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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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https://confluence.cornell.edu/download/attachments/127122289/FlowLarge.png?version=1&modificationDate=1320261775000

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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

First, letLet'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.

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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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where a is the hole radius and Sigma-o is the applied uniform stress (denoted P in the problem specification). At the hole (r=a), this reduces to

This result can be understood by looking at a vanishingly small element at the hole as shown schematically below.

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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 theta x when theta = 0. At theta = 90 deg., Sigma-r ~ 0 which also makes sense since r is now aligned with y.

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:

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at r = a

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at r>>a

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Now we will examine the stress far from the hole at theta = 0 and theta = pi/2

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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:

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at r=a

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This boundary condition can also be validated intuitively. Let's look at an element on the hole

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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

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Now we will examine the values of Tau_r_theta when r>>a and at theta = 0 and theta = pi/2

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and

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We will reexamine all of these calculations so we may estimate the validity of the ANSYS simulation later in this tutorial.

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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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At r = a, this reduces to

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At theta = 90 deg., Sigma-theta = 3*Sigma-o for an infinite plate. This leads to a stress concentration factor of 3 for an infinite plate.

For r>>a,

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At theta = 0 and theta = 90 deg., we get

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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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At r=a,

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By looking at a vanishingly small element at the hole , we see that Tau-r-theta at the hole is the shear stress at the hole. Since the hole is a free surface, this has to be zero.

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For r>>a,

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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.

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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

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You might be thinking: will we ever get to ANSYS? OKNow, let's load the problem numerical solution for the finite plate into ANSYS and see how a computer simulation will compareit compares with the above analytical solution for the infinite plate. 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 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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3. To look at the results, double click on "Results" - This should bring up a new window the ANSYS Mechanical application (again you have to twiddle your thumbs a bit before it opens up).

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We'll investigate the items listed under Solution in the next step in of this tutorial.

Continue Go to Step 2 - : Numerical Results

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