Problem 4 - Concepts and Applications of Finite Element Analysis

Question

Consider a uniform straight bar of leagth \(L\) that is supported against rigid body, motion, and is loaded by a smoothly varying distributed axial force \(q\). The bar is to be modeled using two meshes. The first uses \(n\) linear- displacement bar elements, where \(n\) is an even number. The second mesh uses \(n / 2\) quadratic-displacement bar clements. The number and locations of nodes for each mesh are the same. (a) Which mesh will likely provide the more accurate displacement and stress. results? Explain. (b) If each mesh is refined by subdividing elements into two, by what factors will errors in displacements and stresses likely be reduced for each of the finer meshes?

Step-by-Step Solution

Answer

The quadratic displacement bar elements will likely provide more accurate displacement and stress results due to their ability to capture more complex stress distributions. Upon mesh refinement, the error in displacements and stresses will be reduced by roughly a factor of 4 for linear elements and by a factor of 8 for quadratic elements.

Step 1: Answer (a): Mesh Accuracy

(a) The quadratic displacement bar elements will likely provide more accurate displacement and stress results. This is because quadratic elements can represent a parabolic distribution of stresses across an element, while linear elements can only represent a linear distribution. As the load \(q\) is smoothly varying, it might be better represented by a quadratic distribution. The quadratic elements can capture more complex distribution of stress and displacement within the bar, making it more accurate for stress and displacement predictions.

Step 2: Answer (b): Mesh Refinement Effect

(b) Mesh refinement (subdividing elements into two) generally reduces the error in displacements and stresses. For linear elements, error is proportional to the square of the element size. So, refining a mesh (halving the element size) will reduce the error by approximately a factor of 4. For quadratic elements, error is proportional to the cube of the element size. So, refining a mesh (halving the element size) will reduce the error by approximately a factor of 8. However, this factor can vary, depending upon various details like problem geometry, boundary conditions, and stress state in the structure.

Key Concepts

Mesh Refinement in Finite Element Analysis

Mesh refinement, an essential technique in finite element analysis (FEA), significantly enhances the accuracy of simulation results. It involves dividing elements in a mesh into smaller ones, which provides a more detailed representation of the physical system.

Considering a straight bar subjected to a changing axial force, a refined mesh aids in capturing the variations in stress and displacement with higher precision. A key concept here is that as the mesh becomes finer—with an increase in the number of elements—the possible errors in the calculated results decrease. For linear-displacement elements, if the element size is halved, the estimated errors in displacements and stresses drop by a factor of 4, since the error relates to the square of the element size. Conversely, quadratic-displacement elements benefit even more from refinement. In this case, chopping the element size in half tends to cut down errors by roughly a factor of 8, as the error correlates to the cube of the element size.

However, it's important to strike a balance as overly dense meshes can become computationally expensive and time-consuming without yielding proportionate gains in accuracy.

Displacement and Stress Modeling

Displacement and stress modeling in FEA are fundamental for understanding how a structure behaves under loading conditions. The choice of element type—linear versus quadratic—can significantly affect the fidelity of the simulation results.

Linear-displacement elements assume a straight-line variation of displacement across the element. Though faster to compute, they may not adequately model the true physical behavior if the loading or the geometry is complex. On the other hand, quadratic-displacement elements accommodate a curved, parabolic distribution, which grants them the aptitude to better approximate smooth variations in loads and complex geometries.

In our example of the straight bar under a smoothly varying axial force, quadratic elements would likely offer more accurate results. This is due to their ability to emulate the curvilinear nature of the force distribution, providing a closer match to the actual stress and displacement within the bar.

Quadratic-Displacement Elements

Quadratic-displacement elements are a boon to FEA when the aim is to capture complex deformations with greater accuracy. These elements utilize a second-degree polynomial to describe the element's shape functions, which makes them capable of modeling curved shapes internally, even when the outer shape of the element is straight.

Using quadratic elements, the smooth stress distribution in a straight bar subject to a distributed axial load could be represented more precisely. The higher-order polynomials allow for mid-side nodes, which provide intermediate points between the element's corners to better define the curvature.

The advantage of quadratic over linear elements becomes even more pronounced during mesh refinement, as indicated by a potentially eightfold error reduction for quadratic elements compared to a fourfold reduction for linear ones. Such attributes make quadratic-displacement elements markedly suitable for analyses where the predicted response is sensitive to stress gradients, such as around sharp corners or in areas of high load concentration.