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The governing PDE for ELASTIC ISOTROPIC THIN PLATE bending (Kirchhoff) is:
A{'text': '$\\nabla \\cdot w = 0$', 'label': 'D'}
B{'text': '$D\\nabla^4 w = q(x,y)$ — biharmonic, where D = Et³/[12(1−ν²)] is the plate flexural rigidity', 'label': 'B'}
C{'text': '$EI w_{xxxx} = q$ (beam)', 'label': 'C'}
D{'text': '$\\nabla^2 w = 0$ (Laplace)', 'label': 'A'}
Answer & Solution
Correct answer: B. {'text': '$D\\nabla^4 w = q(x,y)$ — biharmonic, where D = Et³/[12(1−ν²)] is the plate flexural rigidity', 'label': 'B'}
Equilibrium combined with constitutive + Kirchhoff kinematics gives the biharmonic equation D∇⁴w = q. D = Et³/[12(1−ν²)] generalizes beam EI to 2D — note the (1−ν²) factor from plane strain. q is the transverse load per unit area.
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