Numerical Results

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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.

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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.

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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.

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

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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 much 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 r-direction at the midsection of the beam.

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

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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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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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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.

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