Problem 8 - Concepts and Applications of Finite Element Analysis

Question

The sketch represents three nodes on a \(z=\) constant face of an axisymmetric quadratic element. Node 7 is at midside. Determine the consistent nodal load vector for these three nodes if \(z\)-direction surface traction \(\Phi_{z}\) is applied as follows. (a) \(\Phi_{z}\) is the constant value \(p\) over the face. (b) \(\Phi_{z}=\left(\xi^{2}-\xi\right) p_{4} / 2+\left(1-\xi^{2}\right) p_{7}+\left(\xi^{2}+\xi\right) p_{3} / 2\), which is a parabolic variation based on nodal values \(p_{4}, p_{7}\), and \(p_{3}\).

Step-by-Step Solution

Answer

The load vectors for Nodes 4, 7 and 3 under a constant \(\Phi_{z}\) are all evaluated to \(p * \frac{l}{2}\). Under parabolic variation \(\Phi_{z}\), the load vectors are \(\frac{l}{2} p_{4}\), \(l p_{7}\), and \(\frac{l}{2} p_{3}\) for Nodes 4, 7 and 3 respectively.

Step 1: Calculate load vector for constant \(\Phi_{z}\)

For a constant value of \(\Phi_{z} = p\), each nodal load vector naught in the \(\Phi_{z}\) direction is computed as \(t_a = p * \frac{l}{2}\), where \(l\) is the length of the edge. Therefore, \(t_4 = t_7 = t_3 = p \frac{l}{2}\). This is because for constant surface traction, the load is evenly divided among the nodes.

Step 2: Calculate load vector when \(\Phi_{z}\) has parabolic variations

For parabolic variation, we substitute \(\xi = -1\) for Node 4, \(\xi = 0\) for Node 7 and \(\xi = 1\) for Node 3 into the parabolic equation. We obtain:\(t_4 = \left((-1)^{2} - (-1)\right) \frac{p_{4}}{2} + (1 - (-1)^{2}) p_{7} + ((-1)^{2} + (-1)) \frac{p_{3}}{2} = \frac{l}{2} p_{4}\)\(t_7 = \left(0^{2} - 0\right) \frac{p_{4}}{2} + (1 - 0^{2}) p_{7} + (0^{2} + 0) \frac{p_{3}}{2} = l p_{7}\)\(t_3 = \left(1^{2} - 1\right) \frac{p_{4}}{2} + (1 - 1^{2}) p_{7} + (1^{2} + 1) \frac{p_{3}}{2} = \frac{l}{2} p_{3}\)These are the values of the nodal load vectors for Nodes 4, 7, 3 respectively when \(\Phi_{z}\) varies parabolically.

Key Concepts

Nodal Load Vector

In finite element analysis, the nodal load vector represents the distribution of forces or loads at specific points—nodes—of a structural element. The nodal load vector is crucial because it helps engineers understand how external forces translate into stresses and strains within the element.

• A fixed load can be evenly distributed across nodes when the force is constant.
• Complex distributions, such as parabolic loads, require integration to determine how the load affects each node differently.

Understanding the formation of the nodal load vector is vital for accurate stress analysis and ensures that the model accurately reflects real-world conditions.

Axisymmetric Elements

Axisymmetric elements are used in finite element analysis to model structures that have rotational symmetry around an axis. These elements simplify the analysis by reducing three-dimensional problems to two dimensions.

• Commonly used for objects like cones, cylinders, and spheres.
• Effective in analyzing pressures, forces, or tractions that are symmetrical.

By exploiting symmetry, the computational effort is reduced, allowing for more efficient analysis without sacrificing accuracy.

Parabolic Variation

Parabolic variation refers to how a variable load is distributed in a curve, resembling a parabola, across a surface. This is more complex than a constant load as it requires consideration of how the load intensity changes across the element.

• In our exercise, \(\Phi_z\) varies across nodes in a parabolic pattern.
• Each node will experience different force intensities, calculated using the parabolic equation.

This type of variation is vital for accurately modeling real-world conditions where loads aren't uniformly distributed, such as pressure profiles or wind loads.

Surface Traction

Surface traction describes the distribution of force acting across the surface of an element. In finite element analysis, it’s essential to understand how this force is applied to model elements accurately.

• When surface traction is constant, nodal loads are evenly distributed.
• With variable surface traction, each node experiences a different force, reflecting the real distribution pattern.

Understanding how surface traction affects elements is crucial for accurate structural analysis, ensuring that the designs can withstand the loads they will experience.