Problem 18 - Concepts and Applications of Finite Element Analysis

Question

A uniform bar of axial stiffness \(k=A E / L\) is arbitrarily oriented in space. Cosines of angles between the bar and the \(x, y\), and \(z\) coordinate axes are \(\ell, m\), and \(n .\) Nodal d.o.f. are translations \(u, v\), and \(w\) at each end. Derive the 6 by 6 element stiffness matrix.

Step-by-Step Solution

Answer

The derived 6 by 6 element stiffness matrix of the bar, presented in terms of axial stiffness of the bar \(k\), cosines of angles \(\ell, m, n\), and nodal translations \(u, v, w\) is \[ [k] = k \begin{bmatrix} \ell^2 & \ell m & \ell n & -\ell^2 & -\ell m & -\ell n \ \ell m & m^2 & mn & -\ell m & -m^2 & -mn \ \ell n & mn & n^2 & -\ell n & -mn & -n^2 \ -\ell^2 & -\ell m & -\ell n & \ell^2 & \ell m & \ell n \ -\ell m & -m^2 & -mn & \ell m & m^2 & mn \ -\ell n & -mn & -n^2 & \ell n & mn & n^2 \end{bmatrix}\]

Step 1: Create the Matrix

The element stiffness matrix, denoted as \([k]\), should initially be a 6 by 6 matrix filled with zeros because we have 6 Degrees Of Freedom - 3 at each end of the bar.

Step 2: Stiffness Matrix Calculation

The matrix should look like this: \[ [k] = k \begin{bmatrix} \ell^2 & \ell m & \ell n & -\ell^2 & -\ell m & -\ell n \ \ell m & m^2 & mn & -\ell m & -m^2 & -mn \ \ell n & mn & n^2 & -\ell n & -mn & -n^2 \ -\ell^2 & -\ell m & -\ell n & \ell^2 & \ell m & \ell n \ -\ell m & -m^2 & -mn & \ell m & m^2 & mn \ -\ell n & -mn & -n^2 & \ell n & mn & n^2 \end{bmatrix}\] The submatrix \(k \begin{bmatrix} \ell^2 & \ell m & \ell n \ \ell m & m^2 & mn \ \ell n & mn & n^2\end{bmatrix}\) represents the relationship between the translations and the axial stiffness at one end of the bar, while the submatrix \(-k \begin{bmatrix} \ell^2 & \ell m & \ell n \ \ell m & m^2 & mn \ \ell n & mn & n^2\end{bmatrix}\) represent these relations at the other end of the bar.

Step 3: Final Matrix

Putting the pieces together, the final derived 6 by 6 element stiffness matrix \([k]\) elements are presented in terms of axial stiffness of the bar \(k\), cosines of angles between the bar and the \(x, y\), and \(z\) coordinate axes \(\ell, m\), and \(n\), and nodal translations \(u, v\), and \(w\) at each end.

Key Concepts

Finite Element Analysis

Finite Element Analysis (FEA) is a powerful computational tool used to approximate solutions in engineering problems. One of the key applications of FEA is in structural analysis, where it helps in determining how structures behave under load.
The method works by breaking down a complex structure into smaller, simpler parts known as "finite elements." Each of these elements can be easily analyzed, and the results can be assembled to provide a full picture of the entire structure's performance.
In the context of our problem, FEA is used to derive the stiffness matrix of a bar arbitrarily oriented in space. This stiffness matrix is crucial as it defines the relationship between the forces applied on the bar and the resulting displacements.

• Helps in predicting structural performance.
• Utilizes mathematical models to replicate physical behavior.
• An essential tool in engineering design and analysis.

Degrees Of Freedom

Degrees of Freedom (DOF) in engineering are the set of independent displacements that define a system's movement. For structural problems, DOF typically refers to translational and rotational movements.
In our exercise, the uniform bar has six degrees of freedom. That means there are three translational movements at each end of the bar: along the x, y, and z directions. This allows the bar to stretch, compress, and bend in response to forces.
Understanding DOF is essential as it helps to fully capture how the system can move or deform under various loading conditions. The more DOF a system has, the more complex its analysis becomes.

• Defines the possible independent movements.
• Essential for setting up the stiffness matrix.
• Directly impacts the complexity of the analysis.

Axial Stiffness

Axial stiffness refers to a structural element's resistance to deformation in its axial direction, which is the line along its length. It is a crucial property for elements like bars, rods, or beams within structures.
The axial stiffness, denoted as \(k=A E / L\) where \(A\) is the cross-sectional area, \(E\) is the material's Young's modulus, and \(L\) is the length, dictates how much the element will compress or elongate when subjected to an axial force.
Higher axial stiffness means less deformation for a given force, making the material more resistant to changes in length. This concept is fundamental in constructing the element stiffness matrix, which describes how an element will respond under loading.

• Determines resistance to length changes.
• Depends on material stiffness and geometry.
• Central to formulating the stiffness matrix.

Coordinate Axes Cosines

Coordinate Axes Cosines, often referred to simply as direction cosines, are values that describe the orientation of a line in space relative to coordinate axes. These are given in terms of cosines of the angles between the line and the x, y, and z axes, signified by \(\ell, m,\) and \(n\) respectively.
In the derived stiffness matrix, these direction cosines are key in defining how forces and displacements are coordinated along the element's orientation. By incorporating these cosines, the stiffness matrix acknowledges the arbitrary orientation of the bar in 3D space.
Understanding these parameters is essential for correctly representing the structural orientation and behavior in the global coordinate system.

• Defines the orientation in 3D space.
• Integral in aligning stiffness matrix with global coordinates.
• Ensures accurate capture of directional properties.