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Chapter 12: Problem 9

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

Expert verified
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

01

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.
02

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., \(w(x,0) = 0\).
03

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., \(\frac{\partial w}{\partial x}\Big|_{x=0} = 0\).
04

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.

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Most popular questions from this chapter

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

An annular membrane of inner radius \(a\) and outer radius \(b\) is fixed about its inner edge. A uniform force per unit length of magnitude \(P_{0}\) oriented at an angle \(\beta\) from the plane of the membrane is applied around the periphery of the outer edge. Establish the boundary conditions for the structure if it is modeled as an ideal membrane.

A solid circular Kirchhoff plate of radius \(R\) sits atop elastic supports of stiffness \(k\) per unit length about its edge. Establish the boundary conditions for the structure.

A rectangular Mindlin plate is pinned along its edges at \(x=0\) and \(x=a\), and is constrained by elastic supports possessing torsional stiffness \(k_{T}\) along its edges at \(y=0\) and \(y=b\). Establish the boundary conditions for the structure.
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