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Chapter 12: Problem 2

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.

Short Answer

Expert verified
Answer: The boundary conditions for the annular membrane are as follows: - For the inner edge (r = a): 1. u(a) = 0 (no displacement in radial direction) 2. w(a) = 0 (no displacement in vertical direction) - For the outer edge (r = b): 1. t_{rr}(b) = P_0 * cos(𝛽) (radial force balance) 2. t_{rz}(b) = P_0 * sin(𝛽) (vertical force balance)

Step by step solution

01

1. Identify given quantities and given boundary conditions

We are given the following quantities and boundary conditions: - Inner radius: a - Outer radius: b - Uniform force per unit length around the outer edge: P_0 - Angle at which the force is applied: \beta
02

2. Analyze the forces acting on the membrane

In order to establish the boundary conditions for the structure, we must first analyze the forces acting on the membrane. There are two main forces acting on the membrane: 1. Tension in the membrane 2. The force applied around the outer edge, P_0
03

3. Decompose P_0 into horizontal and vertical components

Since the force P_0 is acting at an angle \beta from the plane of the membrane, we will decompose P_0 into its horizontal and vertical components: - Horizontal component of P_0: P_{0x} = P_0 \cos(\beta) - Vertical component of P_0: P_{0y} = P_0 \sin(\beta)
04

4. Establish boundary conditions for the inner edge

As the membrane is fixed at the inner edge, we must have no displacement: 1. Displacement in radial direction (u) at the inner edge (r = a): u(a) = 0 2. Displacement in vertical direction (w) at the inner edge (r = a): w(a) = 0
05

5. Establish boundary conditions for the outer edge

For the outer edge (r = b), we must consider the forces acting on the membrane. Due to the applied force P_0, we can write the boundary conditions as follows: 1. Radial force balance at the outer edge (r = b): t_{rr}(b) = P_{0x} 2. Vertical force balance at the outer edge (r = b): t_{rz}(b) = P_{0y}
06

6. Conclusion

The boundary conditions for the annular membrane of inner radius a and outer radius b under a uniform force per unit length P_0, applied at an angle \beta, are as follows: - For the inner edge (r = a): 1. u(a) = 0 2. w(a) = 0 - For the outer edge (r = b): 1. t_{rr}(b) = P_0 \cos(\beta) 2. t_{rz}(b) = P_0 \sin(\beta)

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Most popular questions from this chapter

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.

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 \(x+y=L\).

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