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At a FREE edge of a thin plate (x = a), Kirchhoff's theory imposes how many independent boundary conditions?
A{'text': 'One (w_xx = 0 only)', 'label': 'A'}
B{'text': 'Four', 'label': 'D'}
C{'text': 'TWO conditions, but ONE is the "effective" shear V_x = Q_x + ∂M_xy/∂y = 0 (combining transverse shear and the twisting-moment gradient — Kirchhoff\'s effective shear); the other is M_x = 0', 'label': 'B'}
D{'text': 'Three: M_x, M_xy, V_x = 0', 'label': 'C'}
Answer & Solution
Correct answer: C. {'text': 'TWO conditions, but ONE is the "effective" shear V_x = Q_x + ∂M_xy/∂y = 0 (combining transverse shear and the twisting-moment gradient — Kirchhoff\'s effective shear); the other is M_x = 0', 'label': 'B'}
The classical Kirchhoff plate equation is 4th order — needs 2 BCs per edge. A free edge has M_x = 0 and V_x + ∂M_xy/∂y = 0 (the "Kirchhoff effective shear"). Note: the third condition M_xy = 0 must also hold but is subsumed by the effective shear via integration by parts.
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