Question

A rectangular Kirchoff plate with sides of length \(a\) and \(b\) is clamped on all four edges. Establish the boundary conditions for the structure.

Short answer

Answer: For a rectangular Kirchoff plate with clamped edges of length a and b, the boundary conditions are: 1. Displacement boundary conditions: - w(0,y) = 0 - w(a,y) = 0 - w(x,0) = 0 - w(x,b) = 0 2. Rotation boundary conditions: - ∂w/∂x(0,y) = 0 - ∂w/∂x(a,y) = 0 - ∂w/∂y(x,0) = 0 - ∂w/∂y(x,b) = 0

Step-by-step solution

Clamped Plate Behavior

A Kirchoff plate is a thin plate whose deformation is governed by the Kirchoff-Love hypothesis. This means the plate is thin compared to its other dimensions, it undergoes small deformations, and the normal to the middle surface remains straight and inextensible during deformation. A clamped edge, also known as a fixed or built-in edge, is one in which the edge of the plate is held in place, meaning it cannot translate or rotate at the boundary.

Boundary Conditions for Displacement

As the plate is clamped along all four edges, its displacement in the out-of-plane direction (\(w\)) must be zero along those edges. For a rectangular plate with sides of length \(a\) and \(b\), these conditions can be expressed mathematically as: - Along the edge \(x=0\): \(w(0,y) = 0\) - Along the edge \(x=a\): \(w(a,y) = 0\) - Along the edge \(y=0\): \(w(x,0) = 0\) - Along the edge \(y=b\): \(w(x,b) = 0\)

Boundary Conditions for Rotation

Along with zero displacement, clamped edges also require zero rotation at the boundary. The rotation of the plate is represented by the angle formed between the plate tangent plane on the edges and the undeformed reference plane. The partial derivatives of \(w\) with respect to \(x\) and \(y\) represent the rotation along the x and y directions, and for a clamped edge, these values must also be zero. Mathematically, these conditions can be expressed as: - Along the edge \(x=0\): \(\frac{\partial w}{\partial x}(0,y) = 0\) - Along the edge \(x=a\): \(\frac{\partial w}{\partial x}(a,y) = 0\) - Along the edge \(y=0\): \(\frac{\partial w}{\partial y}(x,0) = 0\) - Along the edge \(y=b\): \(\frac{\partial w}{\partial y}(x,b) = 0\)

Summary of Boundary Conditions

To summarize, for a rectangular Kirchoff plate with sides of length \(a\) and \(b\) clamped on all four edges, the boundary conditions for the structure are: 1. Displacement boundary conditions: - \(w(0,y) = 0\) - \(w(a,y) = 0\) - \(w(x,0) = 0\) - \(w(x,b) = 0\) 2. Rotation boundary conditions: - \(\frac{\partial w}{\partial x}(0,y) = 0\) - \(\frac{\partial w}{\partial x}(a,y) = 0\) - \(\frac{\partial w}{\partial y}(x,0) = 0\) - \(\frac{\partial w}{\partial y}(x,b) = 0\) These boundary conditions must be satisfied for the plate to be considered clamped on all four edges.

Key concepts

Boundary Conditions

Boundary conditions are important in defining how a plate interacts with its surroundings. For a Kirchoff plate clamped on all four edges, these conditions dictate that both displacement and rotation are restricted.

• The displacement (\(w\)) describes how much the plate moves out of its original plane. On the edges where the plate is clamped, this movement must be zero.
• In technical terms, this means \(w(0, y) = 0\), \(w(a, y) = 0\), \(w(x, 0) = 0\), and \(w(x, b) = 0\).
• The rotation along these edges is also zero, symbolized by the partial derivatives being zero: \(\frac{\partial w}{\partial x}(0, y) = 0\) and \(\frac{\partial w}{\partial y}(x, 0) = 0\)

These boundary conditions ensure that clamped edges remain fixed without any twisting or movement, maintaining the intended structure of the plate.

Clamped Edges

Clamped edges refer to the edges of a plate where both displacement and rotation are restricted. When we say edges are clamped:

• They cannot move up or down, which is the out-of-plane displacement \(w\).
• They cannot rotate; hence, any slope or tilt is not permitted.

Clamping is like firmly fixing the plate at its edges, ensuring the edges remain straight and unmovable. This allows the entire plate to distribute loads more effectively, avoiding any bending or twisting at the edges. Clamping results in strict boundary conditions that guide the structural integrity of the plate.

Kirchhoff-Love Hypothesis

The Kirchhoff-Love hypothesis is a fundamental assumption in plate theory. It simplifies the analysis by making specific assumptions about the behavior of thin plates.

• The plate is thin compared to its lateral dimensions, meaning its thickness is much smaller than its length or width.
• Any deformation the plate undergoes is small, allowing for a linear analysis rather than nonlinear.
• The normal line to the plate's mid-surface stays straight and is inextensible, meaning it does not stretch or bend during deformation.

These assumptions allow for an easier mathematical treatment of plates, especially when calculating their deformation under different forces while adhering to physical constraints.

Displacement and Rotation in Plates

Understanding displacement and rotation is central to analyzing plate behavior. Displacement in plates refers to the movement of points on the plate from their original position.

• For a Kirchoff plate, the displacement out-of-plane is critical, as it indicates how the plate 'bends'.
• Rotation involves the change in angle of the plate surface. It can be represented using partial derivatives along the edges.

In clamped plates, both displacement and rotation are kept to zero at the edges to ensure the plate stays fixed with no lifting or tilting. This creates a rigid boundary condition essential for maintaining the intended structural behavior of the plate under various loads.