Problem 3 - Concepts and Applications of Finite Element Analysis

Question

Load \(P\) applied to the tip of the column shown is called a follower load: it always has the same orientation \(\theta\) as the tip. By considering work done by \(P\) in undergoing small displacements \(v\) and \(\theta\) from initial position to final position, show that the system is not conservative. Neglect work associated with displacement of \(P\) in the \(x\) direction.

Step-by-Step Solution

Answer

The system is not conservative because the work done by the follower load is dependent on the path taken, which depends on the vertical displacement and the angular displacement.

Step 1: Understanding follower loads

A follower load is a specific type of force that maintains its direction relative to a deforming structural element. In this case, the follower load \(P\) always stays tangent to the tip of the column, maintaining the same angle \(\theta\). This implies that the orientation of \(P\) changes along with the displacement of the column.

Step 2: Calculation of work done by the follower load

Work done is generally given by the dot product of force vector and displacement vector. In this case, the work done by the load \(P\) as it moves from initial position to final position can be given by \(P \cdot v + P \cdot \theta \). We can see the work done is influenced by both \(v\) and \(\theta\), where \(v\) is the vertical displacement and \(\theta\) is the angular displacement from the vertical.

Step 3: Determination of conservativeness of the system

For a system to be conservative, the total work done in moving from one state to another must be independent of the path taken. In this case, the work done by \(P\) depends on both the vertical displacement \(v\) and the angular orientation \(\theta\). Therefore, as the work done is dependent on the path taken, the system is not conservative.

Key Concepts

Non-Conservative System

A non-conservative system is one where the work done by forces is dependent on the path taken during displacement. In contrast to conservative systems, where work is solely determined by the initial and final positions, non-conservative forces consider the entire journey.
In this particular exercise, we examine a follower load, which is quite different from traditional forces like gravity or spring forces that are path-independent, hence conservative.
Here, the force maintains its orientation relative to a structural deformation, resulting in work that depends on the specific path of the deformation.

• Non-conservative forces can include friction and air resistance.
• Their work cannot be recuperated when the system returns to its initial state.

This attribute implies energy loss, contributing to phenomena like thermal energy or sound, and affects engineering structures through dynamic responses that are path-sensitive.

Work Done

Work done refers to the process of a force causing a displacement of an object. It's calculated as the product of the force's magnitude, the displacement, and the cosine of the angle between them.
In mathematical terms, it is represented by the dot product of the force and displacement vectors.
In the case of the follower load, the peculiar feature is that the load changes its direction following the tip of the column.

• The work done by such a load is expressed as: \( P \cdot v + P \cdot \theta \)
• Where \( v \) represents vertical movement, and \( \theta \) is the angular change.

The work done is unique here because it varies not only with vertical movement but also with the angular path, reinforcing that this system doesn't follow the typical conservative system rules.

Displacement

Displacement in this context refers to the change in position of the column's tip due to the applied follower load. Displacement is categorized generally into two components: linear (like moving up or down) and angular (rotating around an axis).
Understanding these displacements is crucial in analyzing how forces like the follower load affect a structure.
For the vertical and angular displacements:

• \( v \) denotes the vertical movement of the column's tip.
• \( \theta \) describes the angular rotation, which directly influences the follower load’s orientation.

Both these movements combine to show how the follower load executes its work, providing a detailed understanding of the forces at play. This helps engineers and students alike to appreciate the intricate dynamics of such systems.