Problem 23 - Concepts and Applications of Finite Element Analysis

Question

Consider the uniform, three-node bar element of Problem \(4.3 .\) Let the element be fixed at the left end and loaded by a uniformly distributed axial load of intensity \(q .\) The respective rows of \([k]\) are \(\lfloor 7,-8,1\rfloor,\lfloor-8,16,-8\rfloor\), and \(11,-8,7]\), each times \(A E / 3 L\). Calculate \(u_{2}\) and \(u_{3}\) if nodal loads are (a) calculated in consistent fashion. (b) of magnitude \(q L / 3\) at each of the three nodes.

Step-by-Step Solution

Answer

The displacements \( u_{2} \) and \( u_{3} \) can be found by first determining nodal loads, subsequently formulating a system of linear equations with the stiffness matrix and nodal loads, and solving this system considering the fixed end. The answers will vary for cases (a) and (b) due to different methods of calculating nodal loads.

Step 1: Determination of nodal loads

First, it's necessary to determine the nodal loads in both cases. In case (a), we calculate loads in a consistent fashion by dividing the total load (qL) among the nodes based on their contribution to the stiffness matrix. We find \( F_{1}, F_{2}, F_{3} \) equals \( wl/2, wl, wl/2 \) respectively in this method. For case (b), we distribute the loads equally among nodes resulting in each nodal load to be \(qL / 3\).

Step 2: Formulate system of equations

Next, we form a system of linear equations using the stiffness matrix and the vector of nodal loads. This will allow us to calculate the nodal displacements. For each case (a) and (b), the system of equations will look different due to the different load distributions. However, the general form is \([A] \cdot [x] = [b]\), where [A] is the stiffness matrix, [x] is the vector of nodal displacements, and [b] is the vector of nodal loads.

Step 3: Solve system of equations

Finally, we solve our system of linear equations to find the nodal displacements. Particular attention should be paid to the fact that the element is fixed at the left end - this means displacement \( u_{1} = 0 \). This information allows us to reduce the number of equations and simplifies the process of finding \( u_{2} \) and \( u_{3} \). This can be done by substitution or using matrix manipulation methods.

Key Concepts

Three-node bar element

When working with a three-node bar element, it's essential to understand the basics of finite element analysis. These elements help us model structures subjected to various loads. A three-node bar element has nodes at three distinct points along its length, typically at each end and the midpoint.
This setup is particularly useful for approximating changes in axial forces along a bar.
Key characteristics of a three-node bar include:

• Three degrees of freedom, with a displacement variable associated with each node.
• Ability to model both linear and non-linear behavior, making it versatile.
• Simplified calculations due to its relatively simple geometry and nodal arrangement.

Finite element analysis with a three-node bar can capture more complex variations in displacement and strain than simpler elements. This makes it particularly advantageous when analyzing structural components under varying loads.

Nodal loads calculation

Nodal loads refer to the forces applied at the nodes of an element. Accurate calculation of these loads is crucial for determining how a structure will deform or react under stress.
In our specific exercise, nodal loads were calculated in two different ways.
First, we used a consistent distribution, which divides the load based on each node's influence on the stiffness matrix. For our bar element, this calculated as:

• Load at node 1: \[F_1 = \frac{wL}{2}\]
• Load at node 2: \[F_2 = wL\]
• Load at node 3: \[F_3 = \frac{wL}{2}\]

Alternatively, the loads were distributed equally among the nodes, meaning each node received a load of:\[qL / 3\]
Each method brings unique insights and considerations, depending on the uniformity and characteristics of the load applied to the structure.

Stiffness matrix

The stiffness matrix is at the heart of finite element analysis. It represents a system's resistance to deformation and translates force distributions into displacements at different nodes. In our exercise, the stiffness matrix \([k]\) was a key component for solving displacement values.
For a three-node bar element, the stiffness matrix was configured as follows:

• \[ k_1 = \left[ 7, -8, 1 \right] \frac{AE}{3L} \]
• \[ k_2 = \left[ -8, 16, -8 \right] \frac{AE}{3L} \]
• \[ k_3 = \left[ 11, -8, 7 \right] \frac{AE}{3L} \]

Each row of the stiffness matrix corresponds to one of the nodes’ influence on the overall structural stiffness.
The matrix is used in forming linear equations that relate the applied nodal loads to nodal displacements, allowing us to calculate unknown values. Solving these equations helped find the displacements at nodes 2 and 3, essential for understanding how the bar deforms under load.