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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. 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.

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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. We'll check these trends in the ANSYS results.

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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