Question

A steel deck plate of a tanker is supported by two longitudinal stiffeners as shown in the figure. The width of the plate is \( a \) and its length is 5 times the width. Assume that the long edge is simply supported, and the short edge is free. The plate is loaded by a distributed pressure, \( p = p_0 \sin\left(\frac{\pi y}{a}\right) \), where \( p_0 \) is the pressure at \( y = a/2 \). The flexural rigidity of the plate is \( D \). The plate equation is given by 

• A. \( D \frac{\partial^4 w}{\partial x^4} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0 \)
• B. \( D \frac{\partial^4 w}{\partial x^2 \partial y^2} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0 \)
• C. \( D \frac{\partial^4 w}{\partial y^4} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0 \)
• D. \( D \frac{\partial^4 w}{\partial x^4} + D \frac{\partial^4 w}{\partial y^4} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0 \)

Hint:

For plate problems, simplify the governing equation based on boundary conditions and the direction of load variation. Focus on the dominant bending direction.

Answer 1

The Correct Option is A

Step 1: Understand the governing equation for plate deformation.
For a thin rectangular plate subjected to a distributed transverse load \( p \), the governing equation for bending is: \[ D \left( \frac{\partial^4 w}{\partial x^4} + 2 \frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4} \right) + p = 0, \] where: - \( D \) is the flexural rigidity of the plate, - \( w \) is the transverse deflection, - \( p \) is the distributed transverse load. Step 2: Simplify based on boundary conditions and loading.
1. The short edges of the plate are free, and the long edges are simply supported. 2. The loading \( p = p_0 \sin\left(\frac{\pi y}{a}\right) \) varies only in the \( y \)-direction, implying there is no significant variation along the \( y \)-axis. 3. Since the plate is simply supported along \( x \)-direction, only the bending along \( x \)-direction contributes to the equation. The equation reduces to: \[ D \frac{\partial^4 w}{\partial x^4} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0. \] Step 3: Analyze the options.
Option (A): Correct, as it represents the simplified plate equation considering the variation along \( x \)-direction and the applied load.
Option (B): Incorrect, as it introduces mixed derivatives, which are not applicable here due to the simplified loading condition.
Option (C): Incorrect, as it considers only the \( y \)-direction, which is not relevant for this configuration.
Option (D): Incorrect, as it considers both \( x \)- and \( y \)-direction variations, which is not consistent with the boundary conditions.
Conclusion: The correct plate equation is: \[ D \frac{\partial^4 w}{\partial x^4} + p_0 \sin\left(\frac{\pi y}{a}\right) = 0. \]