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We see that σr at the hole is the normal stress at the hole. Since the hole is a free surface, this has to be zero.

For r >> a,

Far from the hole, σr is a function of �� only. At �� = 0, σr ~ σo. This makes sense since r is aligned with x when �� = 0. At �� = 90 deg., σr ~ 0 which also makes sense since r is now aligned with y. We'll check these trends in the ANSYS AIM results.

σ��

Let's next consider the expected trends for σ��, the circumferential stress, in the vicinity of the hole and far from the hole. The analytical solution for σ�� in an infinite plate is:

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At r = a, this reduces to

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At �� = 0 and �� = 90 deg., we get

Far from the hole, σ�� is a function of �� only but its variation is the opposite of σr (which is not surprising since r and �� are orthogonal coordinates; when r is aligned with x, �� is aligned with y and vice-versa). As one goes around the hole from �� = 0 to �� = 90 deg., σ�� increases from 0 to σo. More trends to check in the ANSYS AIM results!

��r��

The analytical solution for the shear stress, ��r��, in an infinite plate is:

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The concentration factor for an infinite plate with a hole is K = 3. The maximum stress for an infinite plate with a hole is:

Although there is no analytical solution for a finite plate with a hole, there is empirical data available to find a concentration factor. Using a Concentration Factor Chart (Cornell 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:

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Start-Up

Now that we have the pre-calculations, we are ready begin simulating in ANSYS AIM. Open ANSYS AIM by going to Start > All Apps > ANSYS 18.1 > ANSYS AIM 18.1. Once you are at the start page of AIM, select the Structural template in the top left corner as shown below.

 You will be prompted by the Structural Template to either Define new geometry, Import geometry file, or Connect to active CAD session. Select Define new geometry and press Next. For this problem we will be using static calculation type.Press Finish on the next panel.  The Model Editor will launch automatically.

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