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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.

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Multiplying this by the half-width (5 in) gives the expected displacement of the right edge as ~ 0.1724 in. We'll check this against ANSYS AIM.

σr

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

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

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

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