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{include: ANSYS WB - Plate With a Hole Demo - Panel}

h1. Plate With a Hole Tutorial - Pre-Analysis and Start-Up

h3. Pre-Analysis Calculations 

h4. Sigma_x

First, let's solve for the maximum normal stress in the plate. To find the maximum stress, we need to find the stress concentration factor. MAE 3250 students: if you look to page 158 in Deformable Bodies and Their Material Behavior, Figure 4.22 shows 3 different concentration factor tables. Using the dimensions from the problem specification, we find that {latex} $\frac{a}{w} = \frac{.5}{5}= .1$ {latex} Using the chart, we can estimate {latex} $K \approx 2.7473 $ {latex}. The maximum stress can be found using the formula

{latex}
$\sigma_{max} = K*\sigma_{avg} $
{latex}

where K is the concentration factor, and

{latex} 
$\sigma_{avg} = \frac{F}{A_{minimum}} = \frac{F}{t*w} = \frac{100000}{.1 * (5-.5)} = 2.22222*10^{5} $ psi
{latex}

so the maximum stress can be calculated:  {latex} $\sigma_{max}=(2.7473)*(2.22222*10^5)=6.08890667*10^5$ {latex}

h4. Sigma_r

The equation for the radial stress in a plate with a hole is {latex} \large $ \sigma_{r} = \frac{1}{2} \sigma_{o} [(1 - \frac{a^{2}}{r^{2}} ) + (1 + 3 \frac{a^{4}}{r^{4}} - 4 \frac{a^{2}}{r^{2}})cos(2 \theta)] $ {latex} for r >> a, we find that the above equation reduces to {latex} \large $ \frac{1}{2} \sigma_{o} [1+cos(2 \theta)] $ {latex} so {latex} \large $ lim_{r \rightarrow \infty} \sigma_{r}(r,\theta) = \sigma_{r}(\theta) $ {latex}. Using this new function, we find that far from the hole {latex} \large $ \sigma_{r}|_{\theta=0} = \sigma_{o} $ {latex} and {latex} \large $ \sigma_{r}|_{\theta=\frac{\pi}{2}} = 0 $ {latex}. We also find that at r = a, {latex} $ \sigma_{r} = 0 $ {latex}. We will keep this in mind when we do our simulation.

h4. Sigma_theta

We will use a similar analysis for Sigma_theta.

{latex}$ \sigma_{\theta}= \frac{1}{2} \sigma_{o} [ (1 + \frac{a^{2}}{r^{2}}) - (1 + 3 \frac{a^{4}}{r^{4}})cos(2\theta)] \\ \\ \mbox{when r } \gg \mbox{ a} \\ \\ \sigma_{\theta}(r,\theta) = \frac{1}{2} \sigma_{o} [1 - cos(2\theta)] = \sigma_{\theta}(\theta) \\ \\ \sigma_{\theta}|_{\theta = 0} = 0 \mbox{ and } \sigma_{\theta}|_{\theta = \frac{\pi}{2}} = 0 ${latex} 

h4. Tau_r_theta

The shear stress in the r-theta direction is defined as: {latex} \large $   \tau_{r \theta}= - \frac{1}{2} \sigma_{o} (1 - 3 \frac{a^{4}}{r^{4}} + 2 \frac{a^{2}}{r^{2}})sin(2\theta)   ${latex}. For values r >> a, we find:

{latex} \large $ \tau_{r\theta} = - \frac{\sigma_{o}}{2} sin(2 \theta) \\  \tau_{r\theta}|_{\theta = 0} = \tau_{r\theta}|_{\theta = \frac{\pi}{2}} = 0     ${latex}

h4. ANSYS Simulation

Now, let's load the problem into ANSYS and see how a computer simulation will compare. First, start by downloading the file here

The zip should contain a class demo1 folder with the following contents:
- Plate With a Hole Solution folder
- Plate With a Hole.wbpj
Please make sure both these are in the folder, the program would 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). Be sure to extract before use.

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.

!Plate With a Hole Menu.png!

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)".
!Plate With a Hole Outline.png!
 We'll investigate the items listed under Solution in the next step in this tutorial.

Continue to [Step 2 - Results| ANSYS WB - Plate With a Hole Demo - Results]
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