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Pre-Analysis and Start-Up
Analytical vs. Numerical Approaches
We can either assume the geometry as an infinite plate and solve the problem analytically, or approximate the geometry as a collection of "finite elements", and solve the problem numerically. The following flow chart compares the two approaches.
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Let's first review the analytical results for the infinite plate. We'll then use these results to check the numerical solution from ANSYS.
Analytical Results
Displacement
Let's estimate the expected displacement of the right edge relative to the center of the hole. We can get a reasonable estimate by neglecting the hole and approximating the entire plate as being in uniaxial tension. Dividing the applied tensile stress by the Young's modulus gives the uniform strain in the x direction.
Multiplying this by the half-width (5 in) gives the expected displacement of the right edge as ~ 0.17 1724 in. We'll check this against ANSYS.
Sigma-r
Let's consider the expected trends for Sigma-r, the radial stress, in the vicinity of the hole and far from the hole. The analytical solution for Sigma-r in an infinite plate is:
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Far from the hole, Sigma-r is a function of theta only. At theta = 0, Sigma-r ~ Sigma-o. This makes sense since r is aligned with x when theta = 0. At theta = 90 deg., Sigma-r ~ 0 which also makes sense since r is now aligned with y. We'll check these trends in the ANSYS results.
Sigma-theta
Let's next consider the expected trends for Sigma-theta, the circumferential stress, in the vicinity of the hole and far from the hole. The analytical solution for Sigma-theta in an infinite plate is:
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Far from the hole, Sigma-theta is a function of theta only but its variation is the opposite of Sigma-r (which is not surprising since r and theta are orthogonal coordinates; when r is aligned with x, theta is aligned with y and vice-versa). As one goes around the hole from theta = 0 to theta = 90 deg., Sigma-theta increases from 0 to Sigma-o. More trends to check in the ANSYS results!
Tau-r-theta
The analytical solution for the shear stress Tau-r-theta in an infinite plate is:
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We can deduce that, far from the hole, Tau-r-theta = 0 both at theta = 0 and theta = 90 deg. Even more trends to check in ANSYS!
Sigma-x
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
Numerical Solution using ANSYS
You might be thinking: will we ever get to ANSYS? OK, let's load the numerical solution for the finite plate into ANSYS and see how it compares with the above analytical solution for the infinite plate. First, start by downloading the files here.
The zip file should contain the following:
- 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 older 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.
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We'll investigate the items listed under Solution in the next step of this tutorial.
Continue Go to Step 2 - : Numerical Results