Question

Consider a Kirchhoff plate whose major plane has the shape of a right isosceles triangle with legs of length \(L\). Establish the boundary conditions for the structure if it is simply supported along the edge \(y=0\), clamped along the edge \(x=0\), and free along the edge \(x+y=L\).

Short answer

Answer: The boundary conditions for each of the three edges are as follows: 1. For the simply supported edge y=0: a. Zero bending moment: \(\frac{\partial^2 w}{\partial y^2}\Big|_{y=0} = 0\) b. Zero displacement: \(w(x,0) = 0\) 2. For the clamped edge x=0: a. Zero displacement: \(w(0,y) = 0\) b. Zero slope: \(\frac{\partial w}{\partial x}\Big|_{x=0} = 0\) 3. For the free edge x+y=L: a. Zero shear force: \(\frac{\partial^3 w}{\partial x \partial y^2} + \frac{\partial^3 w}{\partial y \partial x^2}\Big|_{x+y=L} = 0\) b. Zero bending moment: \(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2}\Big|_{x+y=L} = 0\)

Step-by-step solution

Set up the equation for the plate

For a Kirchhoff plate problem, the governing equation is given by: \(\nabla^4 w(x, y) = q(x, y)\) where \(\nabla^4\) is the biharmonic operator, and \(w(x, y)\) is the deflection of the plate at point \((x, y)\). \(q(x, y)\) is the distributed load acting on the plate.

Determine the boundary conditions for the simply supported edge

As the edge y = 0 is simply supported, the boundary conditions for this edge will be: 1. Zero bending moment along this line, which corresponds to the second derivative of the deflection with respect to y being zero, i.e., \(\frac{\partial^2 w}{\partial y^2}\Big|_{y=0} = 0\) 2. Zero displacement along this line, i.e., \(w(x,0) = 0\).

Determine the boundary conditions for the clamped edge

As the edge x = 0 is clamped, the boundary conditions for this edge will be: 1. Zero displacement, i.e., \(w(0,y) = 0\). 2. Zero slope, i.e., \(\frac{\partial w}{\partial x}\Big|_{x=0} = 0\).

Determine the boundary conditions for the free edge

As the edge x + y = L is free, the boundary conditions for this edge will be: 1. Zero shear force, which corresponds to the third derivative of deflection being zero, i.e., \(\frac{\partial^3 w}{\partial x \partial y^2} + \frac{\partial^3 w}{\partial y \partial x^2}\Big|_{x+y=L} = 0\) 2. Zero bending moment along this line, i.e., \(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2}\Big|_{x+y=L} = 0\) These are the four boundary conditions for each of the three edges of the Kirchhoff plate.

Key concepts

Boundary Conditions

In structural engineering, boundary conditions are essential for solving problems involving plates, such as the Kirchhoff plate. These conditions define how the edges of a plate interact with their supports or the environment. They influence how a plate deforms under a load.

For the problem at hand, we are considering three different types of boundary conditions: simply supported, clamped, and free edges. Each type will affect the plate's behavior differently. Let's explore these more closely:

• A simply supported edge means the edge can rotate but cannot translate vertically.
• A clamped edge cannot rotate or translate, essentially holding the edge in a fixed position.
• A free edge can experience deformations without constraints along the edge, such as displacements or rotations.

Understanding these boundary conditions is key to accurate modeling of deflection, bending moments, and shear forces in structures.

Bending Moment

The bending moment is an important concept in structural engineering. It refers to the internal moment that causes a section of the plate to bend. In the context of a Kirchhoff plate, the bending moment expresses how the plate will curve in response to applied loads.

For simply supported and free edges, the bending moment is particularly interesting. At a simply supported edge, the bending moment equals zero because the edge can rotate freely. This is expressed mathematically as the second derivative of the deflection being zero at the respective boundary.

Meanwhile, for a free edge like the one along the line \(x+y=L\), the bending moment also vanishes. This is due to the absence of constraints or supports at that edge. Understanding how these moments work helps engineers design plates in a way that can withstand the necessary loads.

Shear Force

Shear force in plate theory refers to a force that causes one part of a flexible material to slide past an adjacent part. In the Kirchhoff plate scenario, shear forces come into play significantly when discussing free edges.

For instance, on a free edge like \(x+y=L\), we assume that the shear forces acting on it are zero. This is a defining characteristic: the shear doesn't "hold" or "pull" at the edge since there are no constraints like in clamped or simply supported edges.

This boundary condition translates to the third derivative of the deflection being zero along the free edge. Recognizing and applying the concept of shear force helps maintain the structural integrity by properly accommodating possible slipping or sliding movements within the plate.

Clamped Support

A clamped support is a rigid attachment used in structures to prevent movement, providing no freedom for rotation or displacement at the edge. It can be thought of as a "fixed" support. In the problem's context, an edge like \(x=0\) behaves this way.

For a clamped edge, the boundary conditions demand zero displacement and zero slope, reflected in zero values for both the deflection and its first derivative along this edge. This means the plate neither moves nor tilts at the boundary. This type of condition is among the strongest constraints and provides great stability in structural setups. Clamped supports are often employed where rigidity is crucial.

Simply Supported Edge

Simply supported edges in plates allow rotation but restrict translation. The edge can bend under loads but remains in a horizontal plane. This is achieved by ensuring zero bending moment and zero displacement along such edges.

For the simply supported edge \(y=0\), we observe the boundary conditions:

• Zero displacement ensures the plate tangent remains horizontal at the support, preventing vertical motion.
• Zero bending moment allows the edge to pivot or rotate, showing some flexibility under loads.

Simply supported edges find common use in structures requiring a balance between stability and flexibility, allowing for some movement to reduce stress.