Problem 10 - Concepts and Applications of Finite Element Analysis

Question

Show that \(\epsilon_{r}=\epsilon_{\theta}\) at \(r=0\) in an axially symmetric problem.

Step-by-Step Solution

Answer

Using the Strain-Displacement relations and given the conditions of an axially symmetric problem, it can be established that both \( \epsilon_{r} \) and \( \epsilon_{\theta} \) approach zero at \( r=0 \). Thus, under aforementioned conditions, \( \epsilon_{r} = \epsilon_{\theta} \) at \( r=0 \).

Step 1: Write down the Strain Displacement Relations

The Strain-Displacement relations in cylindrical coordinates for radial (\( u \)) and circumferential (\( v \)) displacement are: \( \epsilon_{r} = \frac{\partial u}{\partial r} \) and \( \epsilon_{\theta} = \frac{1}{r} \frac{\partial v}{\partial \theta} + \frac{v}{r} \). In an axially symmetric problem, the displacement \( v \) in the circumferential direction \( \theta \) does not vary, thus \( \frac{\partial v}{\partial \theta} = 0 \). This simplifies the strain-displacement relations.

Step 2: Simplify Strain-Displacement Relations

Based on the previous step, the simplified strain-displacement relations in an axially symmetric problem are: \( \epsilon_{r} = \frac{\partial u}{\partial r} \) and \( \epsilon_{\theta} = \frac{v}{r} \). We want to evaluate these expressions at \( r=0 \) and see if they are equal.

Step 3: Evaluate At \(r=0\)

Evaluating \( \epsilon_{r} \) and \( \epsilon_{\theta} \) at \( r=0 \) can pose a problem because \( \epsilon_{\theta} = \frac{v}{r} \) will have an undefined denominator at \( r=0 \). However, addressing the issue through the use of a limit, taking \( \lim_{r \to 0} \epsilon_{\theta} = \lim_{r \to 0} \frac{v}{r} \), if \( v \) is finite, will yield \( 0 \). Since the derivative at \( r=0 \) for \( u \) (radial displacement) should also be \( 0 \), it can be argued that \( \epsilon_{r} = \epsilon_{\theta} \) at \( r=0 \) under these conditions.

Key Concepts

Understanding Strain-Displacement Relations

Strain-displacement relations are crucial when dealing with problems in mechanics, especially those involving deformations. They provide a mathematical way to describe how materials stretch and compress under various loads. In the context of cylindrical coordinates, these relations express the strains in terms of displacements. For an axially symmetric problem, we focus on two types of strains: radial strain (\( \epsilon_r \)) and circumferential strain (\( \epsilon_\theta \)).

Radial strain relates to how much the material stretches or compresses in the radial direction. It's given by the derivative of the radial displacement (\( u \)) with respect to the radial coordinate (\( r \)): \( \epsilon_r = \frac{\partial u}{\partial r} \).

Circumferential strain accounts for material deformation around the circumference. It's affected by both the circumferential displacement (\( v \)) and its change with respect to the angular position (\( \theta \)). In an axially symmetric setup, however, \( v \) doesn't change with \( \theta \), simplifying \( \epsilon_\theta \) to \( \frac{v}{r} \).

These simplified relationships help in evaluating conditions like \( \epsilon_r = \epsilon_\theta \) at \( r=0 \) by using limits and derivatives effectively.

The Role of Cylindrical Coordinates

Cylindrical coordinates provide a natural framework for problems involving symmetry around a central axis. These coordinates are especially useful for axially symmetric problems because they allow us to express positions in three-dimensional space using radial, angular, and axial coordinates:

• Radial (\( r \)): the distance from the axis of symmetry.
• Circumferential or angular (\( \theta \)): the angle around the axis, generally ranging from \( 0 \) to \( 2\pi \).
• Axial (\( z \)): represents the position along the axis of symmetry.

The use of cylindrical coordinates simplifies the mathematical description of deformations, especially in materials like pipes or cylindrical shafts. This localizes the problem, reducing complexity in scenarios where displacements and strains predominantly affect the radial and circumferential directions, rather than lines parallel to the axis.

Radial Displacement in Cylindrical Systems

Radial displacement is an essential component in understanding deformations in cylindrical systems. It refers to how much a material point moves in the radial direction from its original position. This is significant in axially symmetric problems where the displacement occurs uniformly around the axis.

In mathematical terms, radial displacement is denoted by \( u \). Evaluating the change in \( u \) with respect to \( r \) gives us the radial strain (\( \epsilon_r \)). In practical applications, such as designing cylindrical storage tanks or pressure vessels, knowing how the material deforms in the radial direction helps predict structural behavior under pressure.

When \( r \to 0 \), particularly at the centerline of such structures, radial displacement plays a pivotal role as it approaches zero, ensuring that strains remain finite and manageable.

Understanding Circumferential Displacement

Circumferential displacement involves the movement of material points around the axis, denoted by \( v \). In an axially symmetric problem, \( v \) can be constant with respect to the angular position (\( \theta \)), simplifying the analysis significantly.

The circumferential strain, calculated as \( \epsilon_\theta = \frac{v}{r} \), reflects how the material stretches or shrinks around a circle centered on the axial line. In practice, any changes in wall thickness or other inconsistencies can alter circumferential displacement.

Evaluating \( \epsilon_\theta \) at \( r=0 \) poses a challenge due to the division by zero. However, using limits allows us to comprehend how materials behave near the axis, effectively predicting that \( \epsilon_r \) and \( \epsilon_\theta \) match each other by resolving this mathematical limitation.