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Chapter 12: Problem 11

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.

Short Answer

Expert verified
Answer: The boundary conditions for the displacement and rotation of a solid circular Kirchhoff plate with radius R, sitting on elastic supports with stiffness k per unit length around its edge are given by: 1. Force Boundary Condition (FBC): At r = R, the distributed force exerted by the elastic supports along the vertical direction is equal to the force resisted by the plate. So, the force boundary condition is \(F = k \cdot v\). 2. Moment Boundary Condition (MBC): At r = R, the distributed moment exerted by the elastic supports at the plate edge is equal to the moment resisted by the plate. So, the moment boundary condition is \(M = k \cdot R \cdot \phi\).

Step by step solution

01

Understanding Kirchhoff Plate and Elastic Supports

The given plate is a Kirchhoff plate, which means it is a thin plate that obeys Kirchhoff's hypothesis where transverse shear deformation is neglected, and normal stresses are neglected relative to bending and membrane stresses. The elastic supports are characterized by stiffness "k" per unit length, which influences the boundary conditions for the plate.
02

Analyzing Plate Displacement and Rotation

We need to establish the boundary conditions for the displacement and rotation of the plate at its edge. Let 'v' be the displacement of the plate along out-of-plane (vertical) direction, and 'φ' be the rotation of the plate around the edge. These two variables are related, as the rotation of a point at the plate edge is equal to the derivative of the vertical displacement with respect to the radial coordinate 'r', i.e., \(\phi = \frac{\partial v}{\partial r}\).
03

Defining Forces and Moments

Elastic supports exert force and moment on the plate edge. The distributed force exerted by the elastic supports is given by \(F = k \cdot v\), and the distributed moment exerted by the elastic supports is given by \(M = k \cdot R \cdot \phi\). Here, we assumed that the moment-angular rotation relationship is linear.
04

Applying Force and Moment Boundary Conditions

We can now establish the boundary conditions for the displacement and rotation of the plate at its edge: 1. Force Boundary Condition (FBC): The distributed force exerted by the elastic supports along the vertical direction (at r = R) must be equal to the force resisted by the plate. Thus, we have \(F = k \cdot v\). 2. Moment Boundary Condition (MBC): The distributed moment exerted by the elastic supports at the plate edge (at r = R) must be equal to the moment resisted by the plate. Thus, we have \(M = k \cdot R \cdot \phi\). These are the required boundary conditions for the displacement and rotation of the Kirchhoff plate supported by elastic supports with stiffness k per unit length.

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Most popular questions from this chapter

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 \(x+y=L\).

A rectangular Kirchoff plate with sides of length \(a\) and \(b\) is clamped on all four edges. 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.

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.
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