Plate With a Hole Tutorial - Pre-Analysis and Start-Up
How to solve this Problem
There are two main methods to solve this problem. Refer to the diagram below:
As seen in the flow chart, we can either assume the geometry as an infinite plate and solve the problem analytically, or we approximate the geometry as a collection of finite elements, and solve the problem numerically. First, we will assume the plate is infinite and will complete the analytical calculations.
Pre-Analysis Calculations
Displacement
First, let's estimate the total displacement of the bar. To this, we will estimate the plate with a hole as a tensile bar with a constant cross sectional area.
Now, subbing in values
sigma_xx
First, let's begin by finding the average stress, the nominal area stress, and the maximum stress with a concentration factor.
The concentration factor for an infinite plate with a hole is K = 3. The maximum stress for an innite plate with a hole is
Although there is no analytical solution for a nite plate with a hole, there is empirical data available to find a concentration factor. Using a Concentration Factor Chart (3250 Students: See Figure 4.22 on page 158 in Deformable Bodies and Their Material Behavior), we find that d/w = 1 and thus K ~ 2:73 Now we can find the maximum stress using the nominal stress and the concentration factor
sigma_r
Now, let's look at the radial stress varies in the plate:
at r=a
This boundary condition can be validated intuitively. Let's look at an element on the hole
From structural mechanics, we know that any stress on a face must be zero; therefore the analytical solution at r = a is correct.
at r>>a
sigma_theta
Now we will examine how sigma_theta varies in the plate. We will approach this very similarly to how we approached the examination of sigma_r:
at r = a
at r>>a
Now we will examine the stress far from the hole at theta = 0 and theta = pi/2
Tau_r_theta
Finally, we will examine how the shear stress in the r_theta direction varies in the plate. The equation for the shear stress in the plate is:
at r=a
This boundary condition can also be validated intuitively. Let's look at an element on the hole
As mentioned before, we know that any stress on a face must be zero; therefore the analytical solution at r = a is correct.
At r>>a
Now we will examine the values of Tau_r_theta when r>>a and at theta = 0 and theta = pi/2
and
We will reexamine all of these calculations so we may estimate the validity of the ANSYS simulation later in this tutorial.
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.
Continue to Step 2 - Results
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