Problem 4 - Concepts and Applications of Finite Element Analysis

Question

Devise an argument that demonstrates the relation \(\varepsilon_{r}=\varepsilon_{\theta}\) at \(r=0\) in an axisymmetric problem. Suggestion: Consider the radial displacement series \(u=c_{1} r+\) \(c_{2} z+c_{3} r^{2}+c_{4} r z+\cdots\)

Step-by-Step Solution

Answer

At \(r=0\), under the given radial displacement series, we find that \(\varepsilon_{r}=\varepsilon_{\theta}\), consistent with the task.

Step 1: Express the Strains

The radial strain \(\varepsilon_{r}\) and the tangential strain \(\varepsilon_{\theta}\) can be expressed in terms of radial displacement 'u'. They are given by: \(\varepsilon_{r} = \frac{\partial u}{\partial r}\) and \(\varepsilon_{\theta} = \frac{u}{r} \).

Step 2: Calculate the Derivatives

The derivative of \(u\) with respect to \(r\) is calculated using the given radial displacement series \(u=c_{1} r+c_{2} z+c_{3} r^{2}+c_{4} r z+\cdots\). After performing the differentiation procedure, we have: \(\frac{\partial u}{\partial r}=c_{1}+2c_{3} r+c_{4} z+\cdots\).

Step 3: Substituting the Derivatives

Substitute \(\frac{\partial u}{\partial r}\) into \(\varepsilon_{r}\) and \(u\) into \(\varepsilon_{\theta}\). You should obtain \(\varepsilon_{r} = c_{1}+2c_{3} r+c_{4} z+\cdots\) and \(\varepsilon_{\theta} = \frac{c_{1} r+c_{2} z+c_{3} r^{2}+c_{4} r z+\cdots}{r}\).

Step 4: Setting \(r=0\)

Both \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) will be evaluated at \(r=0\). Therefore, substitute \(r=0\) into the expressions for \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\). This will yield: \(\varepsilon_{r}=c_{1}+c_{4} z+\cdots\) and \(\varepsilon_{\theta}=c_{1}+\cdots\).

Step 5: Verifying the Result

At \(r=0\), both \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) give the result \(c_{1}+\cdots\). This implies that \(\varepsilon_{r}=\varepsilon_{\theta}\) at \(r=0\). Hence, the relation is proved.

Key Concepts

Axisymmetric Problem

An axisymmetric problem is a type of engineering challenge where the problem's features are symmetrical around a single axis. Think of a doughnut or a drum – these objects are the same all around their central axis. This concept is crucial in finite element analysis because it implies that the problem can be simplified to a two-dimensional one, dramatically reducing complexity.

Axisymmetry means the physical phenomena being analyzed don't change as you move around in a circle at a constant radius from the axis. This symmetry can be useful when dealing with components like pipes, tanks, or any cylindrical object. For engineers, analyzing an axisymmetric problem means accounting for rotational symmetry. This helps in predicting how stress or strain will behave throughout the material or structure.

By recognizing an axisymmetric scenario, engineers can apply mathematical equations and boundary conditions to simplify calculations and make solutions more manageable and less time-consuming.

Radial Displacement Series

In finite element analysis, the radial displacement series expresses how an object's surface moves or displaces at various radial and axial points. Imagine pressing on the surface of a balloon; the radial displacement series, which breaks down into different terms, tells you how each small region of that balloon moves as a result of the pressure.

The series uses a set of coefficients and functions of the radius (\( r \)) and the axial position (\( z \)) to describe displacement. For example, the term

• \( c_{1} r \) represents a linear movement relative to the radius.
• \( c_{2} z \) accounts for axial movements, such as stretching or compressing along the z-axis.
• \( c_{3} r^{2} \) indicates a parabolic radial movement.

Understanding this series is key to determining how components might deform in axisymmetric conditions and allows for precise strain and stress calculations.

Ultimately, the formula for radial displacement helps engineers predict how physical changes affect the entire structure at every small piece or segment.

Strain Calculation

Calculating strain is about understanding deformation. Strain measures how much a material deforms in response to external forces. In the context of axisymmetric problems, strain calculation becomes central to predicting an object's response to stresses, like pulling or squeezing.

There are two primary strains involved: radial strain (\( \varepsilon_r \)) and tangential strain (\( \varepsilon_\theta \)).

• Radial strain, \( \varepsilon_r \), is the change in radial direction per unit length. It is calculated as \( \varepsilon_r = \frac{\partial u}{\partial r} \), signifying how much the object stretches or contracts in its radial direction.
• Tangential strain, \( \varepsilon_\theta \), represents deformation along a circular path around the axis and is given by \( \varepsilon_\theta = \frac{u}{r} \).

Understanding how to determine these strains at any given radius or position, especially at \( r = 0 \), is vital. The solution suggests checking equality of radial and tangential strain at \( r = 0 \) because, at the axis, components deform equivalently in both radial and tangential directions.

This makes strain calculation indispensable in designing robust and reliable mechanical components.