Problem 10.2. Deflection of an orthotropic plate

The deflection function of an orthotropic plate is given:
${w (x, y) = C x^{2} y^{2} (1 - \frac{x}{a} - \frac{y}{b})}^{2}$
where C is a constant. Determine the bending and torsional moments of the plate in the functions of the plate’s stiffnesses D₁₁, D₂₂, D₁₂, D₆₆. Derive the load function acting on the plate. Give the possible supports where the deflection function satisfies the boundary conditions.

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Derive the load function.

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$p (x, y) = D_{11} 24 C \frac{y^{2}}{a^{2}} + D_{22} 24 C \frac{x^{2}}{b^{2}} + 2 (D_{12} + 2 D_{66}) 2 C (2 - 12 \frac{x}{a} (1 - \frac{x}{a}) - 12 \frac{y}{b} (1 - \frac{y}{b}) + 81 \frac{x y}{a b})$

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Steps

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Step 1. Perform the necessary partial derivations of the deflection function according to the variables, x and y.

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$w (x, y) = C {(x y - \frac{x^{2} y}{a} - \frac{x y^{2}}{b})}^{2} \frac{\partial w}{\partial x} = 2 C (x y - \frac{x^{2} y}{a} - \frac{x y^{2}}{b}) (y - \frac{2 x y}{a} - \frac{y^{2}}{b}) \frac{\partial^{2} w}{\partial x^{2}} = 2 C (y^{2} - \frac{2 y^{3}}{b} + \frac{y^{4}}{b} - \frac{6 x y^{2}}{a} + \frac{6 x^{2} y^{2}}{b} + \frac{6 x y^{3}}{a b}) \frac{\partial^{4} w}{\partial x^{4}} = 24 C \frac{y^{2}}{b} \frac{\partial w}{\partial y} = 2 C (x y - \frac{x y^{2}}{b} - \frac{x^{2} y}{a}) (x - \frac{2 x y}{b} - \frac{x^{2}}{a}) \frac{\partial^{2} w}{\partial y^{2}} = 2 C (x^{2} - \frac{2 x^{3}}{a} + \frac{x^{4}}{a} - \frac{6 x^{2} y}{b} + \frac{6 x^{2} y^{2}}{a} + \frac{6 x^{3} y}{a b}) \frac{\partial^{4} w}{\partial y^{4}} = 24 C \frac{x^{2}}{a} \frac{\partial^{2} w}{\partial y \partial x} = 2 C x y (2 - \frac{6 x}{b} - \frac{6 y}{b} + \frac{4 x^{3} y}{a^{3}} + \frac{4 x y^{3}}{b^{3}} + \frac{9 x^{2} y^{2}}{a b}) \frac{\partial^{3} w}{\partial y^{2}} = 2 C (2 - \frac{12 x}{a} + \frac{12 x^{2}}{a^{2}} - \frac{12 y}{b} + \frac{12 y^{2}}{b^{2}} + \frac{81 x^{2} y^{2}}{a b})$

Step 2. Give the moment functions in the function of the curvatures.

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Eq.(10-34) and Table 34.

M_{x} (x, y) = - D_{11} \frac{\partial^{2} w}{\partial x^{2}} - D_{12} \frac{\partial^{2} w}{\partial y^{2}} = - D_{11} 2 C (y^{2} - \frac{2 y^{3}}{b} + \frac{y^{4}}{b} - \frac{6 x y^{2}}{a} + \frac{6 x^{2} y^{2}}{b} + \frac{6 x y^{3}}{a b}) - D_{12} 2 C (x^{2} - \frac{2 x^{3}}{a} + \frac{x^{4}}{a} - \frac{6 x^{2} y}{b} + \frac{6 x^{2} y^{2}}{a} + \frac{6 x^{3} y}{a b}) M_{y} (x, y) = - D_{12} \frac{\partial^{2} w}{\partial x^{2}} - D_{22} \frac{\partial^{2} w}{\partial y^{2}} = - D_{12} 2 C (y^{2} - \frac{2 y^{3}}{b} + \frac{y^{4}}{b} - \frac{6 x y^{2}}{a} + \frac{6 x^{2} y^{2}}{b} + \frac{6 x y^{3}}{a b}) - D_{22} 2 C (x^{2} - \frac{2 x^{3}}{a} + \frac{x^{4}}{a} - \frac{6 x^{2} y}{b} + \frac{6 x^{2} y^{2}}{a} + \frac{6 x^{3} y}{a b}) M_{x y} (x, y) = - D_{66} \frac{\partial^{2} w}{\partial x \partial x} = - D_{66} 2 C x y (2 - \frac{6 x}{b} - \frac{6 y}{b} + \frac{4 x^{3} y}{a^{3}} + \frac{4 x y^{3}}{b^{3}} + \frac{9 x^{2} y^{2}}{a b})

Step 3. Give the load function with the partial differential equation of the orthotropic plate.

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Eq.(10-37)

p (x, y) = D_{11} \frac{\partial^{4} w}{\partial x^{4}} + D_{22} \frac{\partial^{4} w}{\partial y^{4}} + 2 (D_{12} + 2 D_{66}) \frac{\partial^{4} w}{\partial x^{4}} \to p (x, y) = D_{11} 24 C \frac{y^{2}}{a^{2}} + D_{22} 24 C \frac{x^{2}}{b^{2}} + 2 (D_{12} + 2 D_{66}) 2 C (2 - 12 \frac{x}{a} (1 - \frac{x}{a}) - 12 \frac{y}{b} (1 - \frac{y}{b}) + 81 \frac{x y}{a b})

Step 4. Check boundary conditions.

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At edge x = 0:

$w (x = 0, y) = 0 {\frac{\partial w}{\partial x}}_{x = 0} = 0$

Thus the edge is built-in.

At edge y = 0:

$w (x, y = 0) = 0 {\frac{\partial w}{\partial y}}_{y = 0} = 0$

Thus the edge is built-in.

On the other two edges none of the derivatives are zero, thus the plate must be subjected to line loads and moments or the plate is connected to other structural elements.

Results

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The necessary partial derivations of the deflection function according to the variables, x and y are the following:

The moments are the functions of the curvatures:

Eq.(10-34) and Table 34.

M_{x} (x, y) = - D_{11} \frac{\partial^{2} w}{\partial x^{2}} - D_{12} \frac{\partial^{2} w}{\partial y^{2}} = - D_{11} 2 C (y^{2} - \frac{2 y^{3}}{b} + \frac{y^{4}}{b} - \frac{6 x y^{2}}{a} + \frac{6 x^{2} y^{2}}{b} + \frac{6 x y^{3}}{a b}) - D_{12} 2 C (x^{2} - \frac{2 x^{3}}{a} + \frac{x^{4}}{a} - \frac{6 x^{2} y}{b} + \frac{6 x^{2} y^{2}}{a} + \frac{6 x^{3} y}{a b}) M_{y} (x, y) = - D_{12} \frac{\partial^{2} w}{\partial x^{2}} - D_{22} \frac{\partial^{2} w}{\partial y^{2}} = - D_{12} 2 C (y^{2} - \frac{2 y^{3}}{b} + \frac{y^{4}}{b} - \frac{6 x y^{2}}{a} + \frac{6 x^{2} y^{2}}{b} + \frac{6 x y^{3}}{a b}) - D_{22} 2 C (x^{2} - \frac{2 x^{3}}{a} + \frac{x^{4}}{a} - \frac{6 x^{2} y}{b} + \frac{6 x^{2} y^{2}}{a} + \frac{6 x^{3} y}{a b}) M_{x y} (x, y) = - D_{66} \frac{\partial^{2} w}{\partial x \partial x} = - D_{66} 2 C x y (2 - \frac{6 x}{b} - \frac{6 y}{b} + \frac{4 x^{3} y}{a^{3}} + \frac{4 x y^{3}}{b^{3}} + \frac{9 x^{2} y^{2}}{a b})

The load function is given by the partial differential equation of the orthotropic plate.

Eq.(10-37)

p (x, y) = D_{11} \frac{\partial^{4} w}{\partial x^{4}} + D_{22} \frac{\partial^{4} w}{\partial y^{4}} + 2 (D_{12} + 2 D_{66}) \frac{\partial^{4} w}{\partial x^{4}} \to p (x, y) = D_{11} 24 C \frac{y^{2}}{a^{2}} + D_{22} 24 C \frac{x^{2}}{b^{2}} + 2 (D_{12} + 2 D_{66}) 2 C (2 - 12 \frac{x}{a} (1 - \frac{x}{a}) - 12 \frac{y}{b} (1 - \frac{y}{b}) + 81 \frac{x y}{a b})

Boundary conditions at edge x = 0:

$w (x = 0, y) = 0 {\frac{\partial w}{\partial x}}_{x = 0} = 0$

Thus the edge is built-in.

Boundary conditions at edge y = 0:

$w (x, y = 0) = 0 {\frac{\partial w}{\partial y}}_{y = 0} = 0$

Thus the edge is built-in.

On the other two edges none of the derivatives are zero, thus the plate must be subjected to line loads and moments or the plate is connected to other structural elements.