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First, 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 in. We'll check this against ANSYS.

sigma_xx

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

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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

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sigma_r

Now, let's look at the radial stress varies in the plate:

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at r=a

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

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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-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),

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

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We see that Sigma-r 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,

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

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

sigma_xx

First, let's begin by finding the average stress, the nominal area stress, and the maximum stress with a concentration factor.

Image Added

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

Image Added

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

Image Added

ANSYS Simulation

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