Problem 13 - Concepts and Applications of Finite Element Analysis

Question

A plane grillage is a plane network of straight members that carries loads normal to its plane. Thus the grillage resembles a plane frame, but carries lateral loads. A typical member resists bending and torsional deformation and has six d.o.f., as shown. Write, in terms of angle \(\alpha\) in the \(x y\) plane, the transformation matrix that would be used to convert \(\left[k^{\prime}\right]\) to \([k]\), where [k] operates on d.o.f. \(w\) (lateral deflection), \(\theta_{x}\) (rotation about the \(x\) axis), and \(\theta_{y}\) (rotation about the \(y\) axis) at each node

Step-by-Step Solution

Answer

The transformation matrix in terms of angle \(\alpha\) in the \(x y\) plane will be given as a 3x3 matrix: T = \[ \begin{{matrix}} \cos(\alpha) & \sin(\alpha) & 0 \ -\sin(\alpha) & \cos(\alpha) & 0 \ 0 & 0 & 1 \ \end{{matrix}}\].

Step 1: Identification of Degrees of Freedom

Identify the degrees of freedom in the plane grillage. Each member has six degrees of freedom, but only lateral deflection (\(w\)), rotation about the \(x\) axis (\(\theta_{x}\)), and rotation about the \(y\) axis (\(\theta_{y}\)) are operated on by the stiffness matrix [k]. The transformation matrix will effectively convert this matrix [k] which operates on a local coordinate system to a global coordinate system (or vice versa).

Step 2: Understanding Transformation Matrix

The transformation matrix in structural analysis is a mathematical tool that transforms any vector from the local coordinate system (sometimes referred to as the primed system denoted as [k’]) to the global coordinate system ([k]). In this context, it transforms variables related to the degrees of freedom, such as the displacements and rotations associated with a member.

Step 3: Formulating Transformation Matrix

The transformation matrix for a planar frame element subject to bending and torsion in terms of angle \(\alpha\) in the \(x y\) plane, is given by:T = \[ \begin{{matrix}} \cos(\alpha) & \sin(\alpha) & 0 \ -\sin(\alpha) & \cos(\alpha) & 0 \ 0 & 0 & 1 \ \end{{matrix}}\]The transformation matrix contains information about rotations, essentially rotating the system through angle \(\alpha\). This would simply convert from local coordinate system ([k’]) to global coordinate system ([k]), by post-multiplying the local matrix with the transformation matrix.

Key Concepts

Transformation Matrix

In the realm of structural analysis, the transformation matrix is key. Its purpose is to enable the conversion of vectors from a local coordinate system to a global one, or vice versa.
This is essential for analyzing structures like a plane grillage, which is subject to loads and deformations in its plane.The transformation matrix essentially allows us to translate or "switch" the perspective from a local view, where the analysis may first take place, into a global frame of reference. This includes transforming displacements and rotations tied to each structural member.
Consider the matrix:\[ \begin{matrix} \cos(\alpha) & \sin(\alpha) & 0 \ -\sin(\alpha) & \cos(\alpha) & 0 \ 0 & 0 & 1 \ \end{matrix}\]This particular matrix rotates the points within the system by an angle \(\alpha\) around the z-axis, which aligns with the lateral plane of the grillage. Such transformations are vital for creating accurate stiffness matrices, which help in applying loads and understanding the resultant deformations.

Degrees of Freedom

Degrees of Freedom (DOF) refer to the number of independent ways a dynamic system can move without violating any restraint. For structural elements like those in a plane grillage, this concept defines how elements can deform.In our scenario, each member of the grillage has six degrees of freedom, crucial for simulation:

• Lateral deflection \((w)\) along the plane
• Rotation about the x-axis \((\theta_x)\)
• Rotation about the y-axis \((\theta_y)\)

These DOFs are what the transformation matrix influences when converting between local and global coordinate systems. By narrowing focus to just these three DOFs in the plane grillage, we simplify and streamline analyses, focusing on the most impactful deformations. This approach allows engineers to predict and address potential issues in the structural response to lateral loads effectively.

Local and Global Coordinate Systems

To properly analyze structures such as a plane grillage in engineering, you need to understand Local and Global Coordinate Systems. - **Local Coordinate System**: This system is specific to each structural member, aligning with the natural orientation of elements. The calculations are typically more straightforward here, as they are relative to each component's orientation. - **Global Coordinate System**: This encompasses the entire structure, viewing it as a whole rather than component parts. It allows coordination and consistency across the entire system. These two systems complement one another, with local systems simplifying element-level calculations and global systems offering a comprehensive view.
Using a transformation matrix, engineers convert local coordinate data to a global context. This aids in ensuring consistency and understanding how each member's behavior contributes to the overall structural response. Transitioning between these systems is paramount for achieving effective structural design and analysis.