that there are two types of meshing types on the geometry: a mapped face meshing and a face sizing meshing. The mapped face meshing restricts the type of element shapes that will be used in the mesh - in this case, quadrilaterals. The face sizing controls the size of the elements. The size of each element is limited to at most 0.1 square inches. Another thing to notice about the geometry is that the geometry in the simulation is actually only one half of the geometry from the problem statement. This is done using symmetry constraints which allows the simulation can find the same answer as for the full geometry while saving valuable computation time because it is using fewer elements!

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-r

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

Looking at the distribution, we can see that the stress varies only as a function of r as expected. Also, we can see that there seems to be some sort of stress concentrations in the are where the moment was applied. In our analysis, we ignored these transient stresses, but it is important to know that ANSYS calculated them when we try to obtain the solution to the problem statement. In order to further examine the stresses in the r-direction, lets look at an area far from the transient stresses in order to minimize their effect on the computational solution. Click on Solution > Sigma-r without transience. This solution is the stress in the r-direction at the midpoint of the beam. This line is far enough from the moment that the transient stresses will not affect its local stresses.

Click the Max and Min Tags in the menu bar: they will show the maximum and minimum stresses and their locations. Now, we can see that the maximum r-stress is -.110 psi, and the minimum r-stress is -82.302 psi.

Sigma-theta

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

As with Sigma-r, the stress is a function of r, only. Also, there is some transience near the moment. Again, we will look at an area of the geometry far from the moment to decrease the transience's influence on our solution. Click Solution > Sigma-theta without transience in the outline window to bring up the stress distribution at the middle of the bar.

If they are not on, make sure to click on the max and min tags once again to see the maximum and minimum stresses. The maximum theta-stress is 1697.63 psi and the minimum theta-stress is -1916.2 psi

Tau-r-theta

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

Click the probe tool in the menu bar. This will allow you to hover the cursor over the geometry at see the stress at that point. Hover the probe 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 is 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 r-direction 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).

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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 again want to look far away from the transient stresses 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 midpoint 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.

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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 sigma-theta at this point is 910.950 psi.

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.

Continue to Step 3 - Homework
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