Given:
A long rectangular beam with cross section
Find:
A solution for the displacement and stress fields, using strong boundary conditions on the edges and
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[Hint : Assume that the displacement can be expressed as a second degree polynomial (using the Pascal's triangle to determine the terms) ]
Step 1: Boundary conditions
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Step 2: Assume a solution
Let us assume antiplane strain
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Step 3: Calculate the stresses
The stresses are given by , and .
Therefore,
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Step 4: Satisfy stress BCs
Thus we have,
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Since and can be arbitrary, .
Hence, which gives us
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Assume that the body force is zero. Then the equilibrium condition is . Therefore,
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Therefore, the stresses are given by
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Step 5: Satisfy displacement BCs
The displacement is given by
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If we substitute , we cannot determine the constant uniquely.
Hence the displacement boundary conditions have to be applied in a weak sense,
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Therefore,
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