The (x, y) coordinates of the corners of a sq | vedclass

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(D) Since the edges of the square plate are clamped,the displacement $u(x, y)$ must be zero at all boundaries:$1$. At $y = 0$ (edge $OA$),$u(x, 0) = 0

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Both $(b)$ and $(c)$

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(D) Since the edges of the square plate are clamped,the displacement $u(x, y)$ must be zero at all boundaries:$1$. At $y = 0$ (edge $OA$),$u(x, 0) = 0$ for $0 \le x \le L$.$2$. At $x = L$ (edge $AB$),$u(L, y) = 0$ for $0 \le y \le L$.$3$. At $y = L$ (edge $BC$),$u(x, L) = 0$ for $0 \le x \le L$.$4$. At $x = 0$ (edge $OC$),$u(0, y) = 0$ for $0 \le y \le L$.Evaluating the options:- For option $(a)$: $u(0, y) = a \cos(0) \cos(\frac{\pi y}{2L}) = a \cos(\frac{\pi y}{2L}) 
eq 0$ for all $y$.- For option $(b)$: $u(x, y) = a \sin(\frac{\pi x}{L}) \sin(\frac{\pi y}{L})$. Here,$u = 0$ at $x=0, x=L, y=0, y=L$. This satisfies the boundary conditions.- For option $(c)$: $u(x, y) = a \sin(\frac{\pi x}{L}) \sin(\frac{2\pi y}{L})$. Here,$u = 0$ at $x=0, x=L, y=0, y=L/2, y=L$. This also satisfies the boundary conditions.Thus,both $(b)$ and $(c)$ are valid expressions.

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