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The biharmonic equation $\nabla^4 w = D^{-1} q$ in plate theory has the form:
A{'text': '$\\nabla^4 w = \\partial^4 w/\\partial x^4 + 2\\partial^4 w/(\\partial x^2 \\partial y^2) + \\partial^4 w/\\partial y^4$', 'label': 'B'}
B{'text': '$\\nabla^4 w = \\partial^8 w/\\partial x^8$', 'label': 'D'}
C{'text': '$\\nabla^4 w = \\partial^4 w/\\partial x^4 + \\partial^4 w/\\partial y^4$', 'label': 'A'}
D{'text': '$\\nabla^4 w = \\partial^2 w/\\partial x^2 + \\partial^2 w/\\partial y^2$', 'label': 'C'}
Answer & Solution
Correct answer: A. {'text': '$\\nabla^4 w = \\partial^4 w/\\partial x^4 + 2\\partial^4 w/(\\partial x^2 \\partial y^2) + \\partial^4 w/\\partial y^4$', 'label': 'B'}
The biharmonic operator $\nabla^4 = (\nabla^2)^2$. Expanded: $w_{xxxx} + 2 w_{xxyy} + w_{yyyy}$. The mixed term ${2 w_{xxyy}}$ couples bending in x with bending in y — distinguishing 2D plate behavior from 1D beam bending.
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